{"id": "hv2-combo-disj_n2-i0-none::none", "rec_id": "hv2-combo-disj_n2-i0", "axis": "combo", "difficulty": "disj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition s1 is true. Proposition s0 is true. Proposition etrue is true. Proposition x0b is true if proposition x0a is not true. Proposition x0a is true if proposition x0b is not true. Proposition t1 is true if proposition t2 is true. Proposition t3 is true if proposition t4 is true. Proposition t2 is true if proposition t3 is true. Proposition t4 is true if proposition t5 is true. Proposition s3 is true. Proposition t5 is true if proposition qcore is true. Proposition x1a is true if proposition x1b is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t0 is true if proposition t1 is true. Proposition x1b is true if proposition x1a is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition qcore is true if proposition x0a is true. Proposition qcore is true if proposition x1a is true. Proposition s2 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n2-i0-closed_world::closed_world", "rec_id": "hv2-combo-disj_n2-i0", "axis": "combo", "difficulty": "disj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition s1 is true. Proposition s0 is true. Proposition etrue is true. Proposition x0b is true if proposition x0a is not true. Proposition x0a is true if proposition x0b is not true. Proposition t1 is true if proposition t2 is true. Proposition t3 is true if proposition t4 is true. Proposition t2 is true if proposition t3 is true. Proposition t4 is true if proposition t5 is true. Proposition s3 is true. Proposition t5 is true if proposition qcore is true. Proposition x1a is true if proposition x1b is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t0 is true if proposition t1 is true. Proposition x1b is true if proposition x1a is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition qcore is true if proposition x0a is true. Proposition qcore is true if proposition x1a is true. Proposition s2 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n2-i0-cred::cred", "rec_id": "hv2-combo-disj_n2-i0", "axis": "combo", "difficulty": "disj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition s1 is true. Proposition s0 is true. Proposition etrue is true. Proposition x0b is true if proposition x0a is not true. Proposition x0a is true if proposition x0b is not true. Proposition t1 is true if proposition t2 is true. Proposition t3 is true if proposition t4 is true. Proposition t2 is true if proposition t3 is true. Proposition t4 is true if proposition t5 is true. Proposition s3 is true. Proposition t5 is true if proposition qcore is true. Proposition x1a is true if proposition x1b is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t0 is true if proposition t1 is true. Proposition x1b is true if proposition x1a is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition qcore is true if proposition x0a is true. Proposition qcore is true if proposition x1a is true. Proposition s2 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n2-i0-skept::skept", "rec_id": "hv2-combo-disj_n2-i0", "axis": "combo", "difficulty": "disj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition s1 is true. Proposition s0 is true. Proposition etrue is true. Proposition x0b is true if proposition x0a is not true. Proposition x0a is true if proposition x0b is not true. Proposition t1 is true if proposition t2 is true. Proposition t3 is true if proposition t4 is true. Proposition t2 is true if proposition t3 is true. Proposition t4 is true if proposition t5 is true. Proposition s3 is true. Proposition t5 is true if proposition qcore is true. Proposition x1a is true if proposition x1b is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t0 is true if proposition t1 is true. Proposition x1b is true if proposition x1a is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition qcore is true if proposition x0a is true. Proposition qcore is true if proposition x1a is true. Proposition s2 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n2-i0-wfs::wfs", "rec_id": "hv2-combo-disj_n2-i0", "axis": "combo", "difficulty": "disj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition s1 is true. Proposition s0 is true. Proposition etrue is true. Proposition x0b is true if proposition x0a is not true. Proposition x0a is true if proposition x0b is not true. Proposition t1 is true if proposition t2 is true. Proposition t3 is true if proposition t4 is true. Proposition t2 is true if proposition t3 is true. Proposition t4 is true if proposition t5 is true. Proposition s3 is true. Proposition t5 is true if proposition qcore is true. Proposition x1a is true if proposition x1b is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t0 is true if proposition t1 is true. Proposition x1b is true if proposition x1a is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition qcore is true if proposition x0a is true. Proposition qcore is true if proposition x1a is true. Proposition s2 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n2-i1-none::none", "rec_id": "hv2-combo-disj_n2-i1", "axis": "combo", "difficulty": "disj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition q is true if proposition qcore is true and proposition etrue is true. Proposition x1b is true if proposition x1a is not true. Proposition x1a is true if proposition x1b is not true. Proposition x0a is true if proposition x0b is not true. Proposition etrue is true. Proposition qcore is true if proposition x0a is true. Proposition qcore is true if proposition x1a is true. Proposition x0b is true if proposition x0a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n2-i1-closed_world::closed_world", "rec_id": "hv2-combo-disj_n2-i1", "axis": "combo", "difficulty": "disj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition q is true if proposition qcore is true and proposition etrue is true. Proposition x1b is true if proposition x1a is not true. Proposition x1a is true if proposition x1b is not true. Proposition x0a is true if proposition x0b is not true. Proposition etrue is true. Proposition qcore is true if proposition x0a is true. Proposition qcore is true if proposition x1a is true. Proposition x0b is true if proposition x0a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n2-i1-cred::cred", "rec_id": "hv2-combo-disj_n2-i1", "axis": "combo", "difficulty": "disj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition q is true if proposition qcore is true and proposition etrue is true. Proposition x1b is true if proposition x1a is not true. Proposition x1a is true if proposition x1b is not true. Proposition x0a is true if proposition x0b is not true. Proposition etrue is true. Proposition qcore is true if proposition x0a is true. Proposition qcore is true if proposition x1a is true. Proposition x0b is true if proposition x0a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n2-i1-skept::skept", "rec_id": "hv2-combo-disj_n2-i1", "axis": "combo", "difficulty": "disj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition q is true if proposition qcore is true and proposition etrue is true. Proposition x1b is true if proposition x1a is not true. Proposition x1a is true if proposition x1b is not true. Proposition x0a is true if proposition x0b is not true. Proposition etrue is true. Proposition qcore is true if proposition x0a is true. Proposition qcore is true if proposition x1a is true. Proposition x0b is true if proposition x0a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n2-i1-wfs::wfs", "rec_id": "hv2-combo-disj_n2-i1", "axis": "combo", "difficulty": "disj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition q is true if proposition qcore is true and proposition etrue is true. Proposition x1b is true if proposition x1a is not true. Proposition x1a is true if proposition x1b is not true. Proposition x0a is true if proposition x0b is not true. Proposition etrue is true. Proposition qcore is true if proposition x0a is true. Proposition qcore is true if proposition x1a is true. Proposition x0b is true if proposition x0a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n2-i2-none::none", "rec_id": "hv2-combo-disj_n2-i2", "axis": "combo", "difficulty": "disj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition qcore is true if proposition x1a is true. Proposition x0b is true if proposition x0a is not true. Proposition x1b is true if proposition x1a is not true. Proposition x0a is true if proposition x0b is not true. Proposition qcore is true if proposition x0a is true. Proposition q is true if proposition qcore is true. Proposition x1a is true if proposition x1b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n2-i2-closed_world::closed_world", "rec_id": "hv2-combo-disj_n2-i2", "axis": "combo", "difficulty": "disj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition qcore is true if proposition x1a is true. Proposition x0b is true if proposition x0a is not true. Proposition x1b is true if proposition x1a is not true. Proposition x0a is true if proposition x0b is not true. Proposition qcore is true if proposition x0a is true. Proposition q is true if proposition qcore is true. Proposition x1a is true if proposition x1b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n2-i2-cred::cred", "rec_id": "hv2-combo-disj_n2-i2", "axis": "combo", "difficulty": "disj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition qcore is true if proposition x1a is true. Proposition x0b is true if proposition x0a is not true. Proposition x1b is true if proposition x1a is not true. Proposition x0a is true if proposition x0b is not true. Proposition qcore is true if proposition x0a is true. Proposition q is true if proposition qcore is true. Proposition x1a is true if proposition x1b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n2-i2-skept::skept", "rec_id": "hv2-combo-disj_n2-i2", "axis": "combo", "difficulty": "disj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition qcore is true if proposition x1a is true. Proposition x0b is true if proposition x0a is not true. Proposition x1b is true if proposition x1a is not true. Proposition x0a is true if proposition x0b is not true. Proposition qcore is true if proposition x0a is true. Proposition q is true if proposition qcore is true. Proposition x1a is true if proposition x1b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n2-i2-wfs::wfs", "rec_id": "hv2-combo-disj_n2-i2", "axis": "combo", "difficulty": "disj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition qcore is true if proposition x1a is true. Proposition x0b is true if proposition x0a is not true. Proposition x1b is true if proposition x1a is not true. Proposition x0a is true if proposition x0b is not true. Proposition qcore is true if proposition x0a is true. Proposition q is true if proposition qcore is true. Proposition x1a is true if proposition x1b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n2-i3-none::none", "rec_id": "hv2-combo-disj_n2-i3", "axis": "combo", "difficulty": "disj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition t0 is true if proposition t1 is true. Proposition x1a is true if proposition x1b is not true. Proposition q is true if proposition t0 is true. Proposition x0b is true if proposition x0a is not true. Proposition x0a is true if proposition x0b is not true. Proposition x1b is true if proposition x1a is not true. Proposition qcore is true if proposition x1a is true. Proposition t1 is true if proposition qcore is true. Proposition qcore is true if proposition x0a is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n2-i3-closed_world::closed_world", "rec_id": "hv2-combo-disj_n2-i3", "axis": "combo", "difficulty": "disj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition t0 is true if proposition t1 is true. Proposition x1a is true if proposition x1b is not true. Proposition q is true if proposition t0 is true. Proposition x0b is true if proposition x0a is not true. Proposition x0a is true if proposition x0b is not true. Proposition x1b is true if proposition x1a is not true. Proposition qcore is true if proposition x1a is true. Proposition t1 is true if proposition qcore is true. Proposition qcore is true if proposition x0a is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n2-i3-cred::cred", "rec_id": "hv2-combo-disj_n2-i3", "axis": "combo", "difficulty": "disj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition t0 is true if proposition t1 is true. Proposition x1a is true if proposition x1b is not true. Proposition q is true if proposition t0 is true. Proposition x0b is true if proposition x0a is not true. Proposition x0a is true if proposition x0b is not true. Proposition x1b is true if proposition x1a is not true. Proposition qcore is true if proposition x1a is true. Proposition t1 is true if proposition qcore is true. Proposition qcore is true if proposition x0a is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n2-i3-skept::skept", "rec_id": "hv2-combo-disj_n2-i3", "axis": "combo", "difficulty": "disj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition t0 is true if proposition t1 is true. Proposition x1a is true if proposition x1b is not true. Proposition q is true if proposition t0 is true. Proposition x0b is true if proposition x0a is not true. Proposition x0a is true if proposition x0b is not true. Proposition x1b is true if proposition x1a is not true. Proposition qcore is true if proposition x1a is true. Proposition t1 is true if proposition qcore is true. Proposition qcore is true if proposition x0a is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n2-i3-wfs::wfs", "rec_id": "hv2-combo-disj_n2-i3", "axis": "combo", "difficulty": "disj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition t0 is true if proposition t1 is true. Proposition x1a is true if proposition x1b is not true. Proposition q is true if proposition t0 is true. Proposition x0b is true if proposition x0a is not true. Proposition x0a is true if proposition x0b is not true. Proposition x1b is true if proposition x1a is not true. Proposition qcore is true if proposition x1a is true. Proposition t1 is true if proposition qcore is true. Proposition qcore is true if proposition x0a is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n2-i4-none::none", "rec_id": "hv2-combo-disj_n2-i4", "axis": "combo", "difficulty": "disj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition x0a is true if proposition x0b is not true. Proposition s0 is true. Proposition q is true if proposition qcore is true and proposition wide is true and proposition etrue is true. Proposition qcore is true if proposition x0a is true. Proposition s3 is true. Proposition x1a is true if proposition x1b is not true. Proposition s1 is true. Proposition etrue is true. Proposition x1b is true if proposition x1a is not true. Proposition x0b is true if proposition x0a is not true. Proposition s2 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition qcore is true if proposition x1a is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n2-i4-closed_world::closed_world", "rec_id": "hv2-combo-disj_n2-i4", "axis": "combo", "difficulty": "disj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition x0a is true if proposition x0b is not true. Proposition s0 is true. Proposition q is true if proposition qcore is true and proposition wide is true and proposition etrue is true. Proposition qcore is true if proposition x0a is true. Proposition s3 is true. Proposition x1a is true if proposition x1b is not true. Proposition s1 is true. Proposition etrue is true. Proposition x1b is true if proposition x1a is not true. Proposition x0b is true if proposition x0a is not true. Proposition s2 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition qcore is true if proposition x1a is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n2-i4-cred::cred", "rec_id": "hv2-combo-disj_n2-i4", "axis": "combo", "difficulty": "disj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition x0a is true if proposition x0b is not true. Proposition s0 is true. Proposition q is true if proposition qcore is true and proposition wide is true and proposition etrue is true. Proposition qcore is true if proposition x0a is true. Proposition s3 is true. Proposition x1a is true if proposition x1b is not true. Proposition s1 is true. Proposition etrue is true. Proposition x1b is true if proposition x1a is not true. Proposition x0b is true if proposition x0a is not true. Proposition s2 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition qcore is true if proposition x1a is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n2-i4-skept::skept", "rec_id": "hv2-combo-disj_n2-i4", "axis": "combo", "difficulty": "disj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition x0a is true if proposition x0b is not true. Proposition s0 is true. Proposition q is true if proposition qcore is true and proposition wide is true and proposition etrue is true. Proposition qcore is true if proposition x0a is true. Proposition s3 is true. Proposition x1a is true if proposition x1b is not true. Proposition s1 is true. Proposition etrue is true. Proposition x1b is true if proposition x1a is not true. Proposition x0b is true if proposition x0a is not true. Proposition s2 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition qcore is true if proposition x1a is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n2-i4-wfs::wfs", "rec_id": "hv2-combo-disj_n2-i4", "axis": "combo", "difficulty": "disj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition x0a is true if proposition x0b is not true. Proposition s0 is true. Proposition q is true if proposition qcore is true and proposition wide is true and proposition etrue is true. Proposition qcore is true if proposition x0a is true. Proposition s3 is true. Proposition x1a is true if proposition x1b is not true. Proposition s1 is true. Proposition etrue is true. Proposition x1b is true if proposition x1a is not true. Proposition x0b is true if proposition x0a is not true. Proposition s2 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition qcore is true if proposition x1a is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n2-i5-none::none", "rec_id": "hv2-combo-disj_n2-i5", "axis": "combo", "difficulty": "disj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition s1 is true. Proposition q is true if proposition qcore is true and proposition wide is true. Proposition x1b is true if proposition x1a is not true. Proposition x0a is true if proposition x0b is not true. Proposition x0b is true if proposition x0a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition qcore is true if proposition x0a is true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x1a is true. Proposition s0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n2-i5-closed_world::closed_world", "rec_id": "hv2-combo-disj_n2-i5", "axis": "combo", "difficulty": "disj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition s1 is true. Proposition q is true if proposition qcore is true and proposition wide is true. Proposition x1b is true if proposition x1a is not true. Proposition x0a is true if proposition x0b is not true. Proposition x0b is true if proposition x0a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition qcore is true if proposition x0a is true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x1a is true. Proposition s0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n2-i5-cred::cred", "rec_id": "hv2-combo-disj_n2-i5", "axis": "combo", "difficulty": "disj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition s1 is true. Proposition q is true if proposition qcore is true and proposition wide is true. Proposition x1b is true if proposition x1a is not true. Proposition x0a is true if proposition x0b is not true. Proposition x0b is true if proposition x0a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition qcore is true if proposition x0a is true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x1a is true. Proposition s0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n2-i5-skept::skept", "rec_id": "hv2-combo-disj_n2-i5", "axis": "combo", "difficulty": "disj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition s1 is true. Proposition q is true if proposition qcore is true and proposition wide is true. Proposition x1b is true if proposition x1a is not true. Proposition x0a is true if proposition x0b is not true. Proposition x0b is true if proposition x0a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition qcore is true if proposition x0a is true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x1a is true. Proposition s0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n2-i5-wfs::wfs", "rec_id": "hv2-combo-disj_n2-i5", "axis": "combo", "difficulty": "disj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition s1 is true. Proposition q is true if proposition qcore is true and proposition wide is true. Proposition x1b is true if proposition x1a is not true. Proposition x0a is true if proposition x0b is not true. Proposition x0b is true if proposition x0a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition qcore is true if proposition x0a is true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x1a is true. Proposition s0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n2-i6-none::none", "rec_id": "hv2-combo-disj_n2-i6", "axis": "combo", "difficulty": "disj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition t4 is true if proposition t5 is true. Proposition s1 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition qcore is true if proposition x0a is true. Proposition x0b is true if proposition x0a is not true. Proposition t5 is true if proposition qcore is true. Proposition t0 is true if proposition t1 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x1a is true. Proposition t3 is true if proposition t4 is true. Proposition s0 is true. Proposition t1 is true if proposition t2 is true. Proposition t2 is true if proposition t3 is true. Proposition x0a is true if proposition x0b is not true. Proposition x1b is true if proposition x1a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n2-i6-closed_world::closed_world", "rec_id": "hv2-combo-disj_n2-i6", "axis": "combo", "difficulty": "disj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition t4 is true if proposition t5 is true. Proposition s1 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition qcore is true if proposition x0a is true. Proposition x0b is true if proposition x0a is not true. Proposition t5 is true if proposition qcore is true. Proposition t0 is true if proposition t1 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x1a is true. Proposition t3 is true if proposition t4 is true. Proposition s0 is true. Proposition t1 is true if proposition t2 is true. Proposition t2 is true if proposition t3 is true. Proposition x0a is true if proposition x0b is not true. Proposition x1b is true if proposition x1a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n2-i6-cred::cred", "rec_id": "hv2-combo-disj_n2-i6", "axis": "combo", "difficulty": "disj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition t4 is true if proposition t5 is true. Proposition s1 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition qcore is true if proposition x0a is true. Proposition x0b is true if proposition x0a is not true. Proposition t5 is true if proposition qcore is true. Proposition t0 is true if proposition t1 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x1a is true. Proposition t3 is true if proposition t4 is true. Proposition s0 is true. Proposition t1 is true if proposition t2 is true. Proposition t2 is true if proposition t3 is true. Proposition x0a is true if proposition x0b is not true. Proposition x1b is true if proposition x1a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n2-i6-skept::skept", "rec_id": "hv2-combo-disj_n2-i6", "axis": "combo", "difficulty": "disj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition t4 is true if proposition t5 is true. Proposition s1 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition qcore is true if proposition x0a is true. Proposition x0b is true if proposition x0a is not true. Proposition t5 is true if proposition qcore is true. Proposition t0 is true if proposition t1 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x1a is true. Proposition t3 is true if proposition t4 is true. Proposition s0 is true. Proposition t1 is true if proposition t2 is true. Proposition t2 is true if proposition t3 is true. Proposition x0a is true if proposition x0b is not true. Proposition x1b is true if proposition x1a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n2-i6-wfs::wfs", "rec_id": "hv2-combo-disj_n2-i6", "axis": "combo", "difficulty": "disj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition t4 is true if proposition t5 is true. Proposition s1 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition qcore is true if proposition x0a is true. Proposition x0b is true if proposition x0a is not true. Proposition t5 is true if proposition qcore is true. Proposition t0 is true if proposition t1 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x1a is true. Proposition t3 is true if proposition t4 is true. Proposition s0 is true. Proposition t1 is true if proposition t2 is true. Proposition t2 is true if proposition t3 is true. Proposition x0a is true if proposition x0b is not true. Proposition x1b is true if proposition x1a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n2-i7-none::none", "rec_id": "hv2-combo-disj_n2-i7", "axis": "combo", "difficulty": "disj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition x1b is true if proposition x1a is not true. Proposition t0 is true if proposition t1 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition s2 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition t1 is true if proposition qcore is true. Proposition s1 is true. Proposition x0b is true if proposition x0a is not true. Proposition qcore is true if proposition x0a is true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x1a is true. Proposition x0a is true if proposition x0b is not true. Proposition s0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n2-i7-closed_world::closed_world", "rec_id": "hv2-combo-disj_n2-i7", "axis": "combo", "difficulty": "disj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition x1b is true if proposition x1a is not true. Proposition t0 is true if proposition t1 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition s2 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition t1 is true if proposition qcore is true. Proposition s1 is true. Proposition x0b is true if proposition x0a is not true. Proposition qcore is true if proposition x0a is true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x1a is true. Proposition x0a is true if proposition x0b is not true. Proposition s0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n2-i7-cred::cred", "rec_id": "hv2-combo-disj_n2-i7", "axis": "combo", "difficulty": "disj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition x1b is true if proposition x1a is not true. Proposition t0 is true if proposition t1 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition s2 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition t1 is true if proposition qcore is true. Proposition s1 is true. Proposition x0b is true if proposition x0a is not true. Proposition qcore is true if proposition x0a is true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x1a is true. Proposition x0a is true if proposition x0b is not true. Proposition s0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n2-i7-skept::skept", "rec_id": "hv2-combo-disj_n2-i7", "axis": "combo", "difficulty": "disj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition x1b is true if proposition x1a is not true. Proposition t0 is true if proposition t1 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition s2 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition t1 is true if proposition qcore is true. Proposition s1 is true. Proposition x0b is true if proposition x0a is not true. Proposition qcore is true if proposition x0a is true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x1a is true. Proposition x0a is true if proposition x0b is not true. Proposition s0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n2-i7-wfs::wfs", "rec_id": "hv2-combo-disj_n2-i7", "axis": "combo", "difficulty": "disj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition x1b is true if proposition x1a is not true. Proposition t0 is true if proposition t1 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition s2 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition t1 is true if proposition qcore is true. Proposition s1 is true. Proposition x0b is true if proposition x0a is not true. Proposition qcore is true if proposition x0a is true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x1a is true. Proposition x0a is true if proposition x0b is not true. Proposition s0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n3-i0-none::none", "rec_id": "hv2-combo-disj_n3-i0", "axis": "combo", "difficulty": "disj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition x0a is true if proposition x0b is not true. Proposition etrue is true. Proposition x0b is true if proposition x0a is not true. Proposition qcore is true if proposition x1a is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition qcore is true if proposition x0a is true. Proposition t1 is true if proposition t2 is true. Proposition x2b is true if proposition x2a is not true. Proposition t3 is true if proposition qcore is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition s1 is true. Proposition qcore is true if proposition x2a is true. Proposition x1a is true if proposition x1b is not true. Proposition x1b is true if proposition x1a is not true. Proposition t2 is true if proposition t3 is true. Proposition t0 is true if proposition t1 is true. Proposition x2a is true if proposition x2b is not true. Proposition s0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n3-i0-closed_world::closed_world", "rec_id": "hv2-combo-disj_n3-i0", "axis": "combo", "difficulty": "disj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition x0a is true if proposition x0b is not true. Proposition etrue is true. Proposition x0b is true if proposition x0a is not true. Proposition qcore is true if proposition x1a is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition qcore is true if proposition x0a is true. Proposition t1 is true if proposition t2 is true. Proposition x2b is true if proposition x2a is not true. Proposition t3 is true if proposition qcore is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition s1 is true. Proposition qcore is true if proposition x2a is true. Proposition x1a is true if proposition x1b is not true. Proposition x1b is true if proposition x1a is not true. Proposition t2 is true if proposition t3 is true. Proposition t0 is true if proposition t1 is true. Proposition x2a is true if proposition x2b is not true. Proposition s0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n3-i0-cred::cred", "rec_id": "hv2-combo-disj_n3-i0", "axis": "combo", "difficulty": "disj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition x0a is true if proposition x0b is not true. Proposition etrue is true. Proposition x0b is true if proposition x0a is not true. Proposition qcore is true if proposition x1a is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition qcore is true if proposition x0a is true. Proposition t1 is true if proposition t2 is true. Proposition x2b is true if proposition x2a is not true. Proposition t3 is true if proposition qcore is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition s1 is true. Proposition qcore is true if proposition x2a is true. Proposition x1a is true if proposition x1b is not true. Proposition x1b is true if proposition x1a is not true. Proposition t2 is true if proposition t3 is true. Proposition t0 is true if proposition t1 is true. Proposition x2a is true if proposition x2b is not true. Proposition s0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n3-i0-skept::skept", "rec_id": "hv2-combo-disj_n3-i0", "axis": "combo", "difficulty": "disj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition x0a is true if proposition x0b is not true. Proposition etrue is true. Proposition x0b is true if proposition x0a is not true. Proposition qcore is true if proposition x1a is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition qcore is true if proposition x0a is true. Proposition t1 is true if proposition t2 is true. Proposition x2b is true if proposition x2a is not true. Proposition t3 is true if proposition qcore is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition s1 is true. Proposition qcore is true if proposition x2a is true. Proposition x1a is true if proposition x1b is not true. Proposition x1b is true if proposition x1a is not true. Proposition t2 is true if proposition t3 is true. Proposition t0 is true if proposition t1 is true. Proposition x2a is true if proposition x2b is not true. Proposition s0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n3-i0-wfs::wfs", "rec_id": "hv2-combo-disj_n3-i0", "axis": "combo", "difficulty": "disj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition x0a is true if proposition x0b is not true. Proposition etrue is true. Proposition x0b is true if proposition x0a is not true. Proposition qcore is true if proposition x1a is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition qcore is true if proposition x0a is true. Proposition t1 is true if proposition t2 is true. Proposition x2b is true if proposition x2a is not true. Proposition t3 is true if proposition qcore is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition s1 is true. Proposition qcore is true if proposition x2a is true. Proposition x1a is true if proposition x1b is not true. Proposition x1b is true if proposition x1a is not true. Proposition t2 is true if proposition t3 is true. Proposition t0 is true if proposition t1 is true. Proposition x2a is true if proposition x2b is not true. Proposition s0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n3-i1-none::none", "rec_id": "hv2-combo-disj_n3-i1", "axis": "combo", "difficulty": "disj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition qcore is true if proposition x0a is true. Proposition s0 is true. Proposition x0a is true if proposition x0b is not true. Proposition x0b is true if proposition x0a is not true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x2a is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition x1a is true. Proposition t1 is true if proposition qcore is true. Proposition s1 is true. Proposition x2b is true if proposition x2a is not true. Proposition x2a is true if proposition x2b is not true. Proposition x1b is true if proposition x1a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition s2 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n3-i1-closed_world::closed_world", "rec_id": "hv2-combo-disj_n3-i1", "axis": "combo", "difficulty": "disj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition qcore is true if proposition x0a is true. Proposition s0 is true. Proposition x0a is true if proposition x0b is not true. Proposition x0b is true if proposition x0a is not true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x2a is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition x1a is true. Proposition t1 is true if proposition qcore is true. Proposition s1 is true. Proposition x2b is true if proposition x2a is not true. Proposition x2a is true if proposition x2b is not true. Proposition x1b is true if proposition x1a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition s2 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n3-i1-cred::cred", "rec_id": "hv2-combo-disj_n3-i1", "axis": "combo", "difficulty": "disj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition qcore is true if proposition x0a is true. Proposition s0 is true. Proposition x0a is true if proposition x0b is not true. Proposition x0b is true if proposition x0a is not true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x2a is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition x1a is true. Proposition t1 is true if proposition qcore is true. Proposition s1 is true. Proposition x2b is true if proposition x2a is not true. Proposition x2a is true if proposition x2b is not true. Proposition x1b is true if proposition x1a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition s2 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n3-i1-skept::skept", "rec_id": "hv2-combo-disj_n3-i1", "axis": "combo", "difficulty": "disj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition qcore is true if proposition x0a is true. Proposition s0 is true. Proposition x0a is true if proposition x0b is not true. Proposition x0b is true if proposition x0a is not true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x2a is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition x1a is true. Proposition t1 is true if proposition qcore is true. Proposition s1 is true. Proposition x2b is true if proposition x2a is not true. Proposition x2a is true if proposition x2b is not true. Proposition x1b is true if proposition x1a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition s2 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n3-i1-wfs::wfs", "rec_id": "hv2-combo-disj_n3-i1", "axis": "combo", "difficulty": "disj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition qcore is true if proposition x0a is true. Proposition s0 is true. Proposition x0a is true if proposition x0b is not true. Proposition x0b is true if proposition x0a is not true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x2a is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition x1a is true. Proposition t1 is true if proposition qcore is true. Proposition s1 is true. Proposition x2b is true if proposition x2a is not true. Proposition x2a is true if proposition x2b is not true. Proposition x1b is true if proposition x1a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition s2 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n3-i2-none::none", "rec_id": "hv2-combo-disj_n3-i2", "axis": "combo", "difficulty": "disj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition s1 is true. Proposition qcore is true if proposition x1a is true. Proposition t4 is true if proposition t5 is true. Proposition qcore is true if proposition x0a is true. Proposition x2a is true if proposition x2b is not true. Proposition qcore is true if proposition x2a is true. Proposition x1b is true if proposition x1a is not true. Proposition s0 is true. Proposition t2 is true if proposition t3 is true. Proposition x2b is true if proposition x2a is not true. Proposition x1a is true if proposition x1b is not true. Proposition t0 is true if proposition t1 is true. Proposition t3 is true if proposition t4 is true. Proposition t5 is true if proposition qcore is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t1 is true if proposition t2 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x0b is true if proposition x0a is not true. Proposition x0a is true if proposition x0b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n3-i2-closed_world::closed_world", "rec_id": "hv2-combo-disj_n3-i2", "axis": "combo", "difficulty": "disj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition s1 is true. Proposition qcore is true if proposition x1a is true. Proposition t4 is true if proposition t5 is true. Proposition qcore is true if proposition x0a is true. Proposition x2a is true if proposition x2b is not true. Proposition qcore is true if proposition x2a is true. Proposition x1b is true if proposition x1a is not true. Proposition s0 is true. Proposition t2 is true if proposition t3 is true. Proposition x2b is true if proposition x2a is not true. Proposition x1a is true if proposition x1b is not true. Proposition t0 is true if proposition t1 is true. Proposition t3 is true if proposition t4 is true. Proposition t5 is true if proposition qcore is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t1 is true if proposition t2 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x0b is true if proposition x0a is not true. Proposition x0a is true if proposition x0b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n3-i2-cred::cred", "rec_id": "hv2-combo-disj_n3-i2", "axis": "combo", "difficulty": "disj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition s1 is true. Proposition qcore is true if proposition x1a is true. Proposition t4 is true if proposition t5 is true. Proposition qcore is true if proposition x0a is true. Proposition x2a is true if proposition x2b is not true. Proposition qcore is true if proposition x2a is true. Proposition x1b is true if proposition x1a is not true. Proposition s0 is true. Proposition t2 is true if proposition t3 is true. Proposition x2b is true if proposition x2a is not true. Proposition x1a is true if proposition x1b is not true. Proposition t0 is true if proposition t1 is true. Proposition t3 is true if proposition t4 is true. Proposition t5 is true if proposition qcore is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t1 is true if proposition t2 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x0b is true if proposition x0a is not true. Proposition x0a is true if proposition x0b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n3-i2-skept::skept", "rec_id": "hv2-combo-disj_n3-i2", "axis": "combo", "difficulty": "disj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition s1 is true. Proposition qcore is true if proposition x1a is true. Proposition t4 is true if proposition t5 is true. Proposition qcore is true if proposition x0a is true. Proposition x2a is true if proposition x2b is not true. Proposition qcore is true if proposition x2a is true. Proposition x1b is true if proposition x1a is not true. Proposition s0 is true. Proposition t2 is true if proposition t3 is true. Proposition x2b is true if proposition x2a is not true. Proposition x1a is true if proposition x1b is not true. Proposition t0 is true if proposition t1 is true. Proposition t3 is true if proposition t4 is true. Proposition t5 is true if proposition qcore is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t1 is true if proposition t2 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x0b is true if proposition x0a is not true. Proposition x0a is true if proposition x0b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n3-i2-wfs::wfs", "rec_id": "hv2-combo-disj_n3-i2", "axis": "combo", "difficulty": "disj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition s1 is true. Proposition qcore is true if proposition x1a is true. Proposition t4 is true if proposition t5 is true. Proposition qcore is true if proposition x0a is true. Proposition x2a is true if proposition x2b is not true. Proposition qcore is true if proposition x2a is true. Proposition x1b is true if proposition x1a is not true. Proposition s0 is true. Proposition t2 is true if proposition t3 is true. Proposition x2b is true if proposition x2a is not true. Proposition x1a is true if proposition x1b is not true. Proposition t0 is true if proposition t1 is true. Proposition t3 is true if proposition t4 is true. Proposition t5 is true if proposition qcore is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t1 is true if proposition t2 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x0b is true if proposition x0a is not true. Proposition x0a is true if proposition x0b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n3-i3-none::none", "rec_id": "hv2-combo-disj_n3-i3", "axis": "combo", "difficulty": "disj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition x2a is true if proposition x2b is not true. Proposition s0 is true. Proposition s1 is true. Proposition qcore is true if proposition x0a is true. Proposition t1 is true if proposition t2 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition qcore is true if proposition x1a is true. Proposition t0 is true if proposition t1 is true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x2a is true. Proposition x1b is true if proposition x1a is not true. Proposition s3 is true. Proposition x0a is true if proposition x0b is not true. Proposition x2b is true if proposition x2a is not true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition s2 is true. Proposition t2 is true if proposition t3 is true. Proposition x0b is true if proposition x0a is not true. Proposition t3 is true if proposition qcore is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n3-i3-closed_world::closed_world", "rec_id": "hv2-combo-disj_n3-i3", "axis": "combo", "difficulty": "disj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition x2a is true if proposition x2b is not true. Proposition s0 is true. Proposition s1 is true. Proposition qcore is true if proposition x0a is true. Proposition t1 is true if proposition t2 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition qcore is true if proposition x1a is true. Proposition t0 is true if proposition t1 is true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x2a is true. Proposition x1b is true if proposition x1a is not true. Proposition s3 is true. Proposition x0a is true if proposition x0b is not true. Proposition x2b is true if proposition x2a is not true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition s2 is true. Proposition t2 is true if proposition t3 is true. Proposition x0b is true if proposition x0a is not true. Proposition t3 is true if proposition qcore is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n3-i3-cred::cred", "rec_id": "hv2-combo-disj_n3-i3", "axis": "combo", "difficulty": "disj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition x2a is true if proposition x2b is not true. Proposition s0 is true. Proposition s1 is true. Proposition qcore is true if proposition x0a is true. Proposition t1 is true if proposition t2 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition qcore is true if proposition x1a is true. Proposition t0 is true if proposition t1 is true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x2a is true. Proposition x1b is true if proposition x1a is not true. Proposition s3 is true. Proposition x0a is true if proposition x0b is not true. Proposition x2b is true if proposition x2a is not true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition s2 is true. Proposition t2 is true if proposition t3 is true. Proposition x0b is true if proposition x0a is not true. Proposition t3 is true if proposition qcore is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n3-i3-skept::skept", "rec_id": "hv2-combo-disj_n3-i3", "axis": "combo", "difficulty": "disj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition x2a is true if proposition x2b is not true. Proposition s0 is true. Proposition s1 is true. Proposition qcore is true if proposition x0a is true. Proposition t1 is true if proposition t2 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition qcore is true if proposition x1a is true. Proposition t0 is true if proposition t1 is true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x2a is true. Proposition x1b is true if proposition x1a is not true. Proposition s3 is true. Proposition x0a is true if proposition x0b is not true. Proposition x2b is true if proposition x2a is not true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition s2 is true. Proposition t2 is true if proposition t3 is true. Proposition x0b is true if proposition x0a is not true. Proposition t3 is true if proposition qcore is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n3-i3-wfs::wfs", "rec_id": "hv2-combo-disj_n3-i3", "axis": "combo", "difficulty": "disj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition x2a is true if proposition x2b is not true. Proposition s0 is true. Proposition s1 is true. Proposition qcore is true if proposition x0a is true. Proposition t1 is true if proposition t2 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition qcore is true if proposition x1a is true. Proposition t0 is true if proposition t1 is true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x2a is true. Proposition x1b is true if proposition x1a is not true. Proposition s3 is true. Proposition x0a is true if proposition x0b is not true. Proposition x2b is true if proposition x2a is not true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition s2 is true. Proposition t2 is true if proposition t3 is true. Proposition x0b is true if proposition x0a is not true. Proposition t3 is true if proposition qcore is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n3-i4-none::none", "rec_id": "hv2-combo-disj_n3-i4", "axis": "combo", "difficulty": "disj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition x2a is true if proposition x2b is not true. Proposition s3 is true. Proposition qcore is true if proposition x1a is true. Proposition qcore is true if proposition x2a is true. Proposition s1 is true. Proposition x1b is true if proposition x1a is not true. Proposition qcore is true if proposition x0a is true. Proposition x1a is true if proposition x1b is not true. Proposition etrue is true. Proposition x2b is true if proposition x2a is not true. Proposition s0 is true. Proposition q is true if proposition qcore is true and proposition wide is true and proposition etrue is true. Proposition x0b is true if proposition x0a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s2 is true. Proposition x0a is true if proposition x0b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n3-i4-closed_world::closed_world", "rec_id": "hv2-combo-disj_n3-i4", "axis": "combo", "difficulty": "disj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition x2a is true if proposition x2b is not true. Proposition s3 is true. Proposition qcore is true if proposition x1a is true. Proposition qcore is true if proposition x2a is true. Proposition s1 is true. Proposition x1b is true if proposition x1a is not true. Proposition qcore is true if proposition x0a is true. Proposition x1a is true if proposition x1b is not true. Proposition etrue is true. Proposition x2b is true if proposition x2a is not true. Proposition s0 is true. Proposition q is true if proposition qcore is true and proposition wide is true and proposition etrue is true. Proposition x0b is true if proposition x0a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s2 is true. Proposition x0a is true if proposition x0b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n3-i4-cred::cred", "rec_id": "hv2-combo-disj_n3-i4", "axis": "combo", "difficulty": "disj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition x2a is true if proposition x2b is not true. Proposition s3 is true. Proposition qcore is true if proposition x1a is true. Proposition qcore is true if proposition x2a is true. Proposition s1 is true. Proposition x1b is true if proposition x1a is not true. Proposition qcore is true if proposition x0a is true. Proposition x1a is true if proposition x1b is not true. Proposition etrue is true. Proposition x2b is true if proposition x2a is not true. Proposition s0 is true. Proposition q is true if proposition qcore is true and proposition wide is true and proposition etrue is true. Proposition x0b is true if proposition x0a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s2 is true. Proposition x0a is true if proposition x0b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n3-i4-skept::skept", "rec_id": "hv2-combo-disj_n3-i4", "axis": "combo", "difficulty": "disj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition x2a is true if proposition x2b is not true. Proposition s3 is true. Proposition qcore is true if proposition x1a is true. Proposition qcore is true if proposition x2a is true. Proposition s1 is true. Proposition x1b is true if proposition x1a is not true. Proposition qcore is true if proposition x0a is true. Proposition x1a is true if proposition x1b is not true. Proposition etrue is true. Proposition x2b is true if proposition x2a is not true. Proposition s0 is true. Proposition q is true if proposition qcore is true and proposition wide is true and proposition etrue is true. Proposition x0b is true if proposition x0a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s2 is true. Proposition x0a is true if proposition x0b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n3-i4-wfs::wfs", "rec_id": "hv2-combo-disj_n3-i4", "axis": "combo", "difficulty": "disj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition x2a is true if proposition x2b is not true. Proposition s3 is true. Proposition qcore is true if proposition x1a is true. Proposition qcore is true if proposition x2a is true. Proposition s1 is true. Proposition x1b is true if proposition x1a is not true. Proposition qcore is true if proposition x0a is true. Proposition x1a is true if proposition x1b is not true. Proposition etrue is true. Proposition x2b is true if proposition x2a is not true. Proposition s0 is true. Proposition q is true if proposition qcore is true and proposition wide is true and proposition etrue is true. Proposition x0b is true if proposition x0a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s2 is true. Proposition x0a is true if proposition x0b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n3-i5-none::none", "rec_id": "hv2-combo-disj_n3-i5", "axis": "combo", "difficulty": "disj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition x2a is true if proposition x2b is not true. Proposition t3 is true if proposition qcore is true. Proposition x1a is true if proposition x1b is not true. Proposition t1 is true if proposition t2 is true. Proposition qcore is true if proposition x0a is true. Proposition q is true if proposition t0 is true. Proposition t0 is true if proposition t1 is true. Proposition x2b is true if proposition x2a is not true. Proposition qcore is true if proposition x2a is true. Proposition x0a is true if proposition x0b is not true. Proposition t2 is true if proposition t3 is true. Proposition x0b is true if proposition x0a is not true. Proposition x1b is true if proposition x1a is not true. Proposition qcore is true if proposition x1a is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n3-i5-closed_world::closed_world", "rec_id": "hv2-combo-disj_n3-i5", "axis": "combo", "difficulty": "disj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition x2a is true if proposition x2b is not true. Proposition t3 is true if proposition qcore is true. Proposition x1a is true if proposition x1b is not true. Proposition t1 is true if proposition t2 is true. Proposition qcore is true if proposition x0a is true. Proposition q is true if proposition t0 is true. Proposition t0 is true if proposition t1 is true. Proposition x2b is true if proposition x2a is not true. Proposition qcore is true if proposition x2a is true. Proposition x0a is true if proposition x0b is not true. Proposition t2 is true if proposition t3 is true. Proposition x0b is true if proposition x0a is not true. Proposition x1b is true if proposition x1a is not true. Proposition qcore is true if proposition x1a is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n3-i5-cred::cred", "rec_id": "hv2-combo-disj_n3-i5", "axis": "combo", "difficulty": "disj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition x2a is true if proposition x2b is not true. Proposition t3 is true if proposition qcore is true. Proposition x1a is true if proposition x1b is not true. Proposition t1 is true if proposition t2 is true. Proposition qcore is true if proposition x0a is true. Proposition q is true if proposition t0 is true. Proposition t0 is true if proposition t1 is true. Proposition x2b is true if proposition x2a is not true. Proposition qcore is true if proposition x2a is true. Proposition x0a is true if proposition x0b is not true. Proposition t2 is true if proposition t3 is true. Proposition x0b is true if proposition x0a is not true. Proposition x1b is true if proposition x1a is not true. Proposition qcore is true if proposition x1a is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n3-i5-skept::skept", "rec_id": "hv2-combo-disj_n3-i5", "axis": "combo", "difficulty": "disj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition x2a is true if proposition x2b is not true. Proposition t3 is true if proposition qcore is true. Proposition x1a is true if proposition x1b is not true. Proposition t1 is true if proposition t2 is true. Proposition qcore is true if proposition x0a is true. Proposition q is true if proposition t0 is true. Proposition t0 is true if proposition t1 is true. Proposition x2b is true if proposition x2a is not true. Proposition qcore is true if proposition x2a is true. Proposition x0a is true if proposition x0b is not true. Proposition t2 is true if proposition t3 is true. Proposition x0b is true if proposition x0a is not true. Proposition x1b is true if proposition x1a is not true. Proposition qcore is true if proposition x1a is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n3-i5-wfs::wfs", "rec_id": "hv2-combo-disj_n3-i5", "axis": "combo", "difficulty": "disj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition x2a is true if proposition x2b is not true. Proposition t3 is true if proposition qcore is true. Proposition x1a is true if proposition x1b is not true. Proposition t1 is true if proposition t2 is true. Proposition qcore is true if proposition x0a is true. Proposition q is true if proposition t0 is true. Proposition t0 is true if proposition t1 is true. Proposition x2b is true if proposition x2a is not true. Proposition qcore is true if proposition x2a is true. Proposition x0a is true if proposition x0b is not true. Proposition t2 is true if proposition t3 is true. Proposition x0b is true if proposition x0a is not true. Proposition x1b is true if proposition x1a is not true. Proposition qcore is true if proposition x1a is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n3-i6-none::none", "rec_id": "hv2-combo-disj_n3-i6", "axis": "combo", "difficulty": "disj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition t4 is true if proposition t5 is true. Proposition s1 is true. Proposition s0 is true. Proposition qcore is true if proposition x2a is true. Proposition t0 is true if proposition t1 is true. Proposition t3 is true if proposition t4 is true. Proposition t2 is true if proposition t3 is true. Proposition t5 is true if proposition qcore is true. Proposition x1b is true if proposition x1a is not true. Proposition x0a is true if proposition x0b is not true. Proposition qcore is true if proposition x1a is true. Proposition x2b is true if proposition x2a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition x0b is true if proposition x0a is not true. Proposition x2a is true if proposition x2b is not true. Proposition t1 is true if proposition t2 is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition s2 is true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x0a is true. Proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n3-i6-closed_world::closed_world", "rec_id": "hv2-combo-disj_n3-i6", "axis": "combo", "difficulty": "disj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition t4 is true if proposition t5 is true. Proposition s1 is true. Proposition s0 is true. Proposition qcore is true if proposition x2a is true. Proposition t0 is true if proposition t1 is true. Proposition t3 is true if proposition t4 is true. Proposition t2 is true if proposition t3 is true. Proposition t5 is true if proposition qcore is true. Proposition x1b is true if proposition x1a is not true. Proposition x0a is true if proposition x0b is not true. Proposition qcore is true if proposition x1a is true. Proposition x2b is true if proposition x2a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition x0b is true if proposition x0a is not true. Proposition x2a is true if proposition x2b is not true. Proposition t1 is true if proposition t2 is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition s2 is true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x0a is true. Proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n3-i6-cred::cred", "rec_id": "hv2-combo-disj_n3-i6", "axis": "combo", "difficulty": "disj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition t4 is true if proposition t5 is true. Proposition s1 is true. Proposition s0 is true. Proposition qcore is true if proposition x2a is true. Proposition t0 is true if proposition t1 is true. Proposition t3 is true if proposition t4 is true. Proposition t2 is true if proposition t3 is true. Proposition t5 is true if proposition qcore is true. Proposition x1b is true if proposition x1a is not true. Proposition x0a is true if proposition x0b is not true. Proposition qcore is true if proposition x1a is true. Proposition x2b is true if proposition x2a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition x0b is true if proposition x0a is not true. Proposition x2a is true if proposition x2b is not true. Proposition t1 is true if proposition t2 is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition s2 is true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x0a is true. Proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n3-i6-skept::skept", "rec_id": "hv2-combo-disj_n3-i6", "axis": "combo", "difficulty": "disj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition t4 is true if proposition t5 is true. Proposition s1 is true. Proposition s0 is true. Proposition qcore is true if proposition x2a is true. Proposition t0 is true if proposition t1 is true. Proposition t3 is true if proposition t4 is true. Proposition t2 is true if proposition t3 is true. Proposition t5 is true if proposition qcore is true. Proposition x1b is true if proposition x1a is not true. Proposition x0a is true if proposition x0b is not true. Proposition qcore is true if proposition x1a is true. Proposition x2b is true if proposition x2a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition x0b is true if proposition x0a is not true. Proposition x2a is true if proposition x2b is not true. Proposition t1 is true if proposition t2 is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition s2 is true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x0a is true. Proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n3-i6-wfs::wfs", "rec_id": "hv2-combo-disj_n3-i6", "axis": "combo", "difficulty": "disj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition t4 is true if proposition t5 is true. Proposition s1 is true. Proposition s0 is true. Proposition qcore is true if proposition x2a is true. Proposition t0 is true if proposition t1 is true. Proposition t3 is true if proposition t4 is true. Proposition t2 is true if proposition t3 is true. Proposition t5 is true if proposition qcore is true. Proposition x1b is true if proposition x1a is not true. Proposition x0a is true if proposition x0b is not true. Proposition qcore is true if proposition x1a is true. Proposition x2b is true if proposition x2a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition x0b is true if proposition x0a is not true. Proposition x2a is true if proposition x2b is not true. Proposition t1 is true if proposition t2 is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition s2 is true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x0a is true. Proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n3-i7-none::none", "rec_id": "hv2-combo-disj_n3-i7", "axis": "combo", "difficulty": "disj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition qcore is true if proposition x1a is true. Proposition x0a is true if proposition x0b is not true. Proposition qcore is true if proposition x2a is true. Proposition x1b is true if proposition x1a is not true. Proposition qcore is true if proposition x0a is true. Proposition t1 is true if proposition qcore is true. Proposition x2a is true if proposition x2b is not true. Proposition t0 is true if proposition t1 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition s1 is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition x2b is true if proposition x2a is not true. Proposition s2 is true. Proposition s0 is true. Proposition x0b is true if proposition x0a is not true. Proposition x1a is true if proposition x1b is not true. Proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n3-i7-closed_world::closed_world", "rec_id": "hv2-combo-disj_n3-i7", "axis": "combo", "difficulty": "disj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition qcore is true if proposition x1a is true. Proposition x0a is true if proposition x0b is not true. Proposition qcore is true if proposition x2a is true. Proposition x1b is true if proposition x1a is not true. Proposition qcore is true if proposition x0a is true. Proposition t1 is true if proposition qcore is true. Proposition x2a is true if proposition x2b is not true. Proposition t0 is true if proposition t1 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition s1 is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition x2b is true if proposition x2a is not true. Proposition s2 is true. Proposition s0 is true. Proposition x0b is true if proposition x0a is not true. Proposition x1a is true if proposition x1b is not true. Proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n3-i7-cred::cred", "rec_id": "hv2-combo-disj_n3-i7", "axis": "combo", "difficulty": "disj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition qcore is true if proposition x1a is true. Proposition x0a is true if proposition x0b is not true. Proposition qcore is true if proposition x2a is true. Proposition x1b is true if proposition x1a is not true. Proposition qcore is true if proposition x0a is true. Proposition t1 is true if proposition qcore is true. Proposition x2a is true if proposition x2b is not true. Proposition t0 is true if proposition t1 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition s1 is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition x2b is true if proposition x2a is not true. Proposition s2 is true. Proposition s0 is true. Proposition x0b is true if proposition x0a is not true. Proposition x1a is true if proposition x1b is not true. Proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n3-i7-skept::skept", "rec_id": "hv2-combo-disj_n3-i7", "axis": "combo", "difficulty": "disj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition qcore is true if proposition x1a is true. Proposition x0a is true if proposition x0b is not true. Proposition qcore is true if proposition x2a is true. Proposition x1b is true if proposition x1a is not true. Proposition qcore is true if proposition x0a is true. Proposition t1 is true if proposition qcore is true. Proposition x2a is true if proposition x2b is not true. Proposition t0 is true if proposition t1 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition s1 is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition x2b is true if proposition x2a is not true. Proposition s2 is true. Proposition s0 is true. Proposition x0b is true if proposition x0a is not true. Proposition x1a is true if proposition x1b is not true. Proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n3-i7-wfs::wfs", "rec_id": "hv2-combo-disj_n3-i7", "axis": "combo", "difficulty": "disj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition qcore is true if proposition x1a is true. Proposition x0a is true if proposition x0b is not true. Proposition qcore is true if proposition x2a is true. Proposition x1b is true if proposition x1a is not true. Proposition qcore is true if proposition x0a is true. Proposition t1 is true if proposition qcore is true. Proposition x2a is true if proposition x2b is not true. Proposition t0 is true if proposition t1 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition s1 is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition x2b is true if proposition x2a is not true. Proposition s2 is true. Proposition s0 is true. Proposition x0b is true if proposition x0a is not true. Proposition x1a is true if proposition x1b is not true. Proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n4-i0-none::none", "rec_id": "hv2-combo-disj_n4-i0", "axis": "combo", "difficulty": "disj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition t5 is true if proposition qcore is true. Proposition x1a is true if proposition x1b is not true. Proposition x2a is true if proposition x2b is not true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition qcore is true if proposition x1a is true. Proposition s1 is true. Proposition t1 is true if proposition t2 is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition x0a is true. Proposition x3a is true if proposition x3b is not true. Proposition s3 is true. Proposition x0b is true if proposition x0a is not true. Proposition t2 is true if proposition t3 is true. Proposition x0a is true if proposition x0b is not true. Proposition x2b is true if proposition x2a is not true. Proposition t3 is true if proposition t4 is true. Proposition qcore is true if proposition x3a is true. Proposition x1b is true if proposition x1a is not true. Proposition s0 is true. Proposition qcore is true if proposition x2a is true. Proposition s2 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition x3b is true if proposition x3a is not true. Proposition t4 is true if proposition t5 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n4-i0-closed_world::closed_world", "rec_id": "hv2-combo-disj_n4-i0", "axis": "combo", "difficulty": "disj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition t5 is true if proposition qcore is true. Proposition x1a is true if proposition x1b is not true. Proposition x2a is true if proposition x2b is not true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition qcore is true if proposition x1a is true. Proposition s1 is true. Proposition t1 is true if proposition t2 is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition x0a is true. Proposition x3a is true if proposition x3b is not true. Proposition s3 is true. Proposition x0b is true if proposition x0a is not true. Proposition t2 is true if proposition t3 is true. Proposition x0a is true if proposition x0b is not true. Proposition x2b is true if proposition x2a is not true. Proposition t3 is true if proposition t4 is true. Proposition qcore is true if proposition x3a is true. Proposition x1b is true if proposition x1a is not true. Proposition s0 is true. Proposition qcore is true if proposition x2a is true. Proposition s2 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition x3b is true if proposition x3a is not true. Proposition t4 is true if proposition t5 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n4-i0-cred::cred", "rec_id": "hv2-combo-disj_n4-i0", "axis": "combo", "difficulty": "disj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition t5 is true if proposition qcore is true. Proposition x1a is true if proposition x1b is not true. Proposition x2a is true if proposition x2b is not true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition qcore is true if proposition x1a is true. Proposition s1 is true. Proposition t1 is true if proposition t2 is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition x0a is true. Proposition x3a is true if proposition x3b is not true. Proposition s3 is true. Proposition x0b is true if proposition x0a is not true. Proposition t2 is true if proposition t3 is true. Proposition x0a is true if proposition x0b is not true. Proposition x2b is true if proposition x2a is not true. Proposition t3 is true if proposition t4 is true. Proposition qcore is true if proposition x3a is true. Proposition x1b is true if proposition x1a is not true. Proposition s0 is true. Proposition qcore is true if proposition x2a is true. Proposition s2 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition x3b is true if proposition x3a is not true. Proposition t4 is true if proposition t5 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n4-i0-skept::skept", "rec_id": "hv2-combo-disj_n4-i0", "axis": "combo", "difficulty": "disj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition t5 is true if proposition qcore is true. Proposition x1a is true if proposition x1b is not true. Proposition x2a is true if proposition x2b is not true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition qcore is true if proposition x1a is true. Proposition s1 is true. Proposition t1 is true if proposition t2 is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition x0a is true. Proposition x3a is true if proposition x3b is not true. Proposition s3 is true. Proposition x0b is true if proposition x0a is not true. Proposition t2 is true if proposition t3 is true. Proposition x0a is true if proposition x0b is not true. Proposition x2b is true if proposition x2a is not true. Proposition t3 is true if proposition t4 is true. Proposition qcore is true if proposition x3a is true. Proposition x1b is true if proposition x1a is not true. Proposition s0 is true. Proposition qcore is true if proposition x2a is true. Proposition s2 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition x3b is true if proposition x3a is not true. Proposition t4 is true if proposition t5 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n4-i0-wfs::wfs", "rec_id": "hv2-combo-disj_n4-i0", "axis": "combo", "difficulty": "disj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition t5 is true if proposition qcore is true. Proposition x1a is true if proposition x1b is not true. Proposition x2a is true if proposition x2b is not true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition qcore is true if proposition x1a is true. Proposition s1 is true. Proposition t1 is true if proposition t2 is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition x0a is true. Proposition x3a is true if proposition x3b is not true. Proposition s3 is true. Proposition x0b is true if proposition x0a is not true. Proposition t2 is true if proposition t3 is true. Proposition x0a is true if proposition x0b is not true. Proposition x2b is true if proposition x2a is not true. Proposition t3 is true if proposition t4 is true. Proposition qcore is true if proposition x3a is true. Proposition x1b is true if proposition x1a is not true. Proposition s0 is true. Proposition qcore is true if proposition x2a is true. Proposition s2 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition x3b is true if proposition x3a is not true. Proposition t4 is true if proposition t5 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n4-i1-none::none", "rec_id": "hv2-combo-disj_n4-i1", "axis": "combo", "difficulty": "disj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition qcore is true if proposition x2a is true. Proposition x3a is true if proposition x3b is not true. Proposition x2b is true if proposition x2a is not true. Proposition x1b is true if proposition x1a is not true. Proposition s1 is true. Proposition x1a is true if proposition x1b is not true. Proposition s2 is true. Proposition s0 is true. Proposition qcore is true if proposition x0a is true. Proposition q is true if proposition qcore is true and proposition wide is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition qcore is true if proposition x1a is true. Proposition x2a is true if proposition x2b is not true. Proposition qcore is true if proposition x3a is true. Proposition s3 is true. Proposition x0b is true if proposition x0a is not true. Proposition x3b is true if proposition x3a is not true. Proposition x0a is true if proposition x0b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n4-i1-closed_world::closed_world", "rec_id": "hv2-combo-disj_n4-i1", "axis": "combo", "difficulty": "disj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition qcore is true if proposition x2a is true. Proposition x3a is true if proposition x3b is not true. Proposition x2b is true if proposition x2a is not true. Proposition x1b is true if proposition x1a is not true. Proposition s1 is true. Proposition x1a is true if proposition x1b is not true. Proposition s2 is true. Proposition s0 is true. Proposition qcore is true if proposition x0a is true. Proposition q is true if proposition qcore is true and proposition wide is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition qcore is true if proposition x1a is true. Proposition x2a is true if proposition x2b is not true. Proposition qcore is true if proposition x3a is true. Proposition s3 is true. Proposition x0b is true if proposition x0a is not true. Proposition x3b is true if proposition x3a is not true. Proposition x0a is true if proposition x0b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n4-i1-cred::cred", "rec_id": "hv2-combo-disj_n4-i1", "axis": "combo", "difficulty": "disj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition qcore is true if proposition x2a is true. Proposition x3a is true if proposition x3b is not true. Proposition x2b is true if proposition x2a is not true. Proposition x1b is true if proposition x1a is not true. Proposition s1 is true. Proposition x1a is true if proposition x1b is not true. Proposition s2 is true. Proposition s0 is true. Proposition qcore is true if proposition x0a is true. Proposition q is true if proposition qcore is true and proposition wide is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition qcore is true if proposition x1a is true. Proposition x2a is true if proposition x2b is not true. Proposition qcore is true if proposition x3a is true. Proposition s3 is true. Proposition x0b is true if proposition x0a is not true. Proposition x3b is true if proposition x3a is not true. Proposition x0a is true if proposition x0b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n4-i1-skept::skept", "rec_id": "hv2-combo-disj_n4-i1", "axis": "combo", "difficulty": "disj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition qcore is true if proposition x2a is true. Proposition x3a is true if proposition x3b is not true. Proposition x2b is true if proposition x2a is not true. Proposition x1b is true if proposition x1a is not true. Proposition s1 is true. Proposition x1a is true if proposition x1b is not true. Proposition s2 is true. Proposition s0 is true. Proposition qcore is true if proposition x0a is true. Proposition q is true if proposition qcore is true and proposition wide is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition qcore is true if proposition x1a is true. Proposition x2a is true if proposition x2b is not true. Proposition qcore is true if proposition x3a is true. Proposition s3 is true. Proposition x0b is true if proposition x0a is not true. Proposition x3b is true if proposition x3a is not true. Proposition x0a is true if proposition x0b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n4-i1-wfs::wfs", "rec_id": "hv2-combo-disj_n4-i1", "axis": "combo", "difficulty": "disj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition qcore is true if proposition x2a is true. Proposition x3a is true if proposition x3b is not true. Proposition x2b is true if proposition x2a is not true. Proposition x1b is true if proposition x1a is not true. Proposition s1 is true. Proposition x1a is true if proposition x1b is not true. Proposition s2 is true. Proposition s0 is true. Proposition qcore is true if proposition x0a is true. Proposition q is true if proposition qcore is true and proposition wide is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition qcore is true if proposition x1a is true. Proposition x2a is true if proposition x2b is not true. Proposition qcore is true if proposition x3a is true. Proposition s3 is true. Proposition x0b is true if proposition x0a is not true. Proposition x3b is true if proposition x3a is not true. Proposition x0a is true if proposition x0b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n4-i2-none::none", "rec_id": "hv2-combo-disj_n4-i2", "axis": "combo", "difficulty": "disj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition q is true if proposition t0 is true and proposition wide is true. Proposition qcore is true if proposition x2a is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t0 is true if proposition t1 is true. Proposition x3b is true if proposition x3a is not true. Proposition qcore is true if proposition x1a is true. Proposition x0a is true if proposition x0b is not true. Proposition x2a is true if proposition x2b is not true. Proposition s1 is true. Proposition x2b is true if proposition x2a is not true. Proposition x1a is true if proposition x1b is not true. Proposition s0 is true. Proposition qcore is true if proposition x3a is true. Proposition t1 is true if proposition qcore is true. Proposition x3a is true if proposition x3b is not true. Proposition qcore is true if proposition x0a is true. Proposition x1b is true if proposition x1a is not true. Proposition x0b is true if proposition x0a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n4-i2-closed_world::closed_world", "rec_id": "hv2-combo-disj_n4-i2", "axis": "combo", "difficulty": "disj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition q is true if proposition t0 is true and proposition wide is true. Proposition qcore is true if proposition x2a is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t0 is true if proposition t1 is true. Proposition x3b is true if proposition x3a is not true. Proposition qcore is true if proposition x1a is true. Proposition x0a is true if proposition x0b is not true. Proposition x2a is true if proposition x2b is not true. Proposition s1 is true. Proposition x2b is true if proposition x2a is not true. Proposition x1a is true if proposition x1b is not true. Proposition s0 is true. Proposition qcore is true if proposition x3a is true. Proposition t1 is true if proposition qcore is true. Proposition x3a is true if proposition x3b is not true. Proposition qcore is true if proposition x0a is true. Proposition x1b is true if proposition x1a is not true. Proposition x0b is true if proposition x0a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n4-i2-cred::cred", "rec_id": "hv2-combo-disj_n4-i2", "axis": "combo", "difficulty": "disj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition q is true if proposition t0 is true and proposition wide is true. Proposition qcore is true if proposition x2a is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t0 is true if proposition t1 is true. Proposition x3b is true if proposition x3a is not true. Proposition qcore is true if proposition x1a is true. Proposition x0a is true if proposition x0b is not true. Proposition x2a is true if proposition x2b is not true. Proposition s1 is true. Proposition x2b is true if proposition x2a is not true. Proposition x1a is true if proposition x1b is not true. Proposition s0 is true. Proposition qcore is true if proposition x3a is true. Proposition t1 is true if proposition qcore is true. Proposition x3a is true if proposition x3b is not true. Proposition qcore is true if proposition x0a is true. Proposition x1b is true if proposition x1a is not true. Proposition x0b is true if proposition x0a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n4-i2-skept::skept", "rec_id": "hv2-combo-disj_n4-i2", "axis": "combo", "difficulty": "disj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition q is true if proposition t0 is true and proposition wide is true. Proposition qcore is true if proposition x2a is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t0 is true if proposition t1 is true. Proposition x3b is true if proposition x3a is not true. Proposition qcore is true if proposition x1a is true. Proposition x0a is true if proposition x0b is not true. Proposition x2a is true if proposition x2b is not true. Proposition s1 is true. Proposition x2b is true if proposition x2a is not true. Proposition x1a is true if proposition x1b is not true. Proposition s0 is true. Proposition qcore is true if proposition x3a is true. Proposition t1 is true if proposition qcore is true. Proposition x3a is true if proposition x3b is not true. Proposition qcore is true if proposition x0a is true. Proposition x1b is true if proposition x1a is not true. Proposition x0b is true if proposition x0a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n4-i2-wfs::wfs", "rec_id": "hv2-combo-disj_n4-i2", "axis": "combo", "difficulty": "disj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition q is true if proposition t0 is true and proposition wide is true. Proposition qcore is true if proposition x2a is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t0 is true if proposition t1 is true. Proposition x3b is true if proposition x3a is not true. Proposition qcore is true if proposition x1a is true. Proposition x0a is true if proposition x0b is not true. Proposition x2a is true if proposition x2b is not true. Proposition s1 is true. Proposition x2b is true if proposition x2a is not true. Proposition x1a is true if proposition x1b is not true. Proposition s0 is true. Proposition qcore is true if proposition x3a is true. Proposition t1 is true if proposition qcore is true. Proposition x3a is true if proposition x3b is not true. Proposition qcore is true if proposition x0a is true. Proposition x1b is true if proposition x1a is not true. Proposition x0b is true if proposition x0a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n4-i3-none::none", "rec_id": "hv2-combo-disj_n4-i3", "axis": "combo", "difficulty": "disj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition x1b is true if proposition x1a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition qcore is true if proposition x2a is true. Proposition qcore is true if proposition x0a is true. Proposition x3a is true if proposition x3b is not true. Proposition s1 is true. Proposition x3b is true if proposition x3a is not true. Proposition qcore is true if proposition x3a is true. Proposition x2b is true if proposition x2a is not true. Proposition x1a is true if proposition x1b is not true. Proposition x0b is true if proposition x0a is not true. Proposition s0 is true. Proposition qcore is true if proposition x1a is true. Proposition x0a is true if proposition x0b is not true. Proposition q is true if proposition qcore is true and proposition wide is true. Proposition x2a is true if proposition x2b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n4-i3-closed_world::closed_world", "rec_id": "hv2-combo-disj_n4-i3", "axis": "combo", "difficulty": "disj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition x1b is true if proposition x1a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition qcore is true if proposition x2a is true. Proposition qcore is true if proposition x0a is true. Proposition x3a is true if proposition x3b is not true. Proposition s1 is true. Proposition x3b is true if proposition x3a is not true. Proposition qcore is true if proposition x3a is true. Proposition x2b is true if proposition x2a is not true. Proposition x1a is true if proposition x1b is not true. Proposition x0b is true if proposition x0a is not true. Proposition s0 is true. Proposition qcore is true if proposition x1a is true. Proposition x0a is true if proposition x0b is not true. Proposition q is true if proposition qcore is true and proposition wide is true. Proposition x2a is true if proposition x2b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n4-i3-cred::cred", "rec_id": "hv2-combo-disj_n4-i3", "axis": "combo", "difficulty": "disj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition x1b is true if proposition x1a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition qcore is true if proposition x2a is true. Proposition qcore is true if proposition x0a is true. Proposition x3a is true if proposition x3b is not true. Proposition s1 is true. Proposition x3b is true if proposition x3a is not true. Proposition qcore is true if proposition x3a is true. Proposition x2b is true if proposition x2a is not true. Proposition x1a is true if proposition x1b is not true. Proposition x0b is true if proposition x0a is not true. Proposition s0 is true. Proposition qcore is true if proposition x1a is true. Proposition x0a is true if proposition x0b is not true. Proposition q is true if proposition qcore is true and proposition wide is true. Proposition x2a is true if proposition x2b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n4-i3-skept::skept", "rec_id": "hv2-combo-disj_n4-i3", "axis": "combo", "difficulty": "disj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition x1b is true if proposition x1a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition qcore is true if proposition x2a is true. Proposition qcore is true if proposition x0a is true. Proposition x3a is true if proposition x3b is not true. Proposition s1 is true. Proposition x3b is true if proposition x3a is not true. Proposition qcore is true if proposition x3a is true. Proposition x2b is true if proposition x2a is not true. Proposition x1a is true if proposition x1b is not true. Proposition x0b is true if proposition x0a is not true. Proposition s0 is true. Proposition qcore is true if proposition x1a is true. Proposition x0a is true if proposition x0b is not true. Proposition q is true if proposition qcore is true and proposition wide is true. Proposition x2a is true if proposition x2b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n4-i3-wfs::wfs", "rec_id": "hv2-combo-disj_n4-i3", "axis": "combo", "difficulty": "disj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition x1b is true if proposition x1a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition qcore is true if proposition x2a is true. Proposition qcore is true if proposition x0a is true. Proposition x3a is true if proposition x3b is not true. Proposition s1 is true. Proposition x3b is true if proposition x3a is not true. Proposition qcore is true if proposition x3a is true. Proposition x2b is true if proposition x2a is not true. Proposition x1a is true if proposition x1b is not true. Proposition x0b is true if proposition x0a is not true. Proposition s0 is true. Proposition qcore is true if proposition x1a is true. Proposition x0a is true if proposition x0b is not true. Proposition q is true if proposition qcore is true and proposition wide is true. Proposition x2a is true if proposition x2b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n4-i4-none::none", "rec_id": "hv2-combo-disj_n4-i4", "axis": "combo", "difficulty": "disj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x1b is true if proposition x1a is not true. Proposition t2 is true if proposition t3 is true. Proposition qcore is true if proposition x0a is true. Proposition x0b is true if proposition x0a is not true. Proposition t1 is true if proposition t2 is true. Proposition qcore is true if proposition x3a is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x3a is true if proposition x3b is not true. Proposition qcore is true if proposition x1a is true. Proposition s0 is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition x2a is true. Proposition x3b is true if proposition x3a is not true. Proposition x2b is true if proposition x2a is not true. Proposition t3 is true if proposition qcore is true. Proposition s1 is true. Proposition x0a is true if proposition x0b is not true. Proposition x1a is true if proposition x1b is not true. Proposition x2a is true if proposition x2b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n4-i4-closed_world::closed_world", "rec_id": "hv2-combo-disj_n4-i4", "axis": "combo", "difficulty": "disj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x1b is true if proposition x1a is not true. Proposition t2 is true if proposition t3 is true. Proposition qcore is true if proposition x0a is true. Proposition x0b is true if proposition x0a is not true. Proposition t1 is true if proposition t2 is true. Proposition qcore is true if proposition x3a is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x3a is true if proposition x3b is not true. Proposition qcore is true if proposition x1a is true. Proposition s0 is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition x2a is true. Proposition x3b is true if proposition x3a is not true. Proposition x2b is true if proposition x2a is not true. Proposition t3 is true if proposition qcore is true. Proposition s1 is true. Proposition x0a is true if proposition x0b is not true. Proposition x1a is true if proposition x1b is not true. Proposition x2a is true if proposition x2b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n4-i4-cred::cred", "rec_id": "hv2-combo-disj_n4-i4", "axis": "combo", "difficulty": "disj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x1b is true if proposition x1a is not true. Proposition t2 is true if proposition t3 is true. Proposition qcore is true if proposition x0a is true. Proposition x0b is true if proposition x0a is not true. Proposition t1 is true if proposition t2 is true. Proposition qcore is true if proposition x3a is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x3a is true if proposition x3b is not true. Proposition qcore is true if proposition x1a is true. Proposition s0 is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition x2a is true. Proposition x3b is true if proposition x3a is not true. Proposition x2b is true if proposition x2a is not true. Proposition t3 is true if proposition qcore is true. Proposition s1 is true. Proposition x0a is true if proposition x0b is not true. Proposition x1a is true if proposition x1b is not true. Proposition x2a is true if proposition x2b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n4-i4-skept::skept", "rec_id": "hv2-combo-disj_n4-i4", "axis": "combo", "difficulty": "disj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x1b is true if proposition x1a is not true. Proposition t2 is true if proposition t3 is true. Proposition qcore is true if proposition x0a is true. Proposition x0b is true if proposition x0a is not true. Proposition t1 is true if proposition t2 is true. Proposition qcore is true if proposition x3a is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x3a is true if proposition x3b is not true. Proposition qcore is true if proposition x1a is true. Proposition s0 is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition x2a is true. Proposition x3b is true if proposition x3a is not true. Proposition x2b is true if proposition x2a is not true. Proposition t3 is true if proposition qcore is true. Proposition s1 is true. Proposition x0a is true if proposition x0b is not true. Proposition x1a is true if proposition x1b is not true. Proposition x2a is true if proposition x2b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n4-i4-wfs::wfs", "rec_id": "hv2-combo-disj_n4-i4", "axis": "combo", "difficulty": "disj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x1b is true if proposition x1a is not true. Proposition t2 is true if proposition t3 is true. Proposition qcore is true if proposition x0a is true. Proposition x0b is true if proposition x0a is not true. Proposition t1 is true if proposition t2 is true. Proposition qcore is true if proposition x3a is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x3a is true if proposition x3b is not true. Proposition qcore is true if proposition x1a is true. Proposition s0 is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition x2a is true. Proposition x3b is true if proposition x3a is not true. Proposition x2b is true if proposition x2a is not true. Proposition t3 is true if proposition qcore is true. Proposition s1 is true. Proposition x0a is true if proposition x0b is not true. Proposition x1a is true if proposition x1b is not true. Proposition x2a is true if proposition x2b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n4-i5-none::none", "rec_id": "hv2-combo-disj_n4-i5", "axis": "combo", "difficulty": "disj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition x2b is true if proposition x2a is not true. Proposition x1b is true if proposition x1a is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition x2a is true if proposition x2b is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition x2a is true. Proposition qcore is true if proposition x3a is true. Proposition x0a is true if proposition x0b is not true. Proposition qcore is true if proposition x0a is true. Proposition qcore is true if proposition x1a is true. Proposition x3b is true if proposition x3a is not true. Proposition etrue is true. Proposition t1 is true if proposition qcore is true. Proposition x1a is true if proposition x1b is not true. Proposition x3a is true if proposition x3b is not true. Proposition s1 is true. Proposition x0b is true if proposition x0a is not true. Proposition s0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n4-i5-closed_world::closed_world", "rec_id": "hv2-combo-disj_n4-i5", "axis": "combo", "difficulty": "disj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition x2b is true if proposition x2a is not true. Proposition x1b is true if proposition x1a is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition x2a is true if proposition x2b is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition x2a is true. Proposition qcore is true if proposition x3a is true. Proposition x0a is true if proposition x0b is not true. Proposition qcore is true if proposition x0a is true. Proposition qcore is true if proposition x1a is true. Proposition x3b is true if proposition x3a is not true. Proposition etrue is true. Proposition t1 is true if proposition qcore is true. Proposition x1a is true if proposition x1b is not true. Proposition x3a is true if proposition x3b is not true. Proposition s1 is true. Proposition x0b is true if proposition x0a is not true. Proposition s0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n4-i5-cred::cred", "rec_id": "hv2-combo-disj_n4-i5", "axis": "combo", "difficulty": "disj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition x2b is true if proposition x2a is not true. Proposition x1b is true if proposition x1a is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition x2a is true if proposition x2b is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition x2a is true. Proposition qcore is true if proposition x3a is true. Proposition x0a is true if proposition x0b is not true. Proposition qcore is true if proposition x0a is true. Proposition qcore is true if proposition x1a is true. Proposition x3b is true if proposition x3a is not true. Proposition etrue is true. Proposition t1 is true if proposition qcore is true. Proposition x1a is true if proposition x1b is not true. Proposition x3a is true if proposition x3b is not true. Proposition s1 is true. Proposition x0b is true if proposition x0a is not true. Proposition s0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n4-i5-skept::skept", "rec_id": "hv2-combo-disj_n4-i5", "axis": "combo", "difficulty": "disj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition x2b is true if proposition x2a is not true. Proposition x1b is true if proposition x1a is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition x2a is true if proposition x2b is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition x2a is true. Proposition qcore is true if proposition x3a is true. Proposition x0a is true if proposition x0b is not true. Proposition qcore is true if proposition x0a is true. Proposition qcore is true if proposition x1a is true. Proposition x3b is true if proposition x3a is not true. Proposition etrue is true. Proposition t1 is true if proposition qcore is true. Proposition x1a is true if proposition x1b is not true. Proposition x3a is true if proposition x3b is not true. Proposition s1 is true. Proposition x0b is true if proposition x0a is not true. Proposition s0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n4-i5-wfs::wfs", "rec_id": "hv2-combo-disj_n4-i5", "axis": "combo", "difficulty": "disj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition x2b is true if proposition x2a is not true. Proposition x1b is true if proposition x1a is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition x2a is true if proposition x2b is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition x2a is true. Proposition qcore is true if proposition x3a is true. Proposition x0a is true if proposition x0b is not true. Proposition qcore is true if proposition x0a is true. Proposition qcore is true if proposition x1a is true. Proposition x3b is true if proposition x3a is not true. Proposition etrue is true. Proposition t1 is true if proposition qcore is true. Proposition x1a is true if proposition x1b is not true. Proposition x3a is true if proposition x3b is not true. Proposition s1 is true. Proposition x0b is true if proposition x0a is not true. Proposition s0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n4-i6-none::none", "rec_id": "hv2-combo-disj_n4-i6", "axis": "combo", "difficulty": "disj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition x0a is true if proposition x0b is not true. Proposition x2b is true if proposition x2a is not true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x1a is true. Proposition qcore is true if proposition x0a is true. Proposition qcore is true if proposition x3a is true. Proposition x2a is true if proposition x2b is not true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition qcore is true. Proposition q is true if proposition t0 is true. Proposition t0 is true if proposition t1 is true. Proposition t1 is true if proposition t2 is true. Proposition x3a is true if proposition x3b is not true. Proposition x3b is true if proposition x3a is not true. Proposition qcore is true if proposition x2a is true. Proposition x0b is true if proposition x0a is not true. Proposition x1b is true if proposition x1a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n4-i6-closed_world::closed_world", "rec_id": "hv2-combo-disj_n4-i6", "axis": "combo", "difficulty": "disj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition x0a is true if proposition x0b is not true. Proposition x2b is true if proposition x2a is not true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x1a is true. Proposition qcore is true if proposition x0a is true. Proposition qcore is true if proposition x3a is true. Proposition x2a is true if proposition x2b is not true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition qcore is true. Proposition q is true if proposition t0 is true. Proposition t0 is true if proposition t1 is true. Proposition t1 is true if proposition t2 is true. Proposition x3a is true if proposition x3b is not true. Proposition x3b is true if proposition x3a is not true. Proposition qcore is true if proposition x2a is true. Proposition x0b is true if proposition x0a is not true. Proposition x1b is true if proposition x1a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n4-i6-cred::cred", "rec_id": "hv2-combo-disj_n4-i6", "axis": "combo", "difficulty": "disj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition x0a is true if proposition x0b is not true. Proposition x2b is true if proposition x2a is not true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x1a is true. Proposition qcore is true if proposition x0a is true. Proposition qcore is true if proposition x3a is true. Proposition x2a is true if proposition x2b is not true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition qcore is true. Proposition q is true if proposition t0 is true. Proposition t0 is true if proposition t1 is true. Proposition t1 is true if proposition t2 is true. Proposition x3a is true if proposition x3b is not true. Proposition x3b is true if proposition x3a is not true. Proposition qcore is true if proposition x2a is true. Proposition x0b is true if proposition x0a is not true. Proposition x1b is true if proposition x1a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n4-i6-skept::skept", "rec_id": "hv2-combo-disj_n4-i6", "axis": "combo", "difficulty": "disj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition x0a is true if proposition x0b is not true. Proposition x2b is true if proposition x2a is not true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x1a is true. Proposition qcore is true if proposition x0a is true. Proposition qcore is true if proposition x3a is true. Proposition x2a is true if proposition x2b is not true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition qcore is true. Proposition q is true if proposition t0 is true. Proposition t0 is true if proposition t1 is true. Proposition t1 is true if proposition t2 is true. Proposition x3a is true if proposition x3b is not true. Proposition x3b is true if proposition x3a is not true. Proposition qcore is true if proposition x2a is true. Proposition x0b is true if proposition x0a is not true. Proposition x1b is true if proposition x1a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n4-i6-wfs::wfs", "rec_id": "hv2-combo-disj_n4-i6", "axis": "combo", "difficulty": "disj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition x0a is true if proposition x0b is not true. Proposition x2b is true if proposition x2a is not true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x1a is true. Proposition qcore is true if proposition x0a is true. Proposition qcore is true if proposition x3a is true. Proposition x2a is true if proposition x2b is not true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition qcore is true. Proposition q is true if proposition t0 is true. Proposition t0 is true if proposition t1 is true. Proposition t1 is true if proposition t2 is true. Proposition x3a is true if proposition x3b is not true. Proposition x3b is true if proposition x3a is not true. Proposition qcore is true if proposition x2a is true. Proposition x0b is true if proposition x0a is not true. Proposition x1b is true if proposition x1a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n4-i7-none::none", "rec_id": "hv2-combo-disj_n4-i7", "axis": "combo", "difficulty": "disj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition x1b is true if proposition x1a is not true. Proposition s0 is true. Proposition etrue is true. Proposition s2 is true. Proposition x0a is true if proposition x0b is not true. Proposition qcore is true if proposition x1a is true. Proposition qcore is true if proposition x2a is true. Proposition x3b is true if proposition x3a is not true. Proposition x3a is true if proposition x3b is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t0 is true if proposition t1 is true. Proposition s3 is true. Proposition t1 is true if proposition t2 is true. Proposition t3 is true if proposition qcore is true. Proposition t2 is true if proposition t3 is true. Proposition x2a is true if proposition x2b is not true. Proposition x0b is true if proposition x0a is not true. Proposition qcore is true if proposition x3a is true. Proposition x2b is true if proposition x2a is not true. Proposition qcore is true if proposition x0a is true. Proposition s1 is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition x1a is true if proposition x1b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n4-i7-closed_world::closed_world", "rec_id": "hv2-combo-disj_n4-i7", "axis": "combo", "difficulty": "disj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition x1b is true if proposition x1a is not true. Proposition s0 is true. Proposition etrue is true. Proposition s2 is true. Proposition x0a is true if proposition x0b is not true. Proposition qcore is true if proposition x1a is true. Proposition qcore is true if proposition x2a is true. Proposition x3b is true if proposition x3a is not true. Proposition x3a is true if proposition x3b is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t0 is true if proposition t1 is true. Proposition s3 is true. Proposition t1 is true if proposition t2 is true. Proposition t3 is true if proposition qcore is true. Proposition t2 is true if proposition t3 is true. Proposition x2a is true if proposition x2b is not true. Proposition x0b is true if proposition x0a is not true. Proposition qcore is true if proposition x3a is true. Proposition x2b is true if proposition x2a is not true. Proposition qcore is true if proposition x0a is true. Proposition s1 is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition x1a is true if proposition x1b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n4-i7-cred::cred", "rec_id": "hv2-combo-disj_n4-i7", "axis": "combo", "difficulty": "disj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition x1b is true if proposition x1a is not true. Proposition s0 is true. Proposition etrue is true. Proposition s2 is true. Proposition x0a is true if proposition x0b is not true. Proposition qcore is true if proposition x1a is true. Proposition qcore is true if proposition x2a is true. Proposition x3b is true if proposition x3a is not true. Proposition x3a is true if proposition x3b is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t0 is true if proposition t1 is true. Proposition s3 is true. Proposition t1 is true if proposition t2 is true. Proposition t3 is true if proposition qcore is true. Proposition t2 is true if proposition t3 is true. Proposition x2a is true if proposition x2b is not true. Proposition x0b is true if proposition x0a is not true. Proposition qcore is true if proposition x3a is true. Proposition x2b is true if proposition x2a is not true. Proposition qcore is true if proposition x0a is true. Proposition s1 is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition x1a is true if proposition x1b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n4-i7-skept::skept", "rec_id": "hv2-combo-disj_n4-i7", "axis": "combo", "difficulty": "disj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition x1b is true if proposition x1a is not true. Proposition s0 is true. Proposition etrue is true. Proposition s2 is true. Proposition x0a is true if proposition x0b is not true. Proposition qcore is true if proposition x1a is true. Proposition qcore is true if proposition x2a is true. Proposition x3b is true if proposition x3a is not true. Proposition x3a is true if proposition x3b is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t0 is true if proposition t1 is true. Proposition s3 is true. Proposition t1 is true if proposition t2 is true. Proposition t3 is true if proposition qcore is true. Proposition t2 is true if proposition t3 is true. Proposition x2a is true if proposition x2b is not true. Proposition x0b is true if proposition x0a is not true. Proposition qcore is true if proposition x3a is true. Proposition x2b is true if proposition x2a is not true. Proposition qcore is true if proposition x0a is true. Proposition s1 is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition x1a is true if proposition x1b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n4-i7-wfs::wfs", "rec_id": "hv2-combo-disj_n4-i7", "axis": "combo", "difficulty": "disj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition x1b is true if proposition x1a is not true. Proposition s0 is true. Proposition etrue is true. Proposition s2 is true. Proposition x0a is true if proposition x0b is not true. Proposition qcore is true if proposition x1a is true. Proposition qcore is true if proposition x2a is true. Proposition x3b is true if proposition x3a is not true. Proposition x3a is true if proposition x3b is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t0 is true if proposition t1 is true. Proposition s3 is true. Proposition t1 is true if proposition t2 is true. Proposition t3 is true if proposition qcore is true. Proposition t2 is true if proposition t3 is true. Proposition x2a is true if proposition x2b is not true. Proposition x0b is true if proposition x0a is not true. Proposition qcore is true if proposition x3a is true. Proposition x2b is true if proposition x2a is not true. Proposition qcore is true if proposition x0a is true. Proposition s1 is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition x1a is true if proposition x1b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n5-i0-none::none", "rec_id": "hv2-combo-disj_n5-i0", "axis": "combo", "difficulty": "disj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition qcore is true if proposition x3a is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t5 is true if proposition qcore is true. Proposition x2a is true if proposition x2b is not true. Proposition s0 is true. Proposition qcore is true if proposition x1a is true. Proposition x2b is true if proposition x2a is not true. Proposition t4 is true if proposition t5 is true. Proposition x0a is true if proposition x0b is not true. Proposition x4b is true if proposition x4a is not true. Proposition x3a is true if proposition x3b is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition etrue is true. Proposition x3b is true if proposition x3a is not true. Proposition t1 is true if proposition t2 is true. Proposition x1a is true if proposition x1b is not true. Proposition x4a is true if proposition x4b is not true. Proposition t0 is true if proposition t1 is true. Proposition t2 is true if proposition t3 is true. Proposition qcore is true if proposition x0a is true. Proposition t3 is true if proposition t4 is true. Proposition x0b is true if proposition x0a is not true. Proposition qcore is true if proposition x2a is true. Proposition s1 is true. Proposition x1b is true if proposition x1a is not true. Proposition qcore is true if proposition x4a is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n5-i0-closed_world::closed_world", "rec_id": "hv2-combo-disj_n5-i0", "axis": "combo", "difficulty": "disj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition qcore is true if proposition x3a is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t5 is true if proposition qcore is true. Proposition x2a is true if proposition x2b is not true. Proposition s0 is true. Proposition qcore is true if proposition x1a is true. Proposition x2b is true if proposition x2a is not true. Proposition t4 is true if proposition t5 is true. Proposition x0a is true if proposition x0b is not true. Proposition x4b is true if proposition x4a is not true. Proposition x3a is true if proposition x3b is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition etrue is true. Proposition x3b is true if proposition x3a is not true. Proposition t1 is true if proposition t2 is true. Proposition x1a is true if proposition x1b is not true. Proposition x4a is true if proposition x4b is not true. Proposition t0 is true if proposition t1 is true. Proposition t2 is true if proposition t3 is true. Proposition qcore is true if proposition x0a is true. Proposition t3 is true if proposition t4 is true. Proposition x0b is true if proposition x0a is not true. Proposition qcore is true if proposition x2a is true. Proposition s1 is true. Proposition x1b is true if proposition x1a is not true. Proposition qcore is true if proposition x4a is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n5-i0-cred::cred", "rec_id": "hv2-combo-disj_n5-i0", "axis": "combo", "difficulty": "disj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition qcore is true if proposition x3a is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t5 is true if proposition qcore is true. Proposition x2a is true if proposition x2b is not true. Proposition s0 is true. Proposition qcore is true if proposition x1a is true. Proposition x2b is true if proposition x2a is not true. Proposition t4 is true if proposition t5 is true. Proposition x0a is true if proposition x0b is not true. Proposition x4b is true if proposition x4a is not true. Proposition x3a is true if proposition x3b is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition etrue is true. Proposition x3b is true if proposition x3a is not true. Proposition t1 is true if proposition t2 is true. Proposition x1a is true if proposition x1b is not true. Proposition x4a is true if proposition x4b is not true. Proposition t0 is true if proposition t1 is true. Proposition t2 is true if proposition t3 is true. Proposition qcore is true if proposition x0a is true. Proposition t3 is true if proposition t4 is true. Proposition x0b is true if proposition x0a is not true. Proposition qcore is true if proposition x2a is true. Proposition s1 is true. Proposition x1b is true if proposition x1a is not true. Proposition qcore is true if proposition x4a is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n5-i0-skept::skept", "rec_id": "hv2-combo-disj_n5-i0", "axis": "combo", "difficulty": "disj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition qcore is true if proposition x3a is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t5 is true if proposition qcore is true. Proposition x2a is true if proposition x2b is not true. Proposition s0 is true. Proposition qcore is true if proposition x1a is true. Proposition x2b is true if proposition x2a is not true. Proposition t4 is true if proposition t5 is true. Proposition x0a is true if proposition x0b is not true. Proposition x4b is true if proposition x4a is not true. Proposition x3a is true if proposition x3b is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition etrue is true. Proposition x3b is true if proposition x3a is not true. Proposition t1 is true if proposition t2 is true. Proposition x1a is true if proposition x1b is not true. Proposition x4a is true if proposition x4b is not true. Proposition t0 is true if proposition t1 is true. Proposition t2 is true if proposition t3 is true. Proposition qcore is true if proposition x0a is true. Proposition t3 is true if proposition t4 is true. Proposition x0b is true if proposition x0a is not true. Proposition qcore is true if proposition x2a is true. Proposition s1 is true. Proposition x1b is true if proposition x1a is not true. Proposition qcore is true if proposition x4a is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n5-i0-wfs::wfs", "rec_id": "hv2-combo-disj_n5-i0", "axis": "combo", "difficulty": "disj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition qcore is true if proposition x3a is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t5 is true if proposition qcore is true. Proposition x2a is true if proposition x2b is not true. Proposition s0 is true. Proposition qcore is true if proposition x1a is true. Proposition x2b is true if proposition x2a is not true. Proposition t4 is true if proposition t5 is true. Proposition x0a is true if proposition x0b is not true. Proposition x4b is true if proposition x4a is not true. Proposition x3a is true if proposition x3b is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition etrue is true. Proposition x3b is true if proposition x3a is not true. Proposition t1 is true if proposition t2 is true. Proposition x1a is true if proposition x1b is not true. Proposition x4a is true if proposition x4b is not true. Proposition t0 is true if proposition t1 is true. Proposition t2 is true if proposition t3 is true. Proposition qcore is true if proposition x0a is true. Proposition t3 is true if proposition t4 is true. Proposition x0b is true if proposition x0a is not true. Proposition qcore is true if proposition x2a is true. Proposition s1 is true. Proposition x1b is true if proposition x1a is not true. Proposition qcore is true if proposition x4a is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n5-i1-none::none", "rec_id": "hv2-combo-disj_n5-i1", "axis": "combo", "difficulty": "disj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition q is true if proposition qcore is true and proposition wide is true. Proposition x4a is true if proposition x4b is not true. Proposition x1a is true if proposition x1b is not true. Proposition s1 is true. Proposition s0 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition x2b is true if proposition x2a is not true. Proposition x4b is true if proposition x4a is not true. Proposition x0a is true if proposition x0b is not true. Proposition qcore is true if proposition x0a is true. Proposition s2 is true. Proposition x1b is true if proposition x1a is not true. Proposition qcore is true if proposition x3a is true. Proposition qcore is true if proposition x1a is true. Proposition x3b is true if proposition x3a is not true. Proposition qcore is true if proposition x4a is true. Proposition qcore is true if proposition x2a is true. Proposition x3a is true if proposition x3b is not true. Proposition x2a is true if proposition x2b is not true. Proposition x0b is true if proposition x0a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n5-i1-closed_world::closed_world", "rec_id": "hv2-combo-disj_n5-i1", "axis": "combo", "difficulty": "disj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition q is true if proposition qcore is true and proposition wide is true. Proposition x4a is true if proposition x4b is not true. Proposition x1a is true if proposition x1b is not true. Proposition s1 is true. Proposition s0 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition x2b is true if proposition x2a is not true. Proposition x4b is true if proposition x4a is not true. Proposition x0a is true if proposition x0b is not true. Proposition qcore is true if proposition x0a is true. Proposition s2 is true. Proposition x1b is true if proposition x1a is not true. Proposition qcore is true if proposition x3a is true. Proposition qcore is true if proposition x1a is true. Proposition x3b is true if proposition x3a is not true. Proposition qcore is true if proposition x4a is true. Proposition qcore is true if proposition x2a is true. Proposition x3a is true if proposition x3b is not true. Proposition x2a is true if proposition x2b is not true. Proposition x0b is true if proposition x0a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n5-i1-cred::cred", "rec_id": "hv2-combo-disj_n5-i1", "axis": "combo", "difficulty": "disj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition q is true if proposition qcore is true and proposition wide is true. Proposition x4a is true if proposition x4b is not true. Proposition x1a is true if proposition x1b is not true. Proposition s1 is true. Proposition s0 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition x2b is true if proposition x2a is not true. Proposition x4b is true if proposition x4a is not true. Proposition x0a is true if proposition x0b is not true. Proposition qcore is true if proposition x0a is true. Proposition s2 is true. Proposition x1b is true if proposition x1a is not true. Proposition qcore is true if proposition x3a is true. Proposition qcore is true if proposition x1a is true. Proposition x3b is true if proposition x3a is not true. Proposition qcore is true if proposition x4a is true. Proposition qcore is true if proposition x2a is true. Proposition x3a is true if proposition x3b is not true. Proposition x2a is true if proposition x2b is not true. Proposition x0b is true if proposition x0a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n5-i1-skept::skept", "rec_id": "hv2-combo-disj_n5-i1", "axis": "combo", "difficulty": "disj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition q is true if proposition qcore is true and proposition wide is true. Proposition x4a is true if proposition x4b is not true. Proposition x1a is true if proposition x1b is not true. Proposition s1 is true. Proposition s0 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition x2b is true if proposition x2a is not true. Proposition x4b is true if proposition x4a is not true. Proposition x0a is true if proposition x0b is not true. Proposition qcore is true if proposition x0a is true. Proposition s2 is true. Proposition x1b is true if proposition x1a is not true. Proposition qcore is true if proposition x3a is true. Proposition qcore is true if proposition x1a is true. Proposition x3b is true if proposition x3a is not true. Proposition qcore is true if proposition x4a is true. Proposition qcore is true if proposition x2a is true. Proposition x3a is true if proposition x3b is not true. Proposition x2a is true if proposition x2b is not true. Proposition x0b is true if proposition x0a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n5-i1-wfs::wfs", "rec_id": "hv2-combo-disj_n5-i1", "axis": "combo", "difficulty": "disj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition q is true if proposition qcore is true and proposition wide is true. Proposition x4a is true if proposition x4b is not true. Proposition x1a is true if proposition x1b is not true. Proposition s1 is true. Proposition s0 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition x2b is true if proposition x2a is not true. Proposition x4b is true if proposition x4a is not true. Proposition x0a is true if proposition x0b is not true. Proposition qcore is true if proposition x0a is true. Proposition s2 is true. Proposition x1b is true if proposition x1a is not true. Proposition qcore is true if proposition x3a is true. Proposition qcore is true if proposition x1a is true. Proposition x3b is true if proposition x3a is not true. Proposition qcore is true if proposition x4a is true. Proposition qcore is true if proposition x2a is true. Proposition x3a is true if proposition x3b is not true. Proposition x2a is true if proposition x2b is not true. Proposition x0b is true if proposition x0a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n5-i2-none::none", "rec_id": "hv2-combo-disj_n5-i2", "axis": "combo", "difficulty": "disj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition s0 is true. Proposition x0a is true if proposition x0b is not true. Proposition t5 is true if proposition qcore is true. Proposition qcore is true if proposition x0a is true. Proposition x1a is true if proposition x1b is not true. Proposition t1 is true if proposition t2 is true. Proposition x2b is true if proposition x2a is not true. Proposition x0b is true if proposition x0a is not true. Proposition x4a is true if proposition x4b is not true. Proposition t0 is true if proposition t1 is true. Proposition t4 is true if proposition t5 is true. Proposition x3b is true if proposition x3a is not true. Proposition x2a is true if proposition x2b is not true. Proposition qcore is true if proposition x1a is true. Proposition x1b is true if proposition x1a is not true. Proposition x3a is true if proposition x3b is not true. Proposition t3 is true if proposition t4 is true. Proposition qcore is true if proposition x3a is true. Proposition s1 is true. Proposition qcore is true if proposition x4a is true. Proposition t2 is true if proposition t3 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition qcore is true if proposition x2a is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x4b is true if proposition x4a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n5-i2-closed_world::closed_world", "rec_id": "hv2-combo-disj_n5-i2", "axis": "combo", "difficulty": "disj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition s0 is true. Proposition x0a is true if proposition x0b is not true. Proposition t5 is true if proposition qcore is true. Proposition qcore is true if proposition x0a is true. Proposition x1a is true if proposition x1b is not true. Proposition t1 is true if proposition t2 is true. Proposition x2b is true if proposition x2a is not true. Proposition x0b is true if proposition x0a is not true. Proposition x4a is true if proposition x4b is not true. Proposition t0 is true if proposition t1 is true. Proposition t4 is true if proposition t5 is true. Proposition x3b is true if proposition x3a is not true. Proposition x2a is true if proposition x2b is not true. Proposition qcore is true if proposition x1a is true. Proposition x1b is true if proposition x1a is not true. Proposition x3a is true if proposition x3b is not true. Proposition t3 is true if proposition t4 is true. Proposition qcore is true if proposition x3a is true. Proposition s1 is true. Proposition qcore is true if proposition x4a is true. Proposition t2 is true if proposition t3 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition qcore is true if proposition x2a is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x4b is true if proposition x4a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n5-i2-cred::cred", "rec_id": "hv2-combo-disj_n5-i2", "axis": "combo", "difficulty": "disj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition s0 is true. Proposition x0a is true if proposition x0b is not true. Proposition t5 is true if proposition qcore is true. Proposition qcore is true if proposition x0a is true. Proposition x1a is true if proposition x1b is not true. Proposition t1 is true if proposition t2 is true. Proposition x2b is true if proposition x2a is not true. Proposition x0b is true if proposition x0a is not true. Proposition x4a is true if proposition x4b is not true. Proposition t0 is true if proposition t1 is true. Proposition t4 is true if proposition t5 is true. Proposition x3b is true if proposition x3a is not true. Proposition x2a is true if proposition x2b is not true. Proposition qcore is true if proposition x1a is true. Proposition x1b is true if proposition x1a is not true. Proposition x3a is true if proposition x3b is not true. Proposition t3 is true if proposition t4 is true. Proposition qcore is true if proposition x3a is true. Proposition s1 is true. Proposition qcore is true if proposition x4a is true. Proposition t2 is true if proposition t3 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition qcore is true if proposition x2a is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x4b is true if proposition x4a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n5-i2-skept::skept", "rec_id": "hv2-combo-disj_n5-i2", "axis": "combo", "difficulty": "disj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition s0 is true. Proposition x0a is true if proposition x0b is not true. Proposition t5 is true if proposition qcore is true. Proposition qcore is true if proposition x0a is true. Proposition x1a is true if proposition x1b is not true. Proposition t1 is true if proposition t2 is true. Proposition x2b is true if proposition x2a is not true. Proposition x0b is true if proposition x0a is not true. Proposition x4a is true if proposition x4b is not true. Proposition t0 is true if proposition t1 is true. Proposition t4 is true if proposition t5 is true. Proposition x3b is true if proposition x3a is not true. Proposition x2a is true if proposition x2b is not true. Proposition qcore is true if proposition x1a is true. Proposition x1b is true if proposition x1a is not true. Proposition x3a is true if proposition x3b is not true. Proposition t3 is true if proposition t4 is true. Proposition qcore is true if proposition x3a is true. Proposition s1 is true. Proposition qcore is true if proposition x4a is true. Proposition t2 is true if proposition t3 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition qcore is true if proposition x2a is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x4b is true if proposition x4a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n5-i2-wfs::wfs", "rec_id": "hv2-combo-disj_n5-i2", "axis": "combo", "difficulty": "disj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition s0 is true. Proposition x0a is true if proposition x0b is not true. Proposition t5 is true if proposition qcore is true. Proposition qcore is true if proposition x0a is true. Proposition x1a is true if proposition x1b is not true. Proposition t1 is true if proposition t2 is true. Proposition x2b is true if proposition x2a is not true. Proposition x0b is true if proposition x0a is not true. Proposition x4a is true if proposition x4b is not true. Proposition t0 is true if proposition t1 is true. Proposition t4 is true if proposition t5 is true. Proposition x3b is true if proposition x3a is not true. Proposition x2a is true if proposition x2b is not true. Proposition qcore is true if proposition x1a is true. Proposition x1b is true if proposition x1a is not true. Proposition x3a is true if proposition x3b is not true. Proposition t3 is true if proposition t4 is true. Proposition qcore is true if proposition x3a is true. Proposition s1 is true. Proposition qcore is true if proposition x4a is true. Proposition t2 is true if proposition t3 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition qcore is true if proposition x2a is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x4b is true if proposition x4a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n5-i3-none::none", "rec_id": "hv2-combo-disj_n5-i3", "axis": "combo", "difficulty": "disj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition t0 is true if proposition t1 is true. Proposition t3 is true if proposition qcore is true. Proposition qcore is true if proposition x1a is true. Proposition qcore is true if proposition x3a is true. Proposition s2 is true. Proposition x1b is true if proposition x1a is not true. Proposition x4b is true if proposition x4a is not true. Proposition qcore is true if proposition x0a is true. Proposition qcore is true if proposition x2a is true. Proposition x3b is true if proposition x3a is not true. Proposition t1 is true if proposition t2 is true. Proposition x0a is true if proposition x0b is not true. Proposition s1 is true. Proposition qcore is true if proposition x4a is true. Proposition t2 is true if proposition t3 is true. Proposition x2b is true if proposition x2a is not true. Proposition x1a is true if proposition x1b is not true. Proposition x3a is true if proposition x3b is not true. Proposition x0b is true if proposition x0a is not true. Proposition x4a is true if proposition x4b is not true. Proposition s0 is true. Proposition x2a is true if proposition x2b is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition q is true if proposition t0 is true and proposition wide is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n5-i3-closed_world::closed_world", "rec_id": "hv2-combo-disj_n5-i3", "axis": "combo", "difficulty": "disj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition t0 is true if proposition t1 is true. Proposition t3 is true if proposition qcore is true. Proposition qcore is true if proposition x1a is true. Proposition qcore is true if proposition x3a is true. Proposition s2 is true. Proposition x1b is true if proposition x1a is not true. Proposition x4b is true if proposition x4a is not true. Proposition qcore is true if proposition x0a is true. Proposition qcore is true if proposition x2a is true. Proposition x3b is true if proposition x3a is not true. Proposition t1 is true if proposition t2 is true. Proposition x0a is true if proposition x0b is not true. Proposition s1 is true. Proposition qcore is true if proposition x4a is true. Proposition t2 is true if proposition t3 is true. Proposition x2b is true if proposition x2a is not true. Proposition x1a is true if proposition x1b is not true. Proposition x3a is true if proposition x3b is not true. Proposition x0b is true if proposition x0a is not true. Proposition x4a is true if proposition x4b is not true. Proposition s0 is true. Proposition x2a is true if proposition x2b is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition q is true if proposition t0 is true and proposition wide is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n5-i3-cred::cred", "rec_id": "hv2-combo-disj_n5-i3", "axis": "combo", "difficulty": "disj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition t0 is true if proposition t1 is true. Proposition t3 is true if proposition qcore is true. Proposition qcore is true if proposition x1a is true. Proposition qcore is true if proposition x3a is true. Proposition s2 is true. Proposition x1b is true if proposition x1a is not true. Proposition x4b is true if proposition x4a is not true. Proposition qcore is true if proposition x0a is true. Proposition qcore is true if proposition x2a is true. Proposition x3b is true if proposition x3a is not true. Proposition t1 is true if proposition t2 is true. Proposition x0a is true if proposition x0b is not true. Proposition s1 is true. Proposition qcore is true if proposition x4a is true. Proposition t2 is true if proposition t3 is true. Proposition x2b is true if proposition x2a is not true. Proposition x1a is true if proposition x1b is not true. Proposition x3a is true if proposition x3b is not true. Proposition x0b is true if proposition x0a is not true. Proposition x4a is true if proposition x4b is not true. Proposition s0 is true. Proposition x2a is true if proposition x2b is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition q is true if proposition t0 is true and proposition wide is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n5-i3-skept::skept", "rec_id": "hv2-combo-disj_n5-i3", "axis": "combo", "difficulty": "disj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition t0 is true if proposition t1 is true. Proposition t3 is true if proposition qcore is true. Proposition qcore is true if proposition x1a is true. Proposition qcore is true if proposition x3a is true. Proposition s2 is true. Proposition x1b is true if proposition x1a is not true. Proposition x4b is true if proposition x4a is not true. Proposition qcore is true if proposition x0a is true. Proposition qcore is true if proposition x2a is true. Proposition x3b is true if proposition x3a is not true. Proposition t1 is true if proposition t2 is true. Proposition x0a is true if proposition x0b is not true. Proposition s1 is true. Proposition qcore is true if proposition x4a is true. Proposition t2 is true if proposition t3 is true. Proposition x2b is true if proposition x2a is not true. Proposition x1a is true if proposition x1b is not true. Proposition x3a is true if proposition x3b is not true. Proposition x0b is true if proposition x0a is not true. Proposition x4a is true if proposition x4b is not true. Proposition s0 is true. Proposition x2a is true if proposition x2b is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition q is true if proposition t0 is true and proposition wide is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n5-i3-wfs::wfs", "rec_id": "hv2-combo-disj_n5-i3", "axis": "combo", "difficulty": "disj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition t0 is true if proposition t1 is true. Proposition t3 is true if proposition qcore is true. Proposition qcore is true if proposition x1a is true. Proposition qcore is true if proposition x3a is true. Proposition s2 is true. Proposition x1b is true if proposition x1a is not true. Proposition x4b is true if proposition x4a is not true. Proposition qcore is true if proposition x0a is true. Proposition qcore is true if proposition x2a is true. Proposition x3b is true if proposition x3a is not true. Proposition t1 is true if proposition t2 is true. Proposition x0a is true if proposition x0b is not true. Proposition s1 is true. Proposition qcore is true if proposition x4a is true. Proposition t2 is true if proposition t3 is true. Proposition x2b is true if proposition x2a is not true. Proposition x1a is true if proposition x1b is not true. Proposition x3a is true if proposition x3b is not true. Proposition x0b is true if proposition x0a is not true. Proposition x4a is true if proposition x4b is not true. Proposition s0 is true. Proposition x2a is true if proposition x2b is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition q is true if proposition t0 is true and proposition wide is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n5-i4-none::none", "rec_id": "hv2-combo-disj_n5-i4", "axis": "combo", "difficulty": "disj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition etrue is true. Proposition x2a is true if proposition x2b is not true. Proposition t0 is true if proposition t1 is true. Proposition x1b is true if proposition x1a is not true. Proposition x3a is true if proposition x3b is not true. Proposition x3b is true if proposition x3a is not true. Proposition qcore is true if proposition x2a is true. Proposition x0a is true if proposition x0b is not true. Proposition s1 is true. Proposition x4b is true if proposition x4a is not true. Proposition t1 is true if proposition qcore is true. Proposition qcore is true if proposition x3a is true. Proposition qcore is true if proposition x1a is true. Proposition x0b is true if proposition x0a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x1a is true if proposition x1b is not true. Proposition x2b is true if proposition x2a is not true. Proposition x4a is true if proposition x4b is not true. Proposition qcore is true if proposition x0a is true. Proposition s0 is true. Proposition qcore is true if proposition x4a is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n5-i4-closed_world::closed_world", "rec_id": "hv2-combo-disj_n5-i4", "axis": "combo", "difficulty": "disj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition etrue is true. Proposition x2a is true if proposition x2b is not true. Proposition t0 is true if proposition t1 is true. Proposition x1b is true if proposition x1a is not true. Proposition x3a is true if proposition x3b is not true. Proposition x3b is true if proposition x3a is not true. Proposition qcore is true if proposition x2a is true. Proposition x0a is true if proposition x0b is not true. Proposition s1 is true. Proposition x4b is true if proposition x4a is not true. Proposition t1 is true if proposition qcore is true. Proposition qcore is true if proposition x3a is true. Proposition qcore is true if proposition x1a is true. Proposition x0b is true if proposition x0a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x1a is true if proposition x1b is not true. Proposition x2b is true if proposition x2a is not true. Proposition x4a is true if proposition x4b is not true. Proposition qcore is true if proposition x0a is true. Proposition s0 is true. Proposition qcore is true if proposition x4a is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n5-i4-cred::cred", "rec_id": "hv2-combo-disj_n5-i4", "axis": "combo", "difficulty": "disj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition etrue is true. Proposition x2a is true if proposition x2b is not true. Proposition t0 is true if proposition t1 is true. Proposition x1b is true if proposition x1a is not true. Proposition x3a is true if proposition x3b is not true. Proposition x3b is true if proposition x3a is not true. Proposition qcore is true if proposition x2a is true. Proposition x0a is true if proposition x0b is not true. Proposition s1 is true. Proposition x4b is true if proposition x4a is not true. Proposition t1 is true if proposition qcore is true. Proposition qcore is true if proposition x3a is true. Proposition qcore is true if proposition x1a is true. Proposition x0b is true if proposition x0a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x1a is true if proposition x1b is not true. Proposition x2b is true if proposition x2a is not true. Proposition x4a is true if proposition x4b is not true. Proposition qcore is true if proposition x0a is true. Proposition s0 is true. Proposition qcore is true if proposition x4a is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n5-i4-skept::skept", "rec_id": "hv2-combo-disj_n5-i4", "axis": "combo", "difficulty": "disj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition etrue is true. Proposition x2a is true if proposition x2b is not true. Proposition t0 is true if proposition t1 is true. Proposition x1b is true if proposition x1a is not true. Proposition x3a is true if proposition x3b is not true. Proposition x3b is true if proposition x3a is not true. Proposition qcore is true if proposition x2a is true. Proposition x0a is true if proposition x0b is not true. Proposition s1 is true. Proposition x4b is true if proposition x4a is not true. Proposition t1 is true if proposition qcore is true. Proposition qcore is true if proposition x3a is true. Proposition qcore is true if proposition x1a is true. Proposition x0b is true if proposition x0a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x1a is true if proposition x1b is not true. Proposition x2b is true if proposition x2a is not true. Proposition x4a is true if proposition x4b is not true. Proposition qcore is true if proposition x0a is true. Proposition s0 is true. Proposition qcore is true if proposition x4a is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n5-i4-wfs::wfs", "rec_id": "hv2-combo-disj_n5-i4", "axis": "combo", "difficulty": "disj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition etrue is true. Proposition x2a is true if proposition x2b is not true. Proposition t0 is true if proposition t1 is true. Proposition x1b is true if proposition x1a is not true. Proposition x3a is true if proposition x3b is not true. Proposition x3b is true if proposition x3a is not true. Proposition qcore is true if proposition x2a is true. Proposition x0a is true if proposition x0b is not true. Proposition s1 is true. Proposition x4b is true if proposition x4a is not true. Proposition t1 is true if proposition qcore is true. Proposition qcore is true if proposition x3a is true. Proposition qcore is true if proposition x1a is true. Proposition x0b is true if proposition x0a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x1a is true if proposition x1b is not true. Proposition x2b is true if proposition x2a is not true. Proposition x4a is true if proposition x4b is not true. Proposition qcore is true if proposition x0a is true. Proposition s0 is true. Proposition qcore is true if proposition x4a is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n5-i5-none::none", "rec_id": "hv2-combo-disj_n5-i5", "axis": "combo", "difficulty": "disj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition qcore is true if proposition x0a is true. Proposition x1b is true if proposition x1a is not true. Proposition x2a is true if proposition x2b is not true. Proposition qcore is true if proposition x4a is true. Proposition t0 is true if proposition t1 is true. Proposition t1 is true if proposition qcore is true. Proposition x4b is true if proposition x4a is not true. Proposition x3a is true if proposition x3b is not true. Proposition qcore is true if proposition x2a is true. Proposition s0 is true. Proposition qcore is true if proposition x3a is true. Proposition x4a is true if proposition x4b is not true. Proposition x0b is true if proposition x0a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x0a is true if proposition x0b is not true. Proposition x3b is true if proposition x3a is not true. Proposition qcore is true if proposition x1a is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x2b is true if proposition x2a is not true. Proposition s1 is true. Proposition x1a is true if proposition x1b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n5-i5-closed_world::closed_world", "rec_id": "hv2-combo-disj_n5-i5", "axis": "combo", "difficulty": "disj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition qcore is true if proposition x0a is true. Proposition x1b is true if proposition x1a is not true. Proposition x2a is true if proposition x2b is not true. Proposition qcore is true if proposition x4a is true. Proposition t0 is true if proposition t1 is true. Proposition t1 is true if proposition qcore is true. Proposition x4b is true if proposition x4a is not true. Proposition x3a is true if proposition x3b is not true. Proposition qcore is true if proposition x2a is true. Proposition s0 is true. Proposition qcore is true if proposition x3a is true. Proposition x4a is true if proposition x4b is not true. Proposition x0b is true if proposition x0a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x0a is true if proposition x0b is not true. Proposition x3b is true if proposition x3a is not true. Proposition qcore is true if proposition x1a is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x2b is true if proposition x2a is not true. Proposition s1 is true. Proposition x1a is true if proposition x1b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n5-i5-cred::cred", "rec_id": "hv2-combo-disj_n5-i5", "axis": "combo", "difficulty": "disj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition qcore is true if proposition x0a is true. Proposition x1b is true if proposition x1a is not true. Proposition x2a is true if proposition x2b is not true. Proposition qcore is true if proposition x4a is true. Proposition t0 is true if proposition t1 is true. Proposition t1 is true if proposition qcore is true. Proposition x4b is true if proposition x4a is not true. Proposition x3a is true if proposition x3b is not true. Proposition qcore is true if proposition x2a is true. Proposition s0 is true. Proposition qcore is true if proposition x3a is true. Proposition x4a is true if proposition x4b is not true. Proposition x0b is true if proposition x0a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x0a is true if proposition x0b is not true. Proposition x3b is true if proposition x3a is not true. Proposition qcore is true if proposition x1a is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x2b is true if proposition x2a is not true. Proposition s1 is true. Proposition x1a is true if proposition x1b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n5-i5-skept::skept", "rec_id": "hv2-combo-disj_n5-i5", "axis": "combo", "difficulty": "disj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition qcore is true if proposition x0a is true. Proposition x1b is true if proposition x1a is not true. Proposition x2a is true if proposition x2b is not true. Proposition qcore is true if proposition x4a is true. Proposition t0 is true if proposition t1 is true. Proposition t1 is true if proposition qcore is true. Proposition x4b is true if proposition x4a is not true. Proposition x3a is true if proposition x3b is not true. Proposition qcore is true if proposition x2a is true. Proposition s0 is true. Proposition qcore is true if proposition x3a is true. Proposition x4a is true if proposition x4b is not true. Proposition x0b is true if proposition x0a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x0a is true if proposition x0b is not true. Proposition x3b is true if proposition x3a is not true. Proposition qcore is true if proposition x1a is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x2b is true if proposition x2a is not true. Proposition s1 is true. Proposition x1a is true if proposition x1b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n5-i5-wfs::wfs", "rec_id": "hv2-combo-disj_n5-i5", "axis": "combo", "difficulty": "disj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition qcore is true if proposition x0a is true. Proposition x1b is true if proposition x1a is not true. Proposition x2a is true if proposition x2b is not true. Proposition qcore is true if proposition x4a is true. Proposition t0 is true if proposition t1 is true. Proposition t1 is true if proposition qcore is true. Proposition x4b is true if proposition x4a is not true. Proposition x3a is true if proposition x3b is not true. Proposition qcore is true if proposition x2a is true. Proposition s0 is true. Proposition qcore is true if proposition x3a is true. Proposition x4a is true if proposition x4b is not true. Proposition x0b is true if proposition x0a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x0a is true if proposition x0b is not true. Proposition x3b is true if proposition x3a is not true. Proposition qcore is true if proposition x1a is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x2b is true if proposition x2a is not true. Proposition s1 is true. Proposition x1a is true if proposition x1b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n5-i6-none::none", "rec_id": "hv2-combo-disj_n5-i6", "axis": "combo", "difficulty": "disj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition qcore is true if proposition x3a is true. Proposition x0b is true if proposition x0a is not true. Proposition t0 is true if proposition t1 is true. Proposition x0a is true if proposition x0b is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition qcore is true if proposition x4a is true. Proposition x4a is true if proposition x4b is not true. Proposition x2b is true if proposition x2a is not true. Proposition etrue is true. Proposition t1 is true if proposition t2 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition s0 is true. Proposition x2a is true if proposition x2b is not true. Proposition t3 is true if proposition qcore is true. Proposition x1a is true if proposition x1b is not true. Proposition t2 is true if proposition t3 is true. Proposition qcore is true if proposition x1a is true. Proposition x3b is true if proposition x3a is not true. Proposition x4b is true if proposition x4a is not true. Proposition qcore is true if proposition x0a is true. Proposition x3a is true if proposition x3b is not true. Proposition s1 is true. Proposition x1b is true if proposition x1a is not true. Proposition qcore is true if proposition x2a is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n5-i6-closed_world::closed_world", "rec_id": "hv2-combo-disj_n5-i6", "axis": "combo", "difficulty": "disj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition qcore is true if proposition x3a is true. Proposition x0b is true if proposition x0a is not true. Proposition t0 is true if proposition t1 is true. Proposition x0a is true if proposition x0b is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition qcore is true if proposition x4a is true. Proposition x4a is true if proposition x4b is not true. Proposition x2b is true if proposition x2a is not true. Proposition etrue is true. Proposition t1 is true if proposition t2 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition s0 is true. Proposition x2a is true if proposition x2b is not true. Proposition t3 is true if proposition qcore is true. Proposition x1a is true if proposition x1b is not true. Proposition t2 is true if proposition t3 is true. Proposition qcore is true if proposition x1a is true. Proposition x3b is true if proposition x3a is not true. Proposition x4b is true if proposition x4a is not true. Proposition qcore is true if proposition x0a is true. Proposition x3a is true if proposition x3b is not true. Proposition s1 is true. Proposition x1b is true if proposition x1a is not true. Proposition qcore is true if proposition x2a is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n5-i6-cred::cred", "rec_id": "hv2-combo-disj_n5-i6", "axis": "combo", "difficulty": "disj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition qcore is true if proposition x3a is true. Proposition x0b is true if proposition x0a is not true. Proposition t0 is true if proposition t1 is true. Proposition x0a is true if proposition x0b is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition qcore is true if proposition x4a is true. Proposition x4a is true if proposition x4b is not true. Proposition x2b is true if proposition x2a is not true. Proposition etrue is true. Proposition t1 is true if proposition t2 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition s0 is true. Proposition x2a is true if proposition x2b is not true. Proposition t3 is true if proposition qcore is true. Proposition x1a is true if proposition x1b is not true. Proposition t2 is true if proposition t3 is true. Proposition qcore is true if proposition x1a is true. Proposition x3b is true if proposition x3a is not true. Proposition x4b is true if proposition x4a is not true. Proposition qcore is true if proposition x0a is true. Proposition x3a is true if proposition x3b is not true. Proposition s1 is true. Proposition x1b is true if proposition x1a is not true. Proposition qcore is true if proposition x2a is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n5-i6-skept::skept", "rec_id": "hv2-combo-disj_n5-i6", "axis": "combo", "difficulty": "disj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition qcore is true if proposition x3a is true. Proposition x0b is true if proposition x0a is not true. Proposition t0 is true if proposition t1 is true. Proposition x0a is true if proposition x0b is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition qcore is true if proposition x4a is true. Proposition x4a is true if proposition x4b is not true. Proposition x2b is true if proposition x2a is not true. Proposition etrue is true. Proposition t1 is true if proposition t2 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition s0 is true. Proposition x2a is true if proposition x2b is not true. Proposition t3 is true if proposition qcore is true. Proposition x1a is true if proposition x1b is not true. Proposition t2 is true if proposition t3 is true. Proposition qcore is true if proposition x1a is true. Proposition x3b is true if proposition x3a is not true. Proposition x4b is true if proposition x4a is not true. Proposition qcore is true if proposition x0a is true. Proposition x3a is true if proposition x3b is not true. Proposition s1 is true. Proposition x1b is true if proposition x1a is not true. Proposition qcore is true if proposition x2a is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n5-i6-wfs::wfs", "rec_id": "hv2-combo-disj_n5-i6", "axis": "combo", "difficulty": "disj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition qcore is true if proposition x3a is true. Proposition x0b is true if proposition x0a is not true. Proposition t0 is true if proposition t1 is true. Proposition x0a is true if proposition x0b is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition qcore is true if proposition x4a is true. Proposition x4a is true if proposition x4b is not true. Proposition x2b is true if proposition x2a is not true. Proposition etrue is true. Proposition t1 is true if proposition t2 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition s0 is true. Proposition x2a is true if proposition x2b is not true. Proposition t3 is true if proposition qcore is true. Proposition x1a is true if proposition x1b is not true. Proposition t2 is true if proposition t3 is true. Proposition qcore is true if proposition x1a is true. Proposition x3b is true if proposition x3a is not true. Proposition x4b is true if proposition x4a is not true. Proposition qcore is true if proposition x0a is true. Proposition x3a is true if proposition x3b is not true. Proposition s1 is true. Proposition x1b is true if proposition x1a is not true. Proposition qcore is true if proposition x2a is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n5-i7-none::none", "rec_id": "hv2-combo-disj_n5-i7", "axis": "combo", "difficulty": "disj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition x2a is true if proposition x2b is not true. Proposition t3 is true if proposition qcore is true. Proposition t1 is true if proposition t2 is true. Proposition t2 is true if proposition t3 is true. Proposition x1a is true if proposition x1b is not true. Proposition x0b is true if proposition x0a is not true. Proposition x1b is true if proposition x1a is not true. Proposition x3a is true if proposition x3b is not true. Proposition x4b is true if proposition x4a is not true. Proposition qcore is true if proposition x3a is true. Proposition q is true if proposition t0 is true. Proposition qcore is true if proposition x4a is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition x1a is true. Proposition x4a is true if proposition x4b is not true. Proposition x2b is true if proposition x2a is not true. Proposition qcore is true if proposition x2a is true. Proposition x0a is true if proposition x0b is not true. Proposition qcore is true if proposition x0a is true. Proposition x3b is true if proposition x3a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n5-i7-closed_world::closed_world", "rec_id": "hv2-combo-disj_n5-i7", "axis": "combo", "difficulty": "disj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition x2a is true if proposition x2b is not true. Proposition t3 is true if proposition qcore is true. Proposition t1 is true if proposition t2 is true. Proposition t2 is true if proposition t3 is true. Proposition x1a is true if proposition x1b is not true. Proposition x0b is true if proposition x0a is not true. Proposition x1b is true if proposition x1a is not true. Proposition x3a is true if proposition x3b is not true. Proposition x4b is true if proposition x4a is not true. Proposition qcore is true if proposition x3a is true. Proposition q is true if proposition t0 is true. Proposition qcore is true if proposition x4a is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition x1a is true. Proposition x4a is true if proposition x4b is not true. Proposition x2b is true if proposition x2a is not true. Proposition qcore is true if proposition x2a is true. Proposition x0a is true if proposition x0b is not true. Proposition qcore is true if proposition x0a is true. Proposition x3b is true if proposition x3a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n5-i7-cred::cred", "rec_id": "hv2-combo-disj_n5-i7", "axis": "combo", "difficulty": "disj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition x2a is true if proposition x2b is not true. Proposition t3 is true if proposition qcore is true. Proposition t1 is true if proposition t2 is true. Proposition t2 is true if proposition t3 is true. Proposition x1a is true if proposition x1b is not true. Proposition x0b is true if proposition x0a is not true. Proposition x1b is true if proposition x1a is not true. Proposition x3a is true if proposition x3b is not true. Proposition x4b is true if proposition x4a is not true. Proposition qcore is true if proposition x3a is true. Proposition q is true if proposition t0 is true. Proposition qcore is true if proposition x4a is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition x1a is true. Proposition x4a is true if proposition x4b is not true. Proposition x2b is true if proposition x2a is not true. Proposition qcore is true if proposition x2a is true. Proposition x0a is true if proposition x0b is not true. Proposition qcore is true if proposition x0a is true. Proposition x3b is true if proposition x3a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n5-i7-skept::skept", "rec_id": "hv2-combo-disj_n5-i7", "axis": "combo", "difficulty": "disj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition x2a is true if proposition x2b is not true. Proposition t3 is true if proposition qcore is true. Proposition t1 is true if proposition t2 is true. Proposition t2 is true if proposition t3 is true. Proposition x1a is true if proposition x1b is not true. Proposition x0b is true if proposition x0a is not true. Proposition x1b is true if proposition x1a is not true. Proposition x3a is true if proposition x3b is not true. Proposition x4b is true if proposition x4a is not true. Proposition qcore is true if proposition x3a is true. Proposition q is true if proposition t0 is true. Proposition qcore is true if proposition x4a is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition x1a is true. Proposition x4a is true if proposition x4b is not true. Proposition x2b is true if proposition x2a is not true. Proposition qcore is true if proposition x2a is true. Proposition x0a is true if proposition x0b is not true. Proposition qcore is true if proposition x0a is true. Proposition x3b is true if proposition x3a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n5-i7-wfs::wfs", "rec_id": "hv2-combo-disj_n5-i7", "axis": "combo", "difficulty": "disj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition x2a is true if proposition x2b is not true. Proposition t3 is true if proposition qcore is true. Proposition t1 is true if proposition t2 is true. Proposition t2 is true if proposition t3 is true. Proposition x1a is true if proposition x1b is not true. Proposition x0b is true if proposition x0a is not true. Proposition x1b is true if proposition x1a is not true. Proposition x3a is true if proposition x3b is not true. Proposition x4b is true if proposition x4a is not true. Proposition qcore is true if proposition x3a is true. Proposition q is true if proposition t0 is true. Proposition qcore is true if proposition x4a is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition x1a is true. Proposition x4a is true if proposition x4b is not true. Proposition x2b is true if proposition x2a is not true. Proposition qcore is true if proposition x2a is true. Proposition x0a is true if proposition x0b is not true. Proposition qcore is true if proposition x0a is true. Proposition x3b is true if proposition x3a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n6-i0-none::none", "rec_id": "hv2-combo-disj_n6-i0", "axis": "combo", "difficulty": "disj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition qcore is true if proposition x2a is true. Proposition x3b is true if proposition x3a is not true. Proposition etrue is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition x1b is true if proposition x1a is not true. Proposition x4b is true if proposition x4a is not true. Proposition x1a is true if proposition x1b is not true. Proposition t1 is true if proposition qcore is true. Proposition qcore is true if proposition x3a is true. Proposition s0 is true. Proposition x0a is true if proposition x0b is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x2b is true if proposition x2a is not true. Proposition s1 is true. Proposition x3a is true if proposition x3b is not true. Proposition qcore is true if proposition x4a is true. Proposition qcore is true if proposition x1a is true. Proposition x4a is true if proposition x4b is not true. Proposition qcore is true if proposition x5a is true. Proposition x5b is true if proposition x5a is not true. Proposition qcore is true if proposition x0a is true. Proposition x0b is true if proposition x0a is not true. Proposition x2a is true if proposition x2b is not true. Proposition t0 is true if proposition t1 is true. Proposition x5a is true if proposition x5b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n6-i0-closed_world::closed_world", "rec_id": "hv2-combo-disj_n6-i0", "axis": "combo", "difficulty": "disj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition qcore is true if proposition x2a is true. Proposition x3b is true if proposition x3a is not true. Proposition etrue is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition x1b is true if proposition x1a is not true. Proposition x4b is true if proposition x4a is not true. Proposition x1a is true if proposition x1b is not true. Proposition t1 is true if proposition qcore is true. Proposition qcore is true if proposition x3a is true. Proposition s0 is true. Proposition x0a is true if proposition x0b is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x2b is true if proposition x2a is not true. Proposition s1 is true. Proposition x3a is true if proposition x3b is not true. Proposition qcore is true if proposition x4a is true. Proposition qcore is true if proposition x1a is true. Proposition x4a is true if proposition x4b is not true. Proposition qcore is true if proposition x5a is true. Proposition x5b is true if proposition x5a is not true. Proposition qcore is true if proposition x0a is true. Proposition x0b is true if proposition x0a is not true. Proposition x2a is true if proposition x2b is not true. Proposition t0 is true if proposition t1 is true. Proposition x5a is true if proposition x5b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n6-i0-cred::cred", "rec_id": "hv2-combo-disj_n6-i0", "axis": "combo", "difficulty": "disj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition qcore is true if proposition x2a is true. Proposition x3b is true if proposition x3a is not true. Proposition etrue is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition x1b is true if proposition x1a is not true. Proposition x4b is true if proposition x4a is not true. Proposition x1a is true if proposition x1b is not true. Proposition t1 is true if proposition qcore is true. Proposition qcore is true if proposition x3a is true. Proposition s0 is true. Proposition x0a is true if proposition x0b is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x2b is true if proposition x2a is not true. Proposition s1 is true. Proposition x3a is true if proposition x3b is not true. Proposition qcore is true if proposition x4a is true. Proposition qcore is true if proposition x1a is true. Proposition x4a is true if proposition x4b is not true. Proposition qcore is true if proposition x5a is true. Proposition x5b is true if proposition x5a is not true. Proposition qcore is true if proposition x0a is true. Proposition x0b is true if proposition x0a is not true. Proposition x2a is true if proposition x2b is not true. Proposition t0 is true if proposition t1 is true. Proposition x5a is true if proposition x5b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n6-i0-skept::skept", "rec_id": "hv2-combo-disj_n6-i0", "axis": "combo", "difficulty": "disj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition qcore is true if proposition x2a is true. Proposition x3b is true if proposition x3a is not true. Proposition etrue is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition x1b is true if proposition x1a is not true. Proposition x4b is true if proposition x4a is not true. Proposition x1a is true if proposition x1b is not true. Proposition t1 is true if proposition qcore is true. Proposition qcore is true if proposition x3a is true. Proposition s0 is true. Proposition x0a is true if proposition x0b is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x2b is true if proposition x2a is not true. Proposition s1 is true. Proposition x3a is true if proposition x3b is not true. Proposition qcore is true if proposition x4a is true. Proposition qcore is true if proposition x1a is true. Proposition x4a is true if proposition x4b is not true. Proposition qcore is true if proposition x5a is true. Proposition x5b is true if proposition x5a is not true. Proposition qcore is true if proposition x0a is true. Proposition x0b is true if proposition x0a is not true. Proposition x2a is true if proposition x2b is not true. Proposition t0 is true if proposition t1 is true. Proposition x5a is true if proposition x5b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n6-i0-wfs::wfs", "rec_id": "hv2-combo-disj_n6-i0", "axis": "combo", "difficulty": "disj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition qcore is true if proposition x2a is true. Proposition x3b is true if proposition x3a is not true. Proposition etrue is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition x1b is true if proposition x1a is not true. Proposition x4b is true if proposition x4a is not true. Proposition x1a is true if proposition x1b is not true. Proposition t1 is true if proposition qcore is true. Proposition qcore is true if proposition x3a is true. Proposition s0 is true. Proposition x0a is true if proposition x0b is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x2b is true if proposition x2a is not true. Proposition s1 is true. Proposition x3a is true if proposition x3b is not true. Proposition qcore is true if proposition x4a is true. Proposition qcore is true if proposition x1a is true. Proposition x4a is true if proposition x4b is not true. Proposition qcore is true if proposition x5a is true. Proposition x5b is true if proposition x5a is not true. Proposition qcore is true if proposition x0a is true. Proposition x0b is true if proposition x0a is not true. Proposition x2a is true if proposition x2b is not true. Proposition t0 is true if proposition t1 is true. Proposition x5a is true if proposition x5b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n6-i1-none::none", "rec_id": "hv2-combo-disj_n6-i1", "axis": "combo", "difficulty": "disj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition qcore is true if proposition x5a is true. Proposition s1 is true. Proposition x0b is true if proposition x0a is not true. Proposition x1b is true if proposition x1a is not true. Proposition qcore is true if proposition x0a is true. Proposition x2a is true if proposition x2b is not true. Proposition t1 is true if proposition qcore is true. Proposition qcore is true if proposition x3a is true. Proposition x1a is true if proposition x1b is not true. Proposition t0 is true if proposition t1 is true. Proposition x3b is true if proposition x3a is not true. Proposition x5b is true if proposition x5a is not true. Proposition x4b is true if proposition x4a is not true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x4a is true if proposition x4b is not true. Proposition qcore is true if proposition x1a is true. Proposition x3a is true if proposition x3b is not true. Proposition s0 is true. Proposition x5a is true if proposition x5b is not true. Proposition x2b is true if proposition x2a is not true. Proposition qcore is true if proposition x2a is true. Proposition qcore is true if proposition x4a is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition s2 is true. Proposition x0a is true if proposition x0b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n6-i1-closed_world::closed_world", "rec_id": "hv2-combo-disj_n6-i1", "axis": "combo", "difficulty": "disj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition qcore is true if proposition x5a is true. Proposition s1 is true. Proposition x0b is true if proposition x0a is not true. Proposition x1b is true if proposition x1a is not true. Proposition qcore is true if proposition x0a is true. Proposition x2a is true if proposition x2b is not true. Proposition t1 is true if proposition qcore is true. Proposition qcore is true if proposition x3a is true. Proposition x1a is true if proposition x1b is not true. Proposition t0 is true if proposition t1 is true. Proposition x3b is true if proposition x3a is not true. Proposition x5b is true if proposition x5a is not true. Proposition x4b is true if proposition x4a is not true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x4a is true if proposition x4b is not true. Proposition qcore is true if proposition x1a is true. Proposition x3a is true if proposition x3b is not true. Proposition s0 is true. Proposition x5a is true if proposition x5b is not true. Proposition x2b is true if proposition x2a is not true. Proposition qcore is true if proposition x2a is true. Proposition qcore is true if proposition x4a is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition s2 is true. Proposition x0a is true if proposition x0b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n6-i1-cred::cred", "rec_id": "hv2-combo-disj_n6-i1", "axis": "combo", "difficulty": "disj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition qcore is true if proposition x5a is true. Proposition s1 is true. Proposition x0b is true if proposition x0a is not true. Proposition x1b is true if proposition x1a is not true. Proposition qcore is true if proposition x0a is true. Proposition x2a is true if proposition x2b is not true. Proposition t1 is true if proposition qcore is true. Proposition qcore is true if proposition x3a is true. Proposition x1a is true if proposition x1b is not true. Proposition t0 is true if proposition t1 is true. Proposition x3b is true if proposition x3a is not true. Proposition x5b is true if proposition x5a is not true. Proposition x4b is true if proposition x4a is not true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x4a is true if proposition x4b is not true. Proposition qcore is true if proposition x1a is true. Proposition x3a is true if proposition x3b is not true. Proposition s0 is true. Proposition x5a is true if proposition x5b is not true. Proposition x2b is true if proposition x2a is not true. Proposition qcore is true if proposition x2a is true. Proposition qcore is true if proposition x4a is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition s2 is true. Proposition x0a is true if proposition x0b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n6-i1-skept::skept", "rec_id": "hv2-combo-disj_n6-i1", "axis": "combo", "difficulty": "disj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition qcore is true if proposition x5a is true. Proposition s1 is true. Proposition x0b is true if proposition x0a is not true. Proposition x1b is true if proposition x1a is not true. Proposition qcore is true if proposition x0a is true. Proposition x2a is true if proposition x2b is not true. Proposition t1 is true if proposition qcore is true. Proposition qcore is true if proposition x3a is true. Proposition x1a is true if proposition x1b is not true. Proposition t0 is true if proposition t1 is true. Proposition x3b is true if proposition x3a is not true. Proposition x5b is true if proposition x5a is not true. Proposition x4b is true if proposition x4a is not true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x4a is true if proposition x4b is not true. Proposition qcore is true if proposition x1a is true. Proposition x3a is true if proposition x3b is not true. Proposition s0 is true. Proposition x5a is true if proposition x5b is not true. Proposition x2b is true if proposition x2a is not true. Proposition qcore is true if proposition x2a is true. Proposition qcore is true if proposition x4a is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition s2 is true. Proposition x0a is true if proposition x0b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n6-i1-wfs::wfs", "rec_id": "hv2-combo-disj_n6-i1", "axis": "combo", "difficulty": "disj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition qcore is true if proposition x5a is true. Proposition s1 is true. Proposition x0b is true if proposition x0a is not true. Proposition x1b is true if proposition x1a is not true. Proposition qcore is true if proposition x0a is true. Proposition x2a is true if proposition x2b is not true. Proposition t1 is true if proposition qcore is true. Proposition qcore is true if proposition x3a is true. Proposition x1a is true if proposition x1b is not true. Proposition t0 is true if proposition t1 is true. Proposition x3b is true if proposition x3a is not true. Proposition x5b is true if proposition x5a is not true. Proposition x4b is true if proposition x4a is not true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x4a is true if proposition x4b is not true. Proposition qcore is true if proposition x1a is true. Proposition x3a is true if proposition x3b is not true. Proposition s0 is true. Proposition x5a is true if proposition x5b is not true. Proposition x2b is true if proposition x2a is not true. Proposition qcore is true if proposition x2a is true. Proposition qcore is true if proposition x4a is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition s2 is true. Proposition x0a is true if proposition x0b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n6-i2-none::none", "rec_id": "hv2-combo-disj_n6-i2", "axis": "combo", "difficulty": "disj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition x0b is true if proposition x0a is not true. Proposition s1 is true. Proposition x3a is true if proposition x3b is not true. Proposition q is true if proposition qcore is true and proposition wide is true and proposition etrue is true. Proposition s0 is true. Proposition qcore is true if proposition x3a is true. Proposition x2a is true if proposition x2b is not true. Proposition x1b is true if proposition x1a is not true. Proposition x4a is true if proposition x4b is not true. Proposition qcore is true if proposition x1a is true. Proposition qcore is true if proposition x2a is true. Proposition qcore is true if proposition x4a is true. Proposition qcore is true if proposition x5a is true. Proposition x3b is true if proposition x3a is not true. Proposition qcore is true if proposition x0a is true. Proposition x2b is true if proposition x2a is not true. Proposition x1a is true if proposition x1b is not true. Proposition x5b is true if proposition x5a is not true. Proposition etrue is true. Proposition x0a is true if proposition x0b is not true. Proposition x4b is true if proposition x4a is not true. Proposition x5a is true if proposition x5b is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n6-i2-closed_world::closed_world", "rec_id": "hv2-combo-disj_n6-i2", "axis": "combo", "difficulty": "disj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition x0b is true if proposition x0a is not true. Proposition s1 is true. Proposition x3a is true if proposition x3b is not true. Proposition q is true if proposition qcore is true and proposition wide is true and proposition etrue is true. Proposition s0 is true. Proposition qcore is true if proposition x3a is true. Proposition x2a is true if proposition x2b is not true. Proposition x1b is true if proposition x1a is not true. Proposition x4a is true if proposition x4b is not true. Proposition qcore is true if proposition x1a is true. Proposition qcore is true if proposition x2a is true. Proposition qcore is true if proposition x4a is true. Proposition qcore is true if proposition x5a is true. Proposition x3b is true if proposition x3a is not true. Proposition qcore is true if proposition x0a is true. Proposition x2b is true if proposition x2a is not true. Proposition x1a is true if proposition x1b is not true. Proposition x5b is true if proposition x5a is not true. Proposition etrue is true. Proposition x0a is true if proposition x0b is not true. Proposition x4b is true if proposition x4a is not true. Proposition x5a is true if proposition x5b is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n6-i2-cred::cred", "rec_id": "hv2-combo-disj_n6-i2", "axis": "combo", "difficulty": "disj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition x0b is true if proposition x0a is not true. Proposition s1 is true. Proposition x3a is true if proposition x3b is not true. Proposition q is true if proposition qcore is true and proposition wide is true and proposition etrue is true. Proposition s0 is true. Proposition qcore is true if proposition x3a is true. Proposition x2a is true if proposition x2b is not true. Proposition x1b is true if proposition x1a is not true. Proposition x4a is true if proposition x4b is not true. Proposition qcore is true if proposition x1a is true. Proposition qcore is true if proposition x2a is true. Proposition qcore is true if proposition x4a is true. Proposition qcore is true if proposition x5a is true. Proposition x3b is true if proposition x3a is not true. Proposition qcore is true if proposition x0a is true. Proposition x2b is true if proposition x2a is not true. Proposition x1a is true if proposition x1b is not true. Proposition x5b is true if proposition x5a is not true. Proposition etrue is true. Proposition x0a is true if proposition x0b is not true. Proposition x4b is true if proposition x4a is not true. Proposition x5a is true if proposition x5b is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n6-i2-skept::skept", "rec_id": "hv2-combo-disj_n6-i2", "axis": "combo", "difficulty": "disj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition x0b is true if proposition x0a is not true. Proposition s1 is true. Proposition x3a is true if proposition x3b is not true. Proposition q is true if proposition qcore is true and proposition wide is true and proposition etrue is true. Proposition s0 is true. Proposition qcore is true if proposition x3a is true. Proposition x2a is true if proposition x2b is not true. Proposition x1b is true if proposition x1a is not true. Proposition x4a is true if proposition x4b is not true. Proposition qcore is true if proposition x1a is true. Proposition qcore is true if proposition x2a is true. Proposition qcore is true if proposition x4a is true. Proposition qcore is true if proposition x5a is true. Proposition x3b is true if proposition x3a is not true. Proposition qcore is true if proposition x0a is true. Proposition x2b is true if proposition x2a is not true. Proposition x1a is true if proposition x1b is not true. Proposition x5b is true if proposition x5a is not true. Proposition etrue is true. Proposition x0a is true if proposition x0b is not true. Proposition x4b is true if proposition x4a is not true. Proposition x5a is true if proposition x5b is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n6-i2-wfs::wfs", "rec_id": "hv2-combo-disj_n6-i2", "axis": "combo", "difficulty": "disj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition x0b is true if proposition x0a is not true. Proposition s1 is true. Proposition x3a is true if proposition x3b is not true. Proposition q is true if proposition qcore is true and proposition wide is true and proposition etrue is true. Proposition s0 is true. Proposition qcore is true if proposition x3a is true. Proposition x2a is true if proposition x2b is not true. Proposition x1b is true if proposition x1a is not true. Proposition x4a is true if proposition x4b is not true. Proposition qcore is true if proposition x1a is true. Proposition qcore is true if proposition x2a is true. Proposition qcore is true if proposition x4a is true. Proposition qcore is true if proposition x5a is true. Proposition x3b is true if proposition x3a is not true. Proposition qcore is true if proposition x0a is true. Proposition x2b is true if proposition x2a is not true. Proposition x1a is true if proposition x1b is not true. Proposition x5b is true if proposition x5a is not true. Proposition etrue is true. Proposition x0a is true if proposition x0b is not true. Proposition x4b is true if proposition x4a is not true. Proposition x5a is true if proposition x5b is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n6-i3-none::none", "rec_id": "hv2-combo-disj_n6-i3", "axis": "combo", "difficulty": "disj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition t3 is true if proposition qcore is true. Proposition x4b is true if proposition x4a is not true. Proposition s1 is true. Proposition x3b is true if proposition x3a is not true. Proposition t1 is true if proposition t2 is true. Proposition s2 is true. Proposition x5b is true if proposition x5a is not true. Proposition x0a is true if proposition x0b is not true. Proposition x2a is true if proposition x2b is not true. Proposition x4a is true if proposition x4b is not true. Proposition qcore is true if proposition x1a is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition x1b is true if proposition x1a is not true. Proposition qcore is true if proposition x5a is true. Proposition qcore is true if proposition x3a is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition x4a is true. Proposition s0 is true. Proposition t2 is true if proposition t3 is true. Proposition x2b is true if proposition x2a is not true. Proposition x1a is true if proposition x1b is not true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x0b is true if proposition x0a is not true. Proposition qcore is true if proposition x2a is true. Proposition x5a is true if proposition x5b is not true. Proposition qcore is true if proposition x0a is true. Proposition x3a is true if proposition x3b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n6-i3-closed_world::closed_world", "rec_id": "hv2-combo-disj_n6-i3", "axis": "combo", "difficulty": "disj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition t3 is true if proposition qcore is true. Proposition x4b is true if proposition x4a is not true. Proposition s1 is true. Proposition x3b is true if proposition x3a is not true. Proposition t1 is true if proposition t2 is true. Proposition s2 is true. Proposition x5b is true if proposition x5a is not true. Proposition x0a is true if proposition x0b is not true. Proposition x2a is true if proposition x2b is not true. Proposition x4a is true if proposition x4b is not true. Proposition qcore is true if proposition x1a is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition x1b is true if proposition x1a is not true. Proposition qcore is true if proposition x5a is true. Proposition qcore is true if proposition x3a is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition x4a is true. Proposition s0 is true. Proposition t2 is true if proposition t3 is true. Proposition x2b is true if proposition x2a is not true. Proposition x1a is true if proposition x1b is not true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x0b is true if proposition x0a is not true. Proposition qcore is true if proposition x2a is true. Proposition x5a is true if proposition x5b is not true. Proposition qcore is true if proposition x0a is true. Proposition x3a is true if proposition x3b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n6-i3-cred::cred", "rec_id": "hv2-combo-disj_n6-i3", "axis": "combo", "difficulty": "disj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition t3 is true if proposition qcore is true. Proposition x4b is true if proposition x4a is not true. Proposition s1 is true. Proposition x3b is true if proposition x3a is not true. Proposition t1 is true if proposition t2 is true. Proposition s2 is true. Proposition x5b is true if proposition x5a is not true. Proposition x0a is true if proposition x0b is not true. Proposition x2a is true if proposition x2b is not true. Proposition x4a is true if proposition x4b is not true. Proposition qcore is true if proposition x1a is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition x1b is true if proposition x1a is not true. Proposition qcore is true if proposition x5a is true. Proposition qcore is true if proposition x3a is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition x4a is true. Proposition s0 is true. Proposition t2 is true if proposition t3 is true. Proposition x2b is true if proposition x2a is not true. Proposition x1a is true if proposition x1b is not true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x0b is true if proposition x0a is not true. Proposition qcore is true if proposition x2a is true. Proposition x5a is true if proposition x5b is not true. Proposition qcore is true if proposition x0a is true. Proposition x3a is true if proposition x3b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n6-i3-skept::skept", "rec_id": "hv2-combo-disj_n6-i3", "axis": "combo", "difficulty": "disj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition t3 is true if proposition qcore is true. Proposition x4b is true if proposition x4a is not true. Proposition s1 is true. Proposition x3b is true if proposition x3a is not true. Proposition t1 is true if proposition t2 is true. Proposition s2 is true. Proposition x5b is true if proposition x5a is not true. Proposition x0a is true if proposition x0b is not true. Proposition x2a is true if proposition x2b is not true. Proposition x4a is true if proposition x4b is not true. Proposition qcore is true if proposition x1a is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition x1b is true if proposition x1a is not true. Proposition qcore is true if proposition x5a is true. Proposition qcore is true if proposition x3a is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition x4a is true. Proposition s0 is true. Proposition t2 is true if proposition t3 is true. Proposition x2b is true if proposition x2a is not true. Proposition x1a is true if proposition x1b is not true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x0b is true if proposition x0a is not true. Proposition qcore is true if proposition x2a is true. Proposition x5a is true if proposition x5b is not true. Proposition qcore is true if proposition x0a is true. Proposition x3a is true if proposition x3b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n6-i3-wfs::wfs", "rec_id": "hv2-combo-disj_n6-i3", "axis": "combo", "difficulty": "disj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition t3 is true if proposition qcore is true. Proposition x4b is true if proposition x4a is not true. Proposition s1 is true. Proposition x3b is true if proposition x3a is not true. Proposition t1 is true if proposition t2 is true. Proposition s2 is true. Proposition x5b is true if proposition x5a is not true. Proposition x0a is true if proposition x0b is not true. Proposition x2a is true if proposition x2b is not true. Proposition x4a is true if proposition x4b is not true. Proposition qcore is true if proposition x1a is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition x1b is true if proposition x1a is not true. Proposition qcore is true if proposition x5a is true. Proposition qcore is true if proposition x3a is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition x4a is true. Proposition s0 is true. Proposition t2 is true if proposition t3 is true. Proposition x2b is true if proposition x2a is not true. Proposition x1a is true if proposition x1b is not true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x0b is true if proposition x0a is not true. Proposition qcore is true if proposition x2a is true. Proposition x5a is true if proposition x5b is not true. Proposition qcore is true if proposition x0a is true. Proposition x3a is true if proposition x3b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n6-i4-none::none", "rec_id": "hv2-combo-disj_n6-i4", "axis": "combo", "difficulty": "disj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition t1 is true if proposition t2 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition x0a is true if proposition x0b is not true. Proposition x1a is true if proposition x1b is not true. Proposition x3a is true if proposition x3b is not true. Proposition qcore is true if proposition x0a is true. Proposition x5b is true if proposition x5a is not true. Proposition s0 is true. Proposition t0 is true if proposition t1 is true. Proposition s2 is true. Proposition x2a is true if proposition x2b is not true. Proposition t5 is true if proposition qcore is true. Proposition qcore is true if proposition x5a is true. Proposition x3b is true if proposition x3a is not true. Proposition t2 is true if proposition t3 is true. Proposition qcore is true if proposition x4a is true. Proposition x2b is true if proposition x2a is not true. Proposition qcore is true if proposition x2a is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x4a is true if proposition x4b is not true. Proposition x1b is true if proposition x1a is not true. Proposition x0b is true if proposition x0a is not true. Proposition s1 is true. Proposition qcore is true if proposition x1a is true. Proposition x4b is true if proposition x4a is not true. Proposition x5a is true if proposition x5b is not true. Proposition t4 is true if proposition t5 is true. Proposition t3 is true if proposition t4 is true. Proposition qcore is true if proposition x3a is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n6-i4-closed_world::closed_world", "rec_id": "hv2-combo-disj_n6-i4", "axis": "combo", "difficulty": "disj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition t1 is true if proposition t2 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition x0a is true if proposition x0b is not true. Proposition x1a is true if proposition x1b is not true. Proposition x3a is true if proposition x3b is not true. Proposition qcore is true if proposition x0a is true. Proposition x5b is true if proposition x5a is not true. Proposition s0 is true. Proposition t0 is true if proposition t1 is true. Proposition s2 is true. Proposition x2a is true if proposition x2b is not true. Proposition t5 is true if proposition qcore is true. Proposition qcore is true if proposition x5a is true. Proposition x3b is true if proposition x3a is not true. Proposition t2 is true if proposition t3 is true. Proposition qcore is true if proposition x4a is true. Proposition x2b is true if proposition x2a is not true. Proposition qcore is true if proposition x2a is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x4a is true if proposition x4b is not true. Proposition x1b is true if proposition x1a is not true. Proposition x0b is true if proposition x0a is not true. Proposition s1 is true. Proposition qcore is true if proposition x1a is true. Proposition x4b is true if proposition x4a is not true. Proposition x5a is true if proposition x5b is not true. Proposition t4 is true if proposition t5 is true. Proposition t3 is true if proposition t4 is true. Proposition qcore is true if proposition x3a is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n6-i4-cred::cred", "rec_id": "hv2-combo-disj_n6-i4", "axis": "combo", "difficulty": "disj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition t1 is true if proposition t2 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition x0a is true if proposition x0b is not true. Proposition x1a is true if proposition x1b is not true. Proposition x3a is true if proposition x3b is not true. Proposition qcore is true if proposition x0a is true. Proposition x5b is true if proposition x5a is not true. Proposition s0 is true. Proposition t0 is true if proposition t1 is true. Proposition s2 is true. Proposition x2a is true if proposition x2b is not true. Proposition t5 is true if proposition qcore is true. Proposition qcore is true if proposition x5a is true. Proposition x3b is true if proposition x3a is not true. Proposition t2 is true if proposition t3 is true. Proposition qcore is true if proposition x4a is true. Proposition x2b is true if proposition x2a is not true. Proposition qcore is true if proposition x2a is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x4a is true if proposition x4b is not true. Proposition x1b is true if proposition x1a is not true. Proposition x0b is true if proposition x0a is not true. Proposition s1 is true. Proposition qcore is true if proposition x1a is true. Proposition x4b is true if proposition x4a is not true. Proposition x5a is true if proposition x5b is not true. Proposition t4 is true if proposition t5 is true. Proposition t3 is true if proposition t4 is true. Proposition qcore is true if proposition x3a is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n6-i4-skept::skept", "rec_id": "hv2-combo-disj_n6-i4", "axis": "combo", "difficulty": "disj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition t1 is true if proposition t2 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition x0a is true if proposition x0b is not true. Proposition x1a is true if proposition x1b is not true. Proposition x3a is true if proposition x3b is not true. Proposition qcore is true if proposition x0a is true. Proposition x5b is true if proposition x5a is not true. Proposition s0 is true. Proposition t0 is true if proposition t1 is true. Proposition s2 is true. Proposition x2a is true if proposition x2b is not true. Proposition t5 is true if proposition qcore is true. Proposition qcore is true if proposition x5a is true. Proposition x3b is true if proposition x3a is not true. Proposition t2 is true if proposition t3 is true. Proposition qcore is true if proposition x4a is true. Proposition x2b is true if proposition x2a is not true. Proposition qcore is true if proposition x2a is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x4a is true if proposition x4b is not true. Proposition x1b is true if proposition x1a is not true. Proposition x0b is true if proposition x0a is not true. Proposition s1 is true. Proposition qcore is true if proposition x1a is true. Proposition x4b is true if proposition x4a is not true. Proposition x5a is true if proposition x5b is not true. Proposition t4 is true if proposition t5 is true. Proposition t3 is true if proposition t4 is true. Proposition qcore is true if proposition x3a is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n6-i4-wfs::wfs", "rec_id": "hv2-combo-disj_n6-i4", "axis": "combo", "difficulty": "disj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition t1 is true if proposition t2 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition x0a is true if proposition x0b is not true. Proposition x1a is true if proposition x1b is not true. Proposition x3a is true if proposition x3b is not true. Proposition qcore is true if proposition x0a is true. Proposition x5b is true if proposition x5a is not true. Proposition s0 is true. Proposition t0 is true if proposition t1 is true. Proposition s2 is true. Proposition x2a is true if proposition x2b is not true. Proposition t5 is true if proposition qcore is true. Proposition qcore is true if proposition x5a is true. Proposition x3b is true if proposition x3a is not true. Proposition t2 is true if proposition t3 is true. Proposition qcore is true if proposition x4a is true. Proposition x2b is true if proposition x2a is not true. Proposition qcore is true if proposition x2a is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x4a is true if proposition x4b is not true. Proposition x1b is true if proposition x1a is not true. Proposition x0b is true if proposition x0a is not true. Proposition s1 is true. Proposition qcore is true if proposition x1a is true. Proposition x4b is true if proposition x4a is not true. Proposition x5a is true if proposition x5b is not true. Proposition t4 is true if proposition t5 is true. Proposition t3 is true if proposition t4 is true. Proposition qcore is true if proposition x3a is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n6-i5-none::none", "rec_id": "hv2-combo-disj_n6-i5", "axis": "combo", "difficulty": "disj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition x2b is true if proposition x2a is not true. Proposition t3 is true if proposition t4 is true. Proposition qcore is true if proposition x2a is true. Proposition x5a is true if proposition x5b is not true. Proposition s0 is true. Proposition x5b is true if proposition x5a is not true. Proposition qcore is true if proposition x4a is true. Proposition x0a is true if proposition x0b is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x0b is true if proposition x0a is not true. Proposition t4 is true if proposition t5 is true. Proposition x1b is true if proposition x1a is not true. Proposition x2a is true if proposition x2b is not true. Proposition etrue is true. Proposition x4b is true if proposition x4a is not true. Proposition x4a is true if proposition x4b is not true. Proposition t2 is true if proposition t3 is true. Proposition t1 is true if proposition t2 is true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x1a is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition qcore is true if proposition x0a is true. Proposition qcore is true if proposition x5a is true. Proposition x3a is true if proposition x3b is not true. Proposition s1 is true. Proposition x3b is true if proposition x3a is not true. Proposition qcore is true if proposition x3a is true. Proposition t0 is true if proposition t1 is true. Proposition t5 is true if proposition qcore is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n6-i5-closed_world::closed_world", "rec_id": "hv2-combo-disj_n6-i5", "axis": "combo", "difficulty": "disj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition x2b is true if proposition x2a is not true. Proposition t3 is true if proposition t4 is true. Proposition qcore is true if proposition x2a is true. Proposition x5a is true if proposition x5b is not true. Proposition s0 is true. Proposition x5b is true if proposition x5a is not true. Proposition qcore is true if proposition x4a is true. Proposition x0a is true if proposition x0b is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x0b is true if proposition x0a is not true. Proposition t4 is true if proposition t5 is true. Proposition x1b is true if proposition x1a is not true. Proposition x2a is true if proposition x2b is not true. Proposition etrue is true. Proposition x4b is true if proposition x4a is not true. Proposition x4a is true if proposition x4b is not true. Proposition t2 is true if proposition t3 is true. Proposition t1 is true if proposition t2 is true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x1a is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition qcore is true if proposition x0a is true. Proposition qcore is true if proposition x5a is true. Proposition x3a is true if proposition x3b is not true. Proposition s1 is true. Proposition x3b is true if proposition x3a is not true. Proposition qcore is true if proposition x3a is true. Proposition t0 is true if proposition t1 is true. Proposition t5 is true if proposition qcore is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n6-i5-cred::cred", "rec_id": "hv2-combo-disj_n6-i5", "axis": "combo", "difficulty": "disj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition x2b is true if proposition x2a is not true. Proposition t3 is true if proposition t4 is true. Proposition qcore is true if proposition x2a is true. Proposition x5a is true if proposition x5b is not true. Proposition s0 is true. Proposition x5b is true if proposition x5a is not true. Proposition qcore is true if proposition x4a is true. Proposition x0a is true if proposition x0b is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x0b is true if proposition x0a is not true. Proposition t4 is true if proposition t5 is true. Proposition x1b is true if proposition x1a is not true. Proposition x2a is true if proposition x2b is not true. Proposition etrue is true. Proposition x4b is true if proposition x4a is not true. Proposition x4a is true if proposition x4b is not true. Proposition t2 is true if proposition t3 is true. Proposition t1 is true if proposition t2 is true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x1a is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition qcore is true if proposition x0a is true. Proposition qcore is true if proposition x5a is true. Proposition x3a is true if proposition x3b is not true. Proposition s1 is true. Proposition x3b is true if proposition x3a is not true. Proposition qcore is true if proposition x3a is true. Proposition t0 is true if proposition t1 is true. Proposition t5 is true if proposition qcore is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n6-i5-skept::skept", "rec_id": "hv2-combo-disj_n6-i5", "axis": "combo", "difficulty": "disj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition x2b is true if proposition x2a is not true. Proposition t3 is true if proposition t4 is true. Proposition qcore is true if proposition x2a is true. Proposition x5a is true if proposition x5b is not true. Proposition s0 is true. Proposition x5b is true if proposition x5a is not true. Proposition qcore is true if proposition x4a is true. Proposition x0a is true if proposition x0b is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x0b is true if proposition x0a is not true. Proposition t4 is true if proposition t5 is true. Proposition x1b is true if proposition x1a is not true. Proposition x2a is true if proposition x2b is not true. Proposition etrue is true. Proposition x4b is true if proposition x4a is not true. Proposition x4a is true if proposition x4b is not true. Proposition t2 is true if proposition t3 is true. Proposition t1 is true if proposition t2 is true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x1a is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition qcore is true if proposition x0a is true. Proposition qcore is true if proposition x5a is true. Proposition x3a is true if proposition x3b is not true. Proposition s1 is true. Proposition x3b is true if proposition x3a is not true. Proposition qcore is true if proposition x3a is true. Proposition t0 is true if proposition t1 is true. Proposition t5 is true if proposition qcore is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n6-i5-wfs::wfs", "rec_id": "hv2-combo-disj_n6-i5", "axis": "combo", "difficulty": "disj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition x2b is true if proposition x2a is not true. Proposition t3 is true if proposition t4 is true. Proposition qcore is true if proposition x2a is true. Proposition x5a is true if proposition x5b is not true. Proposition s0 is true. Proposition x5b is true if proposition x5a is not true. Proposition qcore is true if proposition x4a is true. Proposition x0a is true if proposition x0b is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x0b is true if proposition x0a is not true. Proposition t4 is true if proposition t5 is true. Proposition x1b is true if proposition x1a is not true. Proposition x2a is true if proposition x2b is not true. Proposition etrue is true. Proposition x4b is true if proposition x4a is not true. Proposition x4a is true if proposition x4b is not true. Proposition t2 is true if proposition t3 is true. Proposition t1 is true if proposition t2 is true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x1a is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition qcore is true if proposition x0a is true. Proposition qcore is true if proposition x5a is true. Proposition x3a is true if proposition x3b is not true. Proposition s1 is true. Proposition x3b is true if proposition x3a is not true. Proposition qcore is true if proposition x3a is true. Proposition t0 is true if proposition t1 is true. Proposition t5 is true if proposition qcore is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n6-i6-none::none", "rec_id": "hv2-combo-disj_n6-i6", "axis": "combo", "difficulty": "disj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition x2a is true if proposition x2b is not true. Proposition t1 is true if proposition qcore is true. Proposition etrue is true. Proposition x4b is true if proposition x4a is not true. Proposition qcore is true if proposition x0a is true. Proposition x3b is true if proposition x3a is not true. Proposition x1b is true if proposition x1a is not true. Proposition x3a is true if proposition x3b is not true. Proposition qcore is true if proposition x3a is true. Proposition x2b is true if proposition x2a is not true. Proposition qcore is true if proposition x4a is true. Proposition q is true if proposition t0 is true and proposition etrue is true. Proposition x5b is true if proposition x5a is not true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x5a is true. Proposition x5a is true if proposition x5b is not true. Proposition qcore is true if proposition x1a is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition x2a is true. Proposition x4a is true if proposition x4b is not true. Proposition x0a is true if proposition x0b is not true. Proposition x0b is true if proposition x0a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n6-i6-closed_world::closed_world", "rec_id": "hv2-combo-disj_n6-i6", "axis": "combo", "difficulty": "disj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition x2a is true if proposition x2b is not true. Proposition t1 is true if proposition qcore is true. Proposition etrue is true. Proposition x4b is true if proposition x4a is not true. Proposition qcore is true if proposition x0a is true. Proposition x3b is true if proposition x3a is not true. Proposition x1b is true if proposition x1a is not true. Proposition x3a is true if proposition x3b is not true. Proposition qcore is true if proposition x3a is true. Proposition x2b is true if proposition x2a is not true. Proposition qcore is true if proposition x4a is true. Proposition q is true if proposition t0 is true and proposition etrue is true. Proposition x5b is true if proposition x5a is not true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x5a is true. Proposition x5a is true if proposition x5b is not true. Proposition qcore is true if proposition x1a is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition x2a is true. Proposition x4a is true if proposition x4b is not true. Proposition x0a is true if proposition x0b is not true. Proposition x0b is true if proposition x0a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n6-i6-cred::cred", "rec_id": "hv2-combo-disj_n6-i6", "axis": "combo", "difficulty": "disj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition x2a is true if proposition x2b is not true. Proposition t1 is true if proposition qcore is true. Proposition etrue is true. Proposition x4b is true if proposition x4a is not true. Proposition qcore is true if proposition x0a is true. Proposition x3b is true if proposition x3a is not true. Proposition x1b is true if proposition x1a is not true. Proposition x3a is true if proposition x3b is not true. Proposition qcore is true if proposition x3a is true. Proposition x2b is true if proposition x2a is not true. Proposition qcore is true if proposition x4a is true. Proposition q is true if proposition t0 is true and proposition etrue is true. Proposition x5b is true if proposition x5a is not true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x5a is true. Proposition x5a is true if proposition x5b is not true. Proposition qcore is true if proposition x1a is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition x2a is true. Proposition x4a is true if proposition x4b is not true. Proposition x0a is true if proposition x0b is not true. Proposition x0b is true if proposition x0a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n6-i6-skept::skept", "rec_id": "hv2-combo-disj_n6-i6", "axis": "combo", "difficulty": "disj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition x2a is true if proposition x2b is not true. Proposition t1 is true if proposition qcore is true. Proposition etrue is true. Proposition x4b is true if proposition x4a is not true. Proposition qcore is true if proposition x0a is true. Proposition x3b is true if proposition x3a is not true. Proposition x1b is true if proposition x1a is not true. Proposition x3a is true if proposition x3b is not true. Proposition qcore is true if proposition x3a is true. Proposition x2b is true if proposition x2a is not true. Proposition qcore is true if proposition x4a is true. Proposition q is true if proposition t0 is true and proposition etrue is true. Proposition x5b is true if proposition x5a is not true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x5a is true. Proposition x5a is true if proposition x5b is not true. Proposition qcore is true if proposition x1a is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition x2a is true. Proposition x4a is true if proposition x4b is not true. Proposition x0a is true if proposition x0b is not true. Proposition x0b is true if proposition x0a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n6-i6-wfs::wfs", "rec_id": "hv2-combo-disj_n6-i6", "axis": "combo", "difficulty": "disj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition x2a is true if proposition x2b is not true. Proposition t1 is true if proposition qcore is true. Proposition etrue is true. Proposition x4b is true if proposition x4a is not true. Proposition qcore is true if proposition x0a is true. Proposition x3b is true if proposition x3a is not true. Proposition x1b is true if proposition x1a is not true. Proposition x3a is true if proposition x3b is not true. Proposition qcore is true if proposition x3a is true. Proposition x2b is true if proposition x2a is not true. Proposition qcore is true if proposition x4a is true. Proposition q is true if proposition t0 is true and proposition etrue is true. Proposition x5b is true if proposition x5a is not true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x5a is true. Proposition x5a is true if proposition x5b is not true. Proposition qcore is true if proposition x1a is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition x2a is true. Proposition x4a is true if proposition x4b is not true. Proposition x0a is true if proposition x0b is not true. Proposition x0b is true if proposition x0a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n6-i7-none::none", "rec_id": "hv2-combo-disj_n6-i7", "axis": "combo", "difficulty": "disj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition x3b is true if proposition x3a is not true. Proposition etrue is true. Proposition x2b is true if proposition x2a is not true. Proposition qcore is true if proposition x2a is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition qcore is true if proposition x3a is true. Proposition qcore is true if proposition x5a is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition x0b is true if proposition x0a is not true. Proposition x0a is true if proposition x0b is not true. Proposition t1 is true if proposition t2 is true. Proposition x2a is true if proposition x2b is not true. Proposition t2 is true if proposition t3 is true. Proposition x1a is true if proposition x1b is not true. Proposition x4b is true if proposition x4a is not true. Proposition s0 is true. Proposition x5a is true if proposition x5b is not true. Proposition x3a is true if proposition x3b is not true. Proposition x4a is true if proposition x4b is not true. Proposition x1b is true if proposition x1a is not true. Proposition x5b is true if proposition x5a is not true. Proposition t3 is true if proposition qcore is true. Proposition qcore is true if proposition x4a is true. Proposition s2 is true. Proposition qcore is true if proposition x0a is true. Proposition t0 is true if proposition t1 is true. Proposition s1 is true. Proposition qcore is true if proposition x1a is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n6-i7-closed_world::closed_world", "rec_id": "hv2-combo-disj_n6-i7", "axis": "combo", "difficulty": "disj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition x3b is true if proposition x3a is not true. Proposition etrue is true. Proposition x2b is true if proposition x2a is not true. Proposition qcore is true if proposition x2a is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition qcore is true if proposition x3a is true. Proposition qcore is true if proposition x5a is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition x0b is true if proposition x0a is not true. Proposition x0a is true if proposition x0b is not true. Proposition t1 is true if proposition t2 is true. Proposition x2a is true if proposition x2b is not true. Proposition t2 is true if proposition t3 is true. Proposition x1a is true if proposition x1b is not true. Proposition x4b is true if proposition x4a is not true. Proposition s0 is true. Proposition x5a is true if proposition x5b is not true. Proposition x3a is true if proposition x3b is not true. Proposition x4a is true if proposition x4b is not true. Proposition x1b is true if proposition x1a is not true. Proposition x5b is true if proposition x5a is not true. Proposition t3 is true if proposition qcore is true. Proposition qcore is true if proposition x4a is true. Proposition s2 is true. Proposition qcore is true if proposition x0a is true. Proposition t0 is true if proposition t1 is true. Proposition s1 is true. Proposition qcore is true if proposition x1a is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n6-i7-cred::cred", "rec_id": "hv2-combo-disj_n6-i7", "axis": "combo", "difficulty": "disj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition x3b is true if proposition x3a is not true. Proposition etrue is true. Proposition x2b is true if proposition x2a is not true. Proposition qcore is true if proposition x2a is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition qcore is true if proposition x3a is true. Proposition qcore is true if proposition x5a is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition x0b is true if proposition x0a is not true. Proposition x0a is true if proposition x0b is not true. Proposition t1 is true if proposition t2 is true. Proposition x2a is true if proposition x2b is not true. Proposition t2 is true if proposition t3 is true. Proposition x1a is true if proposition x1b is not true. Proposition x4b is true if proposition x4a is not true. Proposition s0 is true. Proposition x5a is true if proposition x5b is not true. Proposition x3a is true if proposition x3b is not true. Proposition x4a is true if proposition x4b is not true. Proposition x1b is true if proposition x1a is not true. Proposition x5b is true if proposition x5a is not true. Proposition t3 is true if proposition qcore is true. Proposition qcore is true if proposition x4a is true. Proposition s2 is true. Proposition qcore is true if proposition x0a is true. Proposition t0 is true if proposition t1 is true. Proposition s1 is true. Proposition qcore is true if proposition x1a is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n6-i7-skept::skept", "rec_id": "hv2-combo-disj_n6-i7", "axis": "combo", "difficulty": "disj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition x3b is true if proposition x3a is not true. Proposition etrue is true. Proposition x2b is true if proposition x2a is not true. Proposition qcore is true if proposition x2a is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition qcore is true if proposition x3a is true. Proposition qcore is true if proposition x5a is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition x0b is true if proposition x0a is not true. Proposition x0a is true if proposition x0b is not true. Proposition t1 is true if proposition t2 is true. Proposition x2a is true if proposition x2b is not true. Proposition t2 is true if proposition t3 is true. Proposition x1a is true if proposition x1b is not true. Proposition x4b is true if proposition x4a is not true. Proposition s0 is true. Proposition x5a is true if proposition x5b is not true. Proposition x3a is true if proposition x3b is not true. Proposition x4a is true if proposition x4b is not true. Proposition x1b is true if proposition x1a is not true. Proposition x5b is true if proposition x5a is not true. Proposition t3 is true if proposition qcore is true. Proposition qcore is true if proposition x4a is true. Proposition s2 is true. Proposition qcore is true if proposition x0a is true. Proposition t0 is true if proposition t1 is true. Proposition s1 is true. Proposition qcore is true if proposition x1a is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-disj_n6-i7-wfs::wfs", "rec_id": "hv2-combo-disj_n6-i7", "axis": "combo", "difficulty": "disj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition x3b is true if proposition x3a is not true. Proposition etrue is true. Proposition x2b is true if proposition x2a is not true. Proposition qcore is true if proposition x2a is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition qcore is true if proposition x3a is true. Proposition qcore is true if proposition x5a is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition x0b is true if proposition x0a is not true. Proposition x0a is true if proposition x0b is not true. Proposition t1 is true if proposition t2 is true. Proposition x2a is true if proposition x2b is not true. Proposition t2 is true if proposition t3 is true. Proposition x1a is true if proposition x1b is not true. Proposition x4b is true if proposition x4a is not true. Proposition s0 is true. Proposition x5a is true if proposition x5b is not true. Proposition x3a is true if proposition x3b is not true. Proposition x4a is true if proposition x4b is not true. Proposition x1b is true if proposition x1a is not true. Proposition x5b is true if proposition x5a is not true. Proposition t3 is true if proposition qcore is true. Proposition qcore is true if proposition x4a is true. Proposition s2 is true. Proposition qcore is true if proposition x0a is true. Proposition t0 is true if proposition t1 is true. Proposition s1 is true. Proposition qcore is true if proposition x1a is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n2-i0-none::none", "rec_id": "hv2-combo-conj_n2-i0", "axis": "combo", "difficulty": "conj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition x0b is true if proposition x0a is not true. Proposition t1 is true if proposition t2 is true. Proposition s2 is true. Proposition x0a is true if proposition x0b is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition t3 is true if proposition t4 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t4 is true if proposition t5 is true. Proposition t5 is true if proposition qcore is true. Proposition x1a is true if proposition x1b is not true. Proposition etrue is true. Proposition x1b is true if proposition x1a is not true. Proposition s1 is true. Proposition t2 is true if proposition t3 is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true. Proposition s0 is true. Proposition s3 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n2-i0-closed_world::closed_world", "rec_id": "hv2-combo-conj_n2-i0", "axis": "combo", "difficulty": "conj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition x0b is true if proposition x0a is not true. Proposition t1 is true if proposition t2 is true. Proposition s2 is true. Proposition x0a is true if proposition x0b is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition t3 is true if proposition t4 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t4 is true if proposition t5 is true. Proposition t5 is true if proposition qcore is true. Proposition x1a is true if proposition x1b is not true. Proposition etrue is true. Proposition x1b is true if proposition x1a is not true. Proposition s1 is true. Proposition t2 is true if proposition t3 is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true. Proposition s0 is true. Proposition s3 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n2-i0-cred::cred", "rec_id": "hv2-combo-conj_n2-i0", "axis": "combo", "difficulty": "conj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition x0b is true if proposition x0a is not true. Proposition t1 is true if proposition t2 is true. Proposition s2 is true. Proposition x0a is true if proposition x0b is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition t3 is true if proposition t4 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t4 is true if proposition t5 is true. Proposition t5 is true if proposition qcore is true. Proposition x1a is true if proposition x1b is not true. Proposition etrue is true. Proposition x1b is true if proposition x1a is not true. Proposition s1 is true. Proposition t2 is true if proposition t3 is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true. Proposition s0 is true. Proposition s3 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n2-i0-skept::skept", "rec_id": "hv2-combo-conj_n2-i0", "axis": "combo", "difficulty": "conj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition x0b is true if proposition x0a is not true. Proposition t1 is true if proposition t2 is true. Proposition s2 is true. Proposition x0a is true if proposition x0b is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition t3 is true if proposition t4 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t4 is true if proposition t5 is true. Proposition t5 is true if proposition qcore is true. Proposition x1a is true if proposition x1b is not true. Proposition etrue is true. Proposition x1b is true if proposition x1a is not true. Proposition s1 is true. Proposition t2 is true if proposition t3 is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true. Proposition s0 is true. Proposition s3 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n2-i0-wfs::wfs", "rec_id": "hv2-combo-conj_n2-i0", "axis": "combo", "difficulty": "conj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition x0b is true if proposition x0a is not true. Proposition t1 is true if proposition t2 is true. Proposition s2 is true. Proposition x0a is true if proposition x0b is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition t3 is true if proposition t4 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t4 is true if proposition t5 is true. Proposition t5 is true if proposition qcore is true. Proposition x1a is true if proposition x1b is not true. Proposition etrue is true. Proposition x1b is true if proposition x1a is not true. Proposition s1 is true. Proposition t2 is true if proposition t3 is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true. Proposition s0 is true. Proposition s3 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n2-i1-none::none", "rec_id": "hv2-combo-conj_n2-i1", "axis": "combo", "difficulty": "conj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition x0b is true if proposition x0a is not true. Proposition x1b is true if proposition x1a is not true. Proposition x1a is true if proposition x1b is not true. Proposition q is true if proposition qcore is true and proposition etrue is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true. Proposition etrue is true. Proposition x0a is true if proposition x0b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n2-i1-closed_world::closed_world", "rec_id": "hv2-combo-conj_n2-i1", "axis": "combo", "difficulty": "conj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition x0b is true if proposition x0a is not true. Proposition x1b is true if proposition x1a is not true. Proposition x1a is true if proposition x1b is not true. Proposition q is true if proposition qcore is true and proposition etrue is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true. Proposition etrue is true. Proposition x0a is true if proposition x0b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n2-i1-cred::cred", "rec_id": "hv2-combo-conj_n2-i1", "axis": "combo", "difficulty": "conj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition x0b is true if proposition x0a is not true. Proposition x1b is true if proposition x1a is not true. Proposition x1a is true if proposition x1b is not true. Proposition q is true if proposition qcore is true and proposition etrue is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true. Proposition etrue is true. Proposition x0a is true if proposition x0b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n2-i1-skept::skept", "rec_id": "hv2-combo-conj_n2-i1", "axis": "combo", "difficulty": "conj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition x0b is true if proposition x0a is not true. Proposition x1b is true if proposition x1a is not true. Proposition x1a is true if proposition x1b is not true. Proposition q is true if proposition qcore is true and proposition etrue is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true. Proposition etrue is true. Proposition x0a is true if proposition x0b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n2-i1-wfs::wfs", "rec_id": "hv2-combo-conj_n2-i1", "axis": "combo", "difficulty": "conj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition x0b is true if proposition x0a is not true. Proposition x1b is true if proposition x1a is not true. Proposition x1a is true if proposition x1b is not true. Proposition q is true if proposition qcore is true and proposition etrue is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true. Proposition etrue is true. Proposition x0a is true if proposition x0b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n2-i2-none::none", "rec_id": "hv2-combo-conj_n2-i2", "axis": "combo", "difficulty": "conj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition qcore is true if proposition x0a is true and proposition x1a is true. Proposition x0b is true if proposition x0a is not true. Proposition x1b is true if proposition x1a is not true. Proposition x0a is true if proposition x0b is not true. Proposition q is true if proposition qcore is true. Proposition x1a is true if proposition x1b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n2-i2-closed_world::closed_world", "rec_id": "hv2-combo-conj_n2-i2", "axis": "combo", "difficulty": "conj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition qcore is true if proposition x0a is true and proposition x1a is true. Proposition x0b is true if proposition x0a is not true. Proposition x1b is true if proposition x1a is not true. Proposition x0a is true if proposition x0b is not true. Proposition q is true if proposition qcore is true. Proposition x1a is true if proposition x1b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n2-i2-cred::cred", "rec_id": "hv2-combo-conj_n2-i2", "axis": "combo", "difficulty": "conj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition qcore is true if proposition x0a is true and proposition x1a is true. Proposition x0b is true if proposition x0a is not true. Proposition x1b is true if proposition x1a is not true. Proposition x0a is true if proposition x0b is not true. Proposition q is true if proposition qcore is true. Proposition x1a is true if proposition x1b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n2-i2-skept::skept", "rec_id": "hv2-combo-conj_n2-i2", "axis": "combo", "difficulty": "conj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition qcore is true if proposition x0a is true and proposition x1a is true. Proposition x0b is true if proposition x0a is not true. Proposition x1b is true if proposition x1a is not true. Proposition x0a is true if proposition x0b is not true. Proposition q is true if proposition qcore is true. Proposition x1a is true if proposition x1b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n2-i2-wfs::wfs", "rec_id": "hv2-combo-conj_n2-i2", "axis": "combo", "difficulty": "conj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition qcore is true if proposition x0a is true and proposition x1a is true. Proposition x0b is true if proposition x0a is not true. Proposition x1b is true if proposition x1a is not true. Proposition x0a is true if proposition x0b is not true. Proposition q is true if proposition qcore is true. Proposition x1a is true if proposition x1b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n2-i3-none::none", "rec_id": "hv2-combo-conj_n2-i3", "axis": "combo", "difficulty": "conj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition x0b is true if proposition x0a is not true. Proposition t1 is true if proposition qcore is true. Proposition t0 is true if proposition t1 is true. Proposition x0a is true if proposition x0b is not true. Proposition x1b is true if proposition x1a is not true. Proposition q is true if proposition t0 is true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n2-i3-closed_world::closed_world", "rec_id": "hv2-combo-conj_n2-i3", "axis": "combo", "difficulty": "conj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition x0b is true if proposition x0a is not true. Proposition t1 is true if proposition qcore is true. Proposition t0 is true if proposition t1 is true. Proposition x0a is true if proposition x0b is not true. Proposition x1b is true if proposition x1a is not true. Proposition q is true if proposition t0 is true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n2-i3-cred::cred", "rec_id": "hv2-combo-conj_n2-i3", "axis": "combo", "difficulty": "conj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition x0b is true if proposition x0a is not true. Proposition t1 is true if proposition qcore is true. Proposition t0 is true if proposition t1 is true. Proposition x0a is true if proposition x0b is not true. Proposition x1b is true if proposition x1a is not true. Proposition q is true if proposition t0 is true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n2-i3-skept::skept", "rec_id": "hv2-combo-conj_n2-i3", "axis": "combo", "difficulty": "conj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition x0b is true if proposition x0a is not true. Proposition t1 is true if proposition qcore is true. Proposition t0 is true if proposition t1 is true. Proposition x0a is true if proposition x0b is not true. Proposition x1b is true if proposition x1a is not true. Proposition q is true if proposition t0 is true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n2-i3-wfs::wfs", "rec_id": "hv2-combo-conj_n2-i3", "axis": "combo", "difficulty": "conj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition x0b is true if proposition x0a is not true. Proposition t1 is true if proposition qcore is true. Proposition t0 is true if proposition t1 is true. Proposition x0a is true if proposition x0b is not true. Proposition x1b is true if proposition x1a is not true. Proposition q is true if proposition t0 is true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n2-i4-none::none", "rec_id": "hv2-combo-conj_n2-i4", "axis": "combo", "difficulty": "conj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition qcore is true if proposition x0a is true and proposition x1a is true. Proposition q is true if proposition qcore is true and proposition wide is true and proposition etrue is true. Proposition x0a is true if proposition x0b is not true. Proposition s1 is true. Proposition x1a is true if proposition x1b is not true. Proposition s2 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition x1b is true if proposition x1a is not true. Proposition x0b is true if proposition x0a is not true. Proposition s3 is true. Proposition etrue is true. Proposition s0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n2-i4-closed_world::closed_world", "rec_id": "hv2-combo-conj_n2-i4", "axis": "combo", "difficulty": "conj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition qcore is true if proposition x0a is true and proposition x1a is true. Proposition q is true if proposition qcore is true and proposition wide is true and proposition etrue is true. Proposition x0a is true if proposition x0b is not true. Proposition s1 is true. Proposition x1a is true if proposition x1b is not true. Proposition s2 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition x1b is true if proposition x1a is not true. Proposition x0b is true if proposition x0a is not true. Proposition s3 is true. Proposition etrue is true. Proposition s0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n2-i4-cred::cred", "rec_id": "hv2-combo-conj_n2-i4", "axis": "combo", "difficulty": "conj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition qcore is true if proposition x0a is true and proposition x1a is true. Proposition q is true if proposition qcore is true and proposition wide is true and proposition etrue is true. Proposition x0a is true if proposition x0b is not true. Proposition s1 is true. Proposition x1a is true if proposition x1b is not true. Proposition s2 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition x1b is true if proposition x1a is not true. Proposition x0b is true if proposition x0a is not true. Proposition s3 is true. Proposition etrue is true. Proposition s0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n2-i4-skept::skept", "rec_id": "hv2-combo-conj_n2-i4", "axis": "combo", "difficulty": "conj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition qcore is true if proposition x0a is true and proposition x1a is true. Proposition q is true if proposition qcore is true and proposition wide is true and proposition etrue is true. Proposition x0a is true if proposition x0b is not true. Proposition s1 is true. Proposition x1a is true if proposition x1b is not true. Proposition s2 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition x1b is true if proposition x1a is not true. Proposition x0b is true if proposition x0a is not true. Proposition s3 is true. Proposition etrue is true. Proposition s0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n2-i4-wfs::wfs", "rec_id": "hv2-combo-conj_n2-i4", "axis": "combo", "difficulty": "conj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition qcore is true if proposition x0a is true and proposition x1a is true. Proposition q is true if proposition qcore is true and proposition wide is true and proposition etrue is true. Proposition x0a is true if proposition x0b is not true. Proposition s1 is true. Proposition x1a is true if proposition x1b is not true. Proposition s2 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition x1b is true if proposition x1a is not true. Proposition x0b is true if proposition x0a is not true. Proposition s3 is true. Proposition etrue is true. Proposition s0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n2-i5-none::none", "rec_id": "hv2-combo-conj_n2-i5", "axis": "combo", "difficulty": "conj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition x1a is true if proposition x1b is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x1b is true if proposition x1a is not true. Proposition x0a is true if proposition x0b is not true. Proposition q is true if proposition qcore is true and proposition wide is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true. Proposition x0b is true if proposition x0a is not true. Proposition s0 is true. Proposition s1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n2-i5-closed_world::closed_world", "rec_id": "hv2-combo-conj_n2-i5", "axis": "combo", "difficulty": "conj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition x1a is true if proposition x1b is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x1b is true if proposition x1a is not true. Proposition x0a is true if proposition x0b is not true. Proposition q is true if proposition qcore is true and proposition wide is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true. Proposition x0b is true if proposition x0a is not true. Proposition s0 is true. Proposition s1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n2-i5-cred::cred", "rec_id": "hv2-combo-conj_n2-i5", "axis": "combo", "difficulty": "conj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition x1a is true if proposition x1b is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x1b is true if proposition x1a is not true. Proposition x0a is true if proposition x0b is not true. Proposition q is true if proposition qcore is true and proposition wide is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true. Proposition x0b is true if proposition x0a is not true. Proposition s0 is true. Proposition s1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n2-i5-skept::skept", "rec_id": "hv2-combo-conj_n2-i5", "axis": "combo", "difficulty": "conj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition x1a is true if proposition x1b is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x1b is true if proposition x1a is not true. Proposition x0a is true if proposition x0b is not true. Proposition q is true if proposition qcore is true and proposition wide is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true. Proposition x0b is true if proposition x0a is not true. Proposition s0 is true. Proposition s1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n2-i5-wfs::wfs", "rec_id": "hv2-combo-conj_n2-i5", "axis": "combo", "difficulty": "conj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition x1a is true if proposition x1b is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x1b is true if proposition x1a is not true. Proposition x0a is true if proposition x0b is not true. Proposition q is true if proposition qcore is true and proposition wide is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true. Proposition x0b is true if proposition x0a is not true. Proposition s0 is true. Proposition s1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n2-i6-none::none", "rec_id": "hv2-combo-conj_n2-i6", "axis": "combo", "difficulty": "conj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition t4 is true if proposition t5 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true. Proposition t5 is true if proposition qcore is true. Proposition x1b is true if proposition x1a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x1a is true if proposition x1b is not true. Proposition s0 is true. Proposition t1 is true if proposition t2 is true. Proposition s1 is true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition t4 is true. Proposition x0a is true if proposition x0b is not true. Proposition x0b is true if proposition x0a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n2-i6-closed_world::closed_world", "rec_id": "hv2-combo-conj_n2-i6", "axis": "combo", "difficulty": "conj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition t4 is true if proposition t5 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true. Proposition t5 is true if proposition qcore is true. Proposition x1b is true if proposition x1a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x1a is true if proposition x1b is not true. Proposition s0 is true. Proposition t1 is true if proposition t2 is true. Proposition s1 is true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition t4 is true. Proposition x0a is true if proposition x0b is not true. Proposition x0b is true if proposition x0a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n2-i6-cred::cred", "rec_id": "hv2-combo-conj_n2-i6", "axis": "combo", "difficulty": "conj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition t4 is true if proposition t5 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true. Proposition t5 is true if proposition qcore is true. Proposition x1b is true if proposition x1a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x1a is true if proposition x1b is not true. Proposition s0 is true. Proposition t1 is true if proposition t2 is true. Proposition s1 is true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition t4 is true. Proposition x0a is true if proposition x0b is not true. Proposition x0b is true if proposition x0a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n2-i6-skept::skept", "rec_id": "hv2-combo-conj_n2-i6", "axis": "combo", "difficulty": "conj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition t4 is true if proposition t5 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true. Proposition t5 is true if proposition qcore is true. Proposition x1b is true if proposition x1a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x1a is true if proposition x1b is not true. Proposition s0 is true. Proposition t1 is true if proposition t2 is true. Proposition s1 is true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition t4 is true. Proposition x0a is true if proposition x0b is not true. Proposition x0b is true if proposition x0a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n2-i6-wfs::wfs", "rec_id": "hv2-combo-conj_n2-i6", "axis": "combo", "difficulty": "conj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition t4 is true if proposition t5 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true. Proposition t5 is true if proposition qcore is true. Proposition x1b is true if proposition x1a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x1a is true if proposition x1b is not true. Proposition s0 is true. Proposition t1 is true if proposition t2 is true. Proposition s1 is true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition t4 is true. Proposition x0a is true if proposition x0b is not true. Proposition x0b is true if proposition x0a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n2-i7-none::none", "rec_id": "hv2-combo-conj_n2-i7", "axis": "combo", "difficulty": "conj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition x1b is true if proposition x1a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition t1 is true if proposition qcore is true. Proposition x0b is true if proposition x0a is not true. Proposition t0 is true if proposition t1 is true. Proposition s2 is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true. Proposition x1a is true if proposition x1b is not true. Proposition s0 is true. Proposition x0a is true if proposition x0b is not true. Proposition s1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n2-i7-closed_world::closed_world", "rec_id": "hv2-combo-conj_n2-i7", "axis": "combo", "difficulty": "conj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition x1b is true if proposition x1a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition t1 is true if proposition qcore is true. Proposition x0b is true if proposition x0a is not true. Proposition t0 is true if proposition t1 is true. Proposition s2 is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true. Proposition x1a is true if proposition x1b is not true. Proposition s0 is true. Proposition x0a is true if proposition x0b is not true. Proposition s1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n2-i7-cred::cred", "rec_id": "hv2-combo-conj_n2-i7", "axis": "combo", "difficulty": "conj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition x1b is true if proposition x1a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition t1 is true if proposition qcore is true. Proposition x0b is true if proposition x0a is not true. Proposition t0 is true if proposition t1 is true. Proposition s2 is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true. Proposition x1a is true if proposition x1b is not true. Proposition s0 is true. Proposition x0a is true if proposition x0b is not true. Proposition s1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n2-i7-skept::skept", "rec_id": "hv2-combo-conj_n2-i7", "axis": "combo", "difficulty": "conj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition x1b is true if proposition x1a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition t1 is true if proposition qcore is true. Proposition x0b is true if proposition x0a is not true. Proposition t0 is true if proposition t1 is true. Proposition s2 is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true. Proposition x1a is true if proposition x1b is not true. Proposition s0 is true. Proposition x0a is true if proposition x0b is not true. Proposition s1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n2-i7-wfs::wfs", "rec_id": "hv2-combo-conj_n2-i7", "axis": "combo", "difficulty": "conj_n2", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition x1b is true if proposition x1a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition t1 is true if proposition qcore is true. Proposition x0b is true if proposition x0a is not true. Proposition t0 is true if proposition t1 is true. Proposition s2 is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true. Proposition x1a is true if proposition x1b is not true. Proposition s0 is true. Proposition x0a is true if proposition x0b is not true. Proposition s1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n3-i0-none::none", "rec_id": "hv2-combo-conj_n3-i0", "axis": "combo", "difficulty": "conj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition x0a is true if proposition x0b is not true. Proposition s1 is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true. Proposition x0b is true if proposition x0a is not true. Proposition x2b is true if proposition x2a is not true. Proposition t1 is true if proposition t2 is true. Proposition t2 is true if proposition t3 is true. Proposition x1b is true if proposition x1a is not true. Proposition x2a is true if proposition x2b is not true. Proposition s0 is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition etrue is true. Proposition t3 is true if proposition qcore is true. Proposition x1a is true if proposition x1b is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n3-i0-closed_world::closed_world", "rec_id": "hv2-combo-conj_n3-i0", "axis": "combo", "difficulty": "conj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition x0a is true if proposition x0b is not true. Proposition s1 is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true. Proposition x0b is true if proposition x0a is not true. Proposition x2b is true if proposition x2a is not true. Proposition t1 is true if proposition t2 is true. Proposition t2 is true if proposition t3 is true. Proposition x1b is true if proposition x1a is not true. Proposition x2a is true if proposition x2b is not true. Proposition s0 is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition etrue is true. Proposition t3 is true if proposition qcore is true. Proposition x1a is true if proposition x1b is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n3-i0-cred::cred", "rec_id": "hv2-combo-conj_n3-i0", "axis": "combo", "difficulty": "conj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition x0a is true if proposition x0b is not true. Proposition s1 is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true. Proposition x0b is true if proposition x0a is not true. Proposition x2b is true if proposition x2a is not true. Proposition t1 is true if proposition t2 is true. Proposition t2 is true if proposition t3 is true. Proposition x1b is true if proposition x1a is not true. Proposition x2a is true if proposition x2b is not true. Proposition s0 is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition etrue is true. Proposition t3 is true if proposition qcore is true. Proposition x1a is true if proposition x1b is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n3-i0-skept::skept", "rec_id": "hv2-combo-conj_n3-i0", "axis": "combo", "difficulty": "conj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition x0a is true if proposition x0b is not true. Proposition s1 is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true. Proposition x0b is true if proposition x0a is not true. Proposition x2b is true if proposition x2a is not true. Proposition t1 is true if proposition t2 is true. Proposition t2 is true if proposition t3 is true. Proposition x1b is true if proposition x1a is not true. Proposition x2a is true if proposition x2b is not true. Proposition s0 is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition etrue is true. Proposition t3 is true if proposition qcore is true. Proposition x1a is true if proposition x1b is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n3-i0-wfs::wfs", "rec_id": "hv2-combo-conj_n3-i0", "axis": "combo", "difficulty": "conj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition x0a is true if proposition x0b is not true. Proposition s1 is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true. Proposition x0b is true if proposition x0a is not true. Proposition x2b is true if proposition x2a is not true. Proposition t1 is true if proposition t2 is true. Proposition t2 is true if proposition t3 is true. Proposition x1b is true if proposition x1a is not true. Proposition x2a is true if proposition x2b is not true. Proposition s0 is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition etrue is true. Proposition t3 is true if proposition qcore is true. Proposition x1a is true if proposition x1b is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n3-i1-none::none", "rec_id": "hv2-combo-conj_n3-i1", "axis": "combo", "difficulty": "conj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition x0b is true if proposition x0a is not true. Proposition s1 is true. Proposition x0a is true if proposition x0b is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x1a is true if proposition x1b is not true. Proposition s0 is true. Proposition s2 is true. Proposition t1 is true if proposition qcore is true. Proposition x2a is true if proposition x2b is not true. Proposition x1b is true if proposition x1a is not true. Proposition t0 is true if proposition t1 is true. Proposition x2b is true if proposition x2a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n3-i1-closed_world::closed_world", "rec_id": "hv2-combo-conj_n3-i1", "axis": "combo", "difficulty": "conj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition x0b is true if proposition x0a is not true. Proposition s1 is true. Proposition x0a is true if proposition x0b is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x1a is true if proposition x1b is not true. Proposition s0 is true. Proposition s2 is true. Proposition t1 is true if proposition qcore is true. Proposition x2a is true if proposition x2b is not true. Proposition x1b is true if proposition x1a is not true. Proposition t0 is true if proposition t1 is true. Proposition x2b is true if proposition x2a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n3-i1-cred::cred", "rec_id": "hv2-combo-conj_n3-i1", "axis": "combo", "difficulty": "conj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition x0b is true if proposition x0a is not true. Proposition s1 is true. Proposition x0a is true if proposition x0b is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x1a is true if proposition x1b is not true. Proposition s0 is true. Proposition s2 is true. Proposition t1 is true if proposition qcore is true. Proposition x2a is true if proposition x2b is not true. Proposition x1b is true if proposition x1a is not true. Proposition t0 is true if proposition t1 is true. Proposition x2b is true if proposition x2a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n3-i1-skept::skept", "rec_id": "hv2-combo-conj_n3-i1", "axis": "combo", "difficulty": "conj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition x0b is true if proposition x0a is not true. Proposition s1 is true. Proposition x0a is true if proposition x0b is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x1a is true if proposition x1b is not true. Proposition s0 is true. Proposition s2 is true. Proposition t1 is true if proposition qcore is true. Proposition x2a is true if proposition x2b is not true. Proposition x1b is true if proposition x1a is not true. Proposition t0 is true if proposition t1 is true. Proposition x2b is true if proposition x2a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n3-i1-wfs::wfs", "rec_id": "hv2-combo-conj_n3-i1", "axis": "combo", "difficulty": "conj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition x0b is true if proposition x0a is not true. Proposition s1 is true. Proposition x0a is true if proposition x0b is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x1a is true if proposition x1b is not true. Proposition s0 is true. Proposition s2 is true. Proposition t1 is true if proposition qcore is true. Proposition x2a is true if proposition x2b is not true. Proposition x1b is true if proposition x1a is not true. Proposition t0 is true if proposition t1 is true. Proposition x2b is true if proposition x2a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n3-i2-none::none", "rec_id": "hv2-combo-conj_n3-i2", "axis": "combo", "difficulty": "conj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition x1b is true if proposition x1a is not true. Proposition x2b is true if proposition x2a is not true. Proposition t4 is true if proposition t5 is true. Proposition t2 is true if proposition t3 is true. Proposition x2a is true if proposition x2b is not true. Proposition s1 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x1a is true if proposition x1b is not true. Proposition t1 is true if proposition t2 is true. Proposition t5 is true if proposition qcore is true. Proposition s0 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t3 is true if proposition t4 is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true. Proposition t0 is true if proposition t1 is true. Proposition x0b is true if proposition x0a is not true. Proposition x0a is true if proposition x0b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n3-i2-closed_world::closed_world", "rec_id": "hv2-combo-conj_n3-i2", "axis": "combo", "difficulty": "conj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition x1b is true if proposition x1a is not true. Proposition x2b is true if proposition x2a is not true. Proposition t4 is true if proposition t5 is true. Proposition t2 is true if proposition t3 is true. Proposition x2a is true if proposition x2b is not true. Proposition s1 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x1a is true if proposition x1b is not true. Proposition t1 is true if proposition t2 is true. Proposition t5 is true if proposition qcore is true. Proposition s0 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t3 is true if proposition t4 is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true. Proposition t0 is true if proposition t1 is true. Proposition x0b is true if proposition x0a is not true. Proposition x0a is true if proposition x0b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n3-i2-cred::cred", "rec_id": "hv2-combo-conj_n3-i2", "axis": "combo", "difficulty": "conj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition x1b is true if proposition x1a is not true. Proposition x2b is true if proposition x2a is not true. Proposition t4 is true if proposition t5 is true. Proposition t2 is true if proposition t3 is true. Proposition x2a is true if proposition x2b is not true. Proposition s1 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x1a is true if proposition x1b is not true. Proposition t1 is true if proposition t2 is true. Proposition t5 is true if proposition qcore is true. Proposition s0 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t3 is true if proposition t4 is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true. Proposition t0 is true if proposition t1 is true. Proposition x0b is true if proposition x0a is not true. Proposition x0a is true if proposition x0b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n3-i2-skept::skept", "rec_id": "hv2-combo-conj_n3-i2", "axis": "combo", "difficulty": "conj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition x1b is true if proposition x1a is not true. Proposition x2b is true if proposition x2a is not true. Proposition t4 is true if proposition t5 is true. Proposition t2 is true if proposition t3 is true. Proposition x2a is true if proposition x2b is not true. Proposition s1 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x1a is true if proposition x1b is not true. Proposition t1 is true if proposition t2 is true. Proposition t5 is true if proposition qcore is true. Proposition s0 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t3 is true if proposition t4 is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true. Proposition t0 is true if proposition t1 is true. Proposition x0b is true if proposition x0a is not true. Proposition x0a is true if proposition x0b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n3-i2-wfs::wfs", "rec_id": "hv2-combo-conj_n3-i2", "axis": "combo", "difficulty": "conj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition x1b is true if proposition x1a is not true. Proposition x2b is true if proposition x2a is not true. Proposition t4 is true if proposition t5 is true. Proposition t2 is true if proposition t3 is true. Proposition x2a is true if proposition x2b is not true. Proposition s1 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x1a is true if proposition x1b is not true. Proposition t1 is true if proposition t2 is true. Proposition t5 is true if proposition qcore is true. Proposition s0 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t3 is true if proposition t4 is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true. Proposition t0 is true if proposition t1 is true. Proposition x0b is true if proposition x0a is not true. Proposition x0a is true if proposition x0b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n3-i3-none::none", "rec_id": "hv2-combo-conj_n3-i3", "axis": "combo", "difficulty": "conj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition x2a is true if proposition x2b is not true. Proposition s2 is true. Proposition s3 is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true. Proposition t2 is true if proposition t3 is true. Proposition t0 is true if proposition t1 is true. Proposition s0 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition x1a is true if proposition x1b is not true. Proposition s1 is true. Proposition x1b is true if proposition x1a is not true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x0a is true if proposition x0b is not true. Proposition t1 is true if proposition t2 is true. Proposition x2b is true if proposition x2a is not true. Proposition t3 is true if proposition qcore is true. Proposition x0b is true if proposition x0a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n3-i3-closed_world::closed_world", "rec_id": "hv2-combo-conj_n3-i3", "axis": "combo", "difficulty": "conj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition x2a is true if proposition x2b is not true. Proposition s2 is true. Proposition s3 is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true. Proposition t2 is true if proposition t3 is true. Proposition t0 is true if proposition t1 is true. Proposition s0 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition x1a is true if proposition x1b is not true. Proposition s1 is true. Proposition x1b is true if proposition x1a is not true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x0a is true if proposition x0b is not true. Proposition t1 is true if proposition t2 is true. Proposition x2b is true if proposition x2a is not true. Proposition t3 is true if proposition qcore is true. Proposition x0b is true if proposition x0a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n3-i3-cred::cred", "rec_id": "hv2-combo-conj_n3-i3", "axis": "combo", "difficulty": "conj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition x2a is true if proposition x2b is not true. Proposition s2 is true. Proposition s3 is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true. Proposition t2 is true if proposition t3 is true. Proposition t0 is true if proposition t1 is true. Proposition s0 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition x1a is true if proposition x1b is not true. Proposition s1 is true. Proposition x1b is true if proposition x1a is not true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x0a is true if proposition x0b is not true. Proposition t1 is true if proposition t2 is true. Proposition x2b is true if proposition x2a is not true. Proposition t3 is true if proposition qcore is true. Proposition x0b is true if proposition x0a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n3-i3-skept::skept", "rec_id": "hv2-combo-conj_n3-i3", "axis": "combo", "difficulty": "conj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition x2a is true if proposition x2b is not true. Proposition s2 is true. Proposition s3 is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true. Proposition t2 is true if proposition t3 is true. Proposition t0 is true if proposition t1 is true. Proposition s0 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition x1a is true if proposition x1b is not true. Proposition s1 is true. Proposition x1b is true if proposition x1a is not true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x0a is true if proposition x0b is not true. Proposition t1 is true if proposition t2 is true. Proposition x2b is true if proposition x2a is not true. Proposition t3 is true if proposition qcore is true. Proposition x0b is true if proposition x0a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n3-i3-wfs::wfs", "rec_id": "hv2-combo-conj_n3-i3", "axis": "combo", "difficulty": "conj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition x2a is true if proposition x2b is not true. Proposition s2 is true. Proposition s3 is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true. Proposition t2 is true if proposition t3 is true. Proposition t0 is true if proposition t1 is true. Proposition s0 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition x1a is true if proposition x1b is not true. Proposition s1 is true. Proposition x1b is true if proposition x1a is not true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x0a is true if proposition x0b is not true. Proposition t1 is true if proposition t2 is true. Proposition x2b is true if proposition x2a is not true. Proposition t3 is true if proposition qcore is true. Proposition x0b is true if proposition x0a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n3-i4-none::none", "rec_id": "hv2-combo-conj_n3-i4", "axis": "combo", "difficulty": "conj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true. Proposition x2a is true if proposition x2b is not true. Proposition x1b is true if proposition x1a is not true. Proposition s2 is true. Proposition s0 is true. Proposition s1 is true. Proposition x2b is true if proposition x2a is not true. Proposition etrue is true. Proposition x0b is true if proposition x0a is not true. Proposition x0a is true if proposition x0b is not true. Proposition s3 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition q is true if proposition qcore is true and proposition wide is true and proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n3-i4-closed_world::closed_world", "rec_id": "hv2-combo-conj_n3-i4", "axis": "combo", "difficulty": "conj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true. Proposition x2a is true if proposition x2b is not true. Proposition x1b is true if proposition x1a is not true. Proposition s2 is true. Proposition s0 is true. Proposition s1 is true. Proposition x2b is true if proposition x2a is not true. Proposition etrue is true. Proposition x0b is true if proposition x0a is not true. Proposition x0a is true if proposition x0b is not true. Proposition s3 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition q is true if proposition qcore is true and proposition wide is true and proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n3-i4-cred::cred", "rec_id": "hv2-combo-conj_n3-i4", "axis": "combo", "difficulty": "conj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true. Proposition x2a is true if proposition x2b is not true. Proposition x1b is true if proposition x1a is not true. Proposition s2 is true. Proposition s0 is true. Proposition s1 is true. Proposition x2b is true if proposition x2a is not true. Proposition etrue is true. Proposition x0b is true if proposition x0a is not true. Proposition x0a is true if proposition x0b is not true. Proposition s3 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition q is true if proposition qcore is true and proposition wide is true and proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n3-i4-skept::skept", "rec_id": "hv2-combo-conj_n3-i4", "axis": "combo", "difficulty": "conj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true. Proposition x2a is true if proposition x2b is not true. Proposition x1b is true if proposition x1a is not true. Proposition s2 is true. Proposition s0 is true. Proposition s1 is true. Proposition x2b is true if proposition x2a is not true. Proposition etrue is true. Proposition x0b is true if proposition x0a is not true. Proposition x0a is true if proposition x0b is not true. Proposition s3 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition q is true if proposition qcore is true and proposition wide is true and proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n3-i4-wfs::wfs", "rec_id": "hv2-combo-conj_n3-i4", "axis": "combo", "difficulty": "conj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true. Proposition x2a is true if proposition x2b is not true. Proposition x1b is true if proposition x1a is not true. Proposition s2 is true. Proposition s0 is true. Proposition s1 is true. Proposition x2b is true if proposition x2a is not true. Proposition etrue is true. Proposition x0b is true if proposition x0a is not true. Proposition x0a is true if proposition x0b is not true. Proposition s3 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition q is true if proposition qcore is true and proposition wide is true and proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n3-i5-none::none", "rec_id": "hv2-combo-conj_n3-i5", "axis": "combo", "difficulty": "conj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true. Proposition t1 is true if proposition t2 is true. Proposition t3 is true if proposition qcore is true. Proposition t0 is true if proposition t1 is true. Proposition x1a is true if proposition x1b is not true. Proposition x2a is true if proposition x2b is not true. Proposition x2b is true if proposition x2a is not true. Proposition x0a is true if proposition x0b is not true. Proposition t2 is true if proposition t3 is true. Proposition x0b is true if proposition x0a is not true. Proposition x1b is true if proposition x1a is not true. Proposition q is true if proposition t0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n3-i5-closed_world::closed_world", "rec_id": "hv2-combo-conj_n3-i5", "axis": "combo", "difficulty": "conj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true. Proposition t1 is true if proposition t2 is true. Proposition t3 is true if proposition qcore is true. Proposition t0 is true if proposition t1 is true. Proposition x1a is true if proposition x1b is not true. Proposition x2a is true if proposition x2b is not true. Proposition x2b is true if proposition x2a is not true. Proposition x0a is true if proposition x0b is not true. Proposition t2 is true if proposition t3 is true. Proposition x0b is true if proposition x0a is not true. Proposition x1b is true if proposition x1a is not true. Proposition q is true if proposition t0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n3-i5-cred::cred", "rec_id": "hv2-combo-conj_n3-i5", "axis": "combo", "difficulty": "conj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true. Proposition t1 is true if proposition t2 is true. Proposition t3 is true if proposition qcore is true. Proposition t0 is true if proposition t1 is true. Proposition x1a is true if proposition x1b is not true. Proposition x2a is true if proposition x2b is not true. Proposition x2b is true if proposition x2a is not true. Proposition x0a is true if proposition x0b is not true. Proposition t2 is true if proposition t3 is true. Proposition x0b is true if proposition x0a is not true. Proposition x1b is true if proposition x1a is not true. Proposition q is true if proposition t0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n3-i5-skept::skept", "rec_id": "hv2-combo-conj_n3-i5", "axis": "combo", "difficulty": "conj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true. Proposition t1 is true if proposition t2 is true. Proposition t3 is true if proposition qcore is true. Proposition t0 is true if proposition t1 is true. Proposition x1a is true if proposition x1b is not true. Proposition x2a is true if proposition x2b is not true. Proposition x2b is true if proposition x2a is not true. Proposition x0a is true if proposition x0b is not true. Proposition t2 is true if proposition t3 is true. Proposition x0b is true if proposition x0a is not true. Proposition x1b is true if proposition x1a is not true. Proposition q is true if proposition t0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n3-i5-wfs::wfs", "rec_id": "hv2-combo-conj_n3-i5", "axis": "combo", "difficulty": "conj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true. Proposition t1 is true if proposition t2 is true. Proposition t3 is true if proposition qcore is true. Proposition t0 is true if proposition t1 is true. Proposition x1a is true if proposition x1b is not true. Proposition x2a is true if proposition x2b is not true. Proposition x2b is true if proposition x2a is not true. Proposition x0a is true if proposition x0b is not true. Proposition t2 is true if proposition t3 is true. Proposition x0b is true if proposition x0a is not true. Proposition x1b is true if proposition x1a is not true. Proposition q is true if proposition t0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n3-i6-none::none", "rec_id": "hv2-combo-conj_n3-i6", "axis": "combo", "difficulty": "conj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition t2 is true if proposition t3 is true. Proposition t4 is true if proposition t5 is true. Proposition s2 is true. Proposition t3 is true if proposition t4 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition s1 is true. Proposition x1b is true if proposition x1a is not true. Proposition x0a is true if proposition x0b is not true. Proposition t5 is true if proposition qcore is true. Proposition x2b is true if proposition x2a is not true. Proposition x0b is true if proposition x0a is not true. Proposition x2a is true if proposition x2b is not true. Proposition t1 is true if proposition t2 is true. Proposition s0 is true. Proposition etrue is true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true. Proposition t0 is true if proposition t1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n3-i6-closed_world::closed_world", "rec_id": "hv2-combo-conj_n3-i6", "axis": "combo", "difficulty": "conj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition t2 is true if proposition t3 is true. Proposition t4 is true if proposition t5 is true. Proposition s2 is true. Proposition t3 is true if proposition t4 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition s1 is true. Proposition x1b is true if proposition x1a is not true. Proposition x0a is true if proposition x0b is not true. Proposition t5 is true if proposition qcore is true. Proposition x2b is true if proposition x2a is not true. Proposition x0b is true if proposition x0a is not true. Proposition x2a is true if proposition x2b is not true. Proposition t1 is true if proposition t2 is true. Proposition s0 is true. Proposition etrue is true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true. Proposition t0 is true if proposition t1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n3-i6-cred::cred", "rec_id": "hv2-combo-conj_n3-i6", "axis": "combo", "difficulty": "conj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition t2 is true if proposition t3 is true. Proposition t4 is true if proposition t5 is true. Proposition s2 is true. Proposition t3 is true if proposition t4 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition s1 is true. Proposition x1b is true if proposition x1a is not true. Proposition x0a is true if proposition x0b is not true. Proposition t5 is true if proposition qcore is true. Proposition x2b is true if proposition x2a is not true. Proposition x0b is true if proposition x0a is not true. Proposition x2a is true if proposition x2b is not true. Proposition t1 is true if proposition t2 is true. Proposition s0 is true. Proposition etrue is true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true. Proposition t0 is true if proposition t1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n3-i6-skept::skept", "rec_id": "hv2-combo-conj_n3-i6", "axis": "combo", "difficulty": "conj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition t2 is true if proposition t3 is true. Proposition t4 is true if proposition t5 is true. Proposition s2 is true. Proposition t3 is true if proposition t4 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition s1 is true. Proposition x1b is true if proposition x1a is not true. Proposition x0a is true if proposition x0b is not true. Proposition t5 is true if proposition qcore is true. Proposition x2b is true if proposition x2a is not true. Proposition x0b is true if proposition x0a is not true. Proposition x2a is true if proposition x2b is not true. Proposition t1 is true if proposition t2 is true. Proposition s0 is true. Proposition etrue is true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true. Proposition t0 is true if proposition t1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n3-i6-wfs::wfs", "rec_id": "hv2-combo-conj_n3-i6", "axis": "combo", "difficulty": "conj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition t2 is true if proposition t3 is true. Proposition t4 is true if proposition t5 is true. Proposition s2 is true. Proposition t3 is true if proposition t4 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition s1 is true. Proposition x1b is true if proposition x1a is not true. Proposition x0a is true if proposition x0b is not true. Proposition t5 is true if proposition qcore is true. Proposition x2b is true if proposition x2a is not true. Proposition x0b is true if proposition x0a is not true. Proposition x2a is true if proposition x2b is not true. Proposition t1 is true if proposition t2 is true. Proposition s0 is true. Proposition etrue is true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true. Proposition t0 is true if proposition t1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n3-i7-none::none", "rec_id": "hv2-combo-conj_n3-i7", "axis": "combo", "difficulty": "conj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition x0a is true if proposition x0b is not true. Proposition x1b is true if proposition x1a is not true. Proposition x1a is true if proposition x1b is not true. Proposition s0 is true. Proposition x2a is true if proposition x2b is not true. Proposition s1 is true. Proposition t1 is true if proposition qcore is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition etrue is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition x2b is true if proposition x2a is not true. Proposition s2 is true. Proposition t0 is true if proposition t1 is true. Proposition x0b is true if proposition x0a is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n3-i7-closed_world::closed_world", "rec_id": "hv2-combo-conj_n3-i7", "axis": "combo", "difficulty": "conj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition x0a is true if proposition x0b is not true. Proposition x1b is true if proposition x1a is not true. Proposition x1a is true if proposition x1b is not true. Proposition s0 is true. Proposition x2a is true if proposition x2b is not true. Proposition s1 is true. Proposition t1 is true if proposition qcore is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition etrue is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition x2b is true if proposition x2a is not true. Proposition s2 is true. Proposition t0 is true if proposition t1 is true. Proposition x0b is true if proposition x0a is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n3-i7-cred::cred", "rec_id": "hv2-combo-conj_n3-i7", "axis": "combo", "difficulty": "conj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition x0a is true if proposition x0b is not true. Proposition x1b is true if proposition x1a is not true. Proposition x1a is true if proposition x1b is not true. Proposition s0 is true. Proposition x2a is true if proposition x2b is not true. Proposition s1 is true. Proposition t1 is true if proposition qcore is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition etrue is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition x2b is true if proposition x2a is not true. Proposition s2 is true. Proposition t0 is true if proposition t1 is true. Proposition x0b is true if proposition x0a is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n3-i7-skept::skept", "rec_id": "hv2-combo-conj_n3-i7", "axis": "combo", "difficulty": "conj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition x0a is true if proposition x0b is not true. Proposition x1b is true if proposition x1a is not true. Proposition x1a is true if proposition x1b is not true. Proposition s0 is true. Proposition x2a is true if proposition x2b is not true. Proposition s1 is true. Proposition t1 is true if proposition qcore is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition etrue is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition x2b is true if proposition x2a is not true. Proposition s2 is true. Proposition t0 is true if proposition t1 is true. Proposition x0b is true if proposition x0a is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n3-i7-wfs::wfs", "rec_id": "hv2-combo-conj_n3-i7", "axis": "combo", "difficulty": "conj_n3", "n_stable_models": 8, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition x0a is true if proposition x0b is not true. Proposition x1b is true if proposition x1a is not true. Proposition x1a is true if proposition x1b is not true. Proposition s0 is true. Proposition x2a is true if proposition x2b is not true. Proposition s1 is true. Proposition t1 is true if proposition qcore is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition etrue is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition x2b is true if proposition x2a is not true. Proposition s2 is true. Proposition t0 is true if proposition t1 is true. Proposition x0b is true if proposition x0a is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n4-i0-none::none", "rec_id": "hv2-combo-conj_n4-i0", "axis": "combo", "difficulty": "conj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true. Proposition t0 is true if proposition t1 is true. Proposition s2 is true. Proposition x1a is true if proposition x1b is not true. Proposition t5 is true if proposition qcore is true. Proposition t2 is true if proposition t3 is true. Proposition x2a is true if proposition x2b is not true. Proposition x3a is true if proposition x3b is not true. Proposition s0 is true. Proposition x0b is true if proposition x0a is not true. Proposition t4 is true if proposition t5 is true. Proposition x0a is true if proposition x0b is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t3 is true if proposition t4 is true. Proposition x2b is true if proposition x2a is not true. Proposition x1b is true if proposition x1a is not true. Proposition s3 is true. Proposition s1 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition t1 is true if proposition t2 is true. Proposition x3b is true if proposition x3a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n4-i0-closed_world::closed_world", "rec_id": "hv2-combo-conj_n4-i0", "axis": "combo", "difficulty": "conj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true. Proposition t0 is true if proposition t1 is true. Proposition s2 is true. Proposition x1a is true if proposition x1b is not true. Proposition t5 is true if proposition qcore is true. Proposition t2 is true if proposition t3 is true. Proposition x2a is true if proposition x2b is not true. Proposition x3a is true if proposition x3b is not true. Proposition s0 is true. Proposition x0b is true if proposition x0a is not true. Proposition t4 is true if proposition t5 is true. Proposition x0a is true if proposition x0b is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t3 is true if proposition t4 is true. Proposition x2b is true if proposition x2a is not true. Proposition x1b is true if proposition x1a is not true. Proposition s3 is true. Proposition s1 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition t1 is true if proposition t2 is true. Proposition x3b is true if proposition x3a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n4-i0-cred::cred", "rec_id": "hv2-combo-conj_n4-i0", "axis": "combo", "difficulty": "conj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true. Proposition t0 is true if proposition t1 is true. Proposition s2 is true. Proposition x1a is true if proposition x1b is not true. Proposition t5 is true if proposition qcore is true. Proposition t2 is true if proposition t3 is true. Proposition x2a is true if proposition x2b is not true. Proposition x3a is true if proposition x3b is not true. Proposition s0 is true. Proposition x0b is true if proposition x0a is not true. Proposition t4 is true if proposition t5 is true. Proposition x0a is true if proposition x0b is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t3 is true if proposition t4 is true. Proposition x2b is true if proposition x2a is not true. Proposition x1b is true if proposition x1a is not true. Proposition s3 is true. Proposition s1 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition t1 is true if proposition t2 is true. Proposition x3b is true if proposition x3a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n4-i0-skept::skept", "rec_id": "hv2-combo-conj_n4-i0", "axis": "combo", "difficulty": "conj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true. Proposition t0 is true if proposition t1 is true. Proposition s2 is true. Proposition x1a is true if proposition x1b is not true. Proposition t5 is true if proposition qcore is true. Proposition t2 is true if proposition t3 is true. Proposition x2a is true if proposition x2b is not true. Proposition x3a is true if proposition x3b is not true. Proposition s0 is true. Proposition x0b is true if proposition x0a is not true. Proposition t4 is true if proposition t5 is true. Proposition x0a is true if proposition x0b is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t3 is true if proposition t4 is true. Proposition x2b is true if proposition x2a is not true. Proposition x1b is true if proposition x1a is not true. Proposition s3 is true. Proposition s1 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition t1 is true if proposition t2 is true. Proposition x3b is true if proposition x3a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n4-i0-wfs::wfs", "rec_id": "hv2-combo-conj_n4-i0", "axis": "combo", "difficulty": "conj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true. Proposition t0 is true if proposition t1 is true. Proposition s2 is true. Proposition x1a is true if proposition x1b is not true. Proposition t5 is true if proposition qcore is true. Proposition t2 is true if proposition t3 is true. Proposition x2a is true if proposition x2b is not true. Proposition x3a is true if proposition x3b is not true. Proposition s0 is true. Proposition x0b is true if proposition x0a is not true. Proposition t4 is true if proposition t5 is true. Proposition x0a is true if proposition x0b is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t3 is true if proposition t4 is true. Proposition x2b is true if proposition x2a is not true. Proposition x1b is true if proposition x1a is not true. Proposition s3 is true. Proposition s1 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition t1 is true if proposition t2 is true. Proposition x3b is true if proposition x3a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n4-i1-none::none", "rec_id": "hv2-combo-conj_n4-i1", "axis": "combo", "difficulty": "conj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s2 is true. Proposition x2b is true if proposition x2a is not true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true. Proposition s1 is true. Proposition x3a is true if proposition x3b is not true. Proposition x3b is true if proposition x3a is not true. Proposition x2a is true if proposition x2b is not true. Proposition x0b is true if proposition x0a is not true. Proposition q is true if proposition qcore is true and proposition wide is true. Proposition s0 is true. Proposition s3 is true. Proposition x1b is true if proposition x1a is not true. Proposition x0a is true if proposition x0b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n4-i1-closed_world::closed_world", "rec_id": "hv2-combo-conj_n4-i1", "axis": "combo", "difficulty": "conj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s2 is true. Proposition x2b is true if proposition x2a is not true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true. Proposition s1 is true. Proposition x3a is true if proposition x3b is not true. Proposition x3b is true if proposition x3a is not true. Proposition x2a is true if proposition x2b is not true. Proposition x0b is true if proposition x0a is not true. Proposition q is true if proposition qcore is true and proposition wide is true. Proposition s0 is true. Proposition s3 is true. Proposition x1b is true if proposition x1a is not true. Proposition x0a is true if proposition x0b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n4-i1-cred::cred", "rec_id": "hv2-combo-conj_n4-i1", "axis": "combo", "difficulty": "conj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s2 is true. Proposition x2b is true if proposition x2a is not true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true. Proposition s1 is true. Proposition x3a is true if proposition x3b is not true. Proposition x3b is true if proposition x3a is not true. Proposition x2a is true if proposition x2b is not true. Proposition x0b is true if proposition x0a is not true. Proposition q is true if proposition qcore is true and proposition wide is true. Proposition s0 is true. Proposition s3 is true. Proposition x1b is true if proposition x1a is not true. Proposition x0a is true if proposition x0b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n4-i1-skept::skept", "rec_id": "hv2-combo-conj_n4-i1", "axis": "combo", "difficulty": "conj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s2 is true. Proposition x2b is true if proposition x2a is not true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true. Proposition s1 is true. Proposition x3a is true if proposition x3b is not true. Proposition x3b is true if proposition x3a is not true. Proposition x2a is true if proposition x2b is not true. Proposition x0b is true if proposition x0a is not true. Proposition q is true if proposition qcore is true and proposition wide is true. Proposition s0 is true. Proposition s3 is true. Proposition x1b is true if proposition x1a is not true. Proposition x0a is true if proposition x0b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n4-i1-wfs::wfs", "rec_id": "hv2-combo-conj_n4-i1", "axis": "combo", "difficulty": "conj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s2 is true. Proposition x2b is true if proposition x2a is not true. Proposition x1a is true if proposition x1b is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true. Proposition s1 is true. Proposition x3a is true if proposition x3b is not true. Proposition x3b is true if proposition x3a is not true. Proposition x2a is true if proposition x2b is not true. Proposition x0b is true if proposition x0a is not true. Proposition q is true if proposition qcore is true and proposition wide is true. Proposition s0 is true. Proposition s3 is true. Proposition x1b is true if proposition x1a is not true. Proposition x0a is true if proposition x0b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n4-i2-none::none", "rec_id": "hv2-combo-conj_n4-i2", "axis": "combo", "difficulty": "conj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition x1b is true if proposition x1a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x3b is true if proposition x3a is not true. Proposition x1a is true if proposition x1b is not true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x2b is true if proposition x2a is not true. Proposition t1 is true if proposition qcore is true. Proposition x3a is true if proposition x3b is not true. Proposition x2a is true if proposition x2b is not true. Proposition s0 is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true. Proposition s1 is true. Proposition x0b is true if proposition x0a is not true. Proposition t0 is true if proposition t1 is true. Proposition x0a is true if proposition x0b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n4-i2-closed_world::closed_world", "rec_id": "hv2-combo-conj_n4-i2", "axis": "combo", "difficulty": "conj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition x1b is true if proposition x1a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x3b is true if proposition x3a is not true. Proposition x1a is true if proposition x1b is not true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x2b is true if proposition x2a is not true. Proposition t1 is true if proposition qcore is true. Proposition x3a is true if proposition x3b is not true. Proposition x2a is true if proposition x2b is not true. Proposition s0 is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true. Proposition s1 is true. Proposition x0b is true if proposition x0a is not true. Proposition t0 is true if proposition t1 is true. Proposition x0a is true if proposition x0b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n4-i2-cred::cred", "rec_id": "hv2-combo-conj_n4-i2", "axis": "combo", "difficulty": "conj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition x1b is true if proposition x1a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x3b is true if proposition x3a is not true. Proposition x1a is true if proposition x1b is not true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x2b is true if proposition x2a is not true. Proposition t1 is true if proposition qcore is true. Proposition x3a is true if proposition x3b is not true. Proposition x2a is true if proposition x2b is not true. Proposition s0 is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true. Proposition s1 is true. Proposition x0b is true if proposition x0a is not true. Proposition t0 is true if proposition t1 is true. Proposition x0a is true if proposition x0b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n4-i2-skept::skept", "rec_id": "hv2-combo-conj_n4-i2", "axis": "combo", "difficulty": "conj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition x1b is true if proposition x1a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x3b is true if proposition x3a is not true. Proposition x1a is true if proposition x1b is not true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x2b is true if proposition x2a is not true. Proposition t1 is true if proposition qcore is true. Proposition x3a is true if proposition x3b is not true. Proposition x2a is true if proposition x2b is not true. Proposition s0 is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true. Proposition s1 is true. Proposition x0b is true if proposition x0a is not true. Proposition t0 is true if proposition t1 is true. Proposition x0a is true if proposition x0b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n4-i2-wfs::wfs", "rec_id": "hv2-combo-conj_n4-i2", "axis": "combo", "difficulty": "conj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition x1b is true if proposition x1a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x3b is true if proposition x3a is not true. Proposition x1a is true if proposition x1b is not true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x2b is true if proposition x2a is not true. Proposition t1 is true if proposition qcore is true. Proposition x3a is true if proposition x3b is not true. Proposition x2a is true if proposition x2b is not true. Proposition s0 is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true. Proposition s1 is true. Proposition x0b is true if proposition x0a is not true. Proposition t0 is true if proposition t1 is true. Proposition x0a is true if proposition x0b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n4-i3-none::none", "rec_id": "hv2-combo-conj_n4-i3", "axis": "combo", "difficulty": "conj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition x1b is true if proposition x1a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition s1 is true. Proposition x3a is true if proposition x3b is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true. Proposition x3b is true if proposition x3a is not true. Proposition x2b is true if proposition x2a is not true. Proposition q is true if proposition qcore is true and proposition wide is true. Proposition x0b is true if proposition x0a is not true. Proposition s0 is true. Proposition x0a is true if proposition x0b is not true. Proposition x2a is true if proposition x2b is not true. Proposition x1a is true if proposition x1b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n4-i3-closed_world::closed_world", "rec_id": "hv2-combo-conj_n4-i3", "axis": "combo", "difficulty": "conj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition x1b is true if proposition x1a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition s1 is true. Proposition x3a is true if proposition x3b is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true. Proposition x3b is true if proposition x3a is not true. Proposition x2b is true if proposition x2a is not true. Proposition q is true if proposition qcore is true and proposition wide is true. Proposition x0b is true if proposition x0a is not true. Proposition s0 is true. Proposition x0a is true if proposition x0b is not true. Proposition x2a is true if proposition x2b is not true. Proposition x1a is true if proposition x1b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n4-i3-cred::cred", "rec_id": "hv2-combo-conj_n4-i3", "axis": "combo", "difficulty": "conj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition x1b is true if proposition x1a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition s1 is true. Proposition x3a is true if proposition x3b is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true. Proposition x3b is true if proposition x3a is not true. Proposition x2b is true if proposition x2a is not true. Proposition q is true if proposition qcore is true and proposition wide is true. Proposition x0b is true if proposition x0a is not true. Proposition s0 is true. Proposition x0a is true if proposition x0b is not true. Proposition x2a is true if proposition x2b is not true. Proposition x1a is true if proposition x1b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n4-i3-skept::skept", "rec_id": "hv2-combo-conj_n4-i3", "axis": "combo", "difficulty": "conj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition x1b is true if proposition x1a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition s1 is true. Proposition x3a is true if proposition x3b is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true. Proposition x3b is true if proposition x3a is not true. Proposition x2b is true if proposition x2a is not true. Proposition q is true if proposition qcore is true and proposition wide is true. Proposition x0b is true if proposition x0a is not true. Proposition s0 is true. Proposition x0a is true if proposition x0b is not true. Proposition x2a is true if proposition x2b is not true. Proposition x1a is true if proposition x1b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n4-i3-wfs::wfs", "rec_id": "hv2-combo-conj_n4-i3", "axis": "combo", "difficulty": "conj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition x1b is true if proposition x1a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition s1 is true. Proposition x3a is true if proposition x3b is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true. Proposition x3b is true if proposition x3a is not true. Proposition x2b is true if proposition x2a is not true. Proposition q is true if proposition qcore is true and proposition wide is true. Proposition x0b is true if proposition x0a is not true. Proposition s0 is true. Proposition x0a is true if proposition x0b is not true. Proposition x2a is true if proposition x2b is not true. Proposition x1a is true if proposition x1b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n4-i4-none::none", "rec_id": "hv2-combo-conj_n4-i4", "axis": "combo", "difficulty": "conj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition q is true if proposition t0 is true and proposition wide is true. Proposition x0b is true if proposition x0a is not true. Proposition t1 is true if proposition t2 is true. Proposition x1b is true if proposition x1a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t3 is true if proposition qcore is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true. Proposition x3b is true if proposition x3a is not true. Proposition x2b is true if proposition x2a is not true. Proposition t2 is true if proposition t3 is true. Proposition x3a is true if proposition x3b is not true. Proposition x0a is true if proposition x0b is not true. Proposition s0 is true. Proposition s1 is true. Proposition x1a is true if proposition x1b is not true. Proposition x2a is true if proposition x2b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n4-i4-closed_world::closed_world", "rec_id": "hv2-combo-conj_n4-i4", "axis": "combo", "difficulty": "conj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition q is true if proposition t0 is true and proposition wide is true. Proposition x0b is true if proposition x0a is not true. Proposition t1 is true if proposition t2 is true. Proposition x1b is true if proposition x1a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t3 is true if proposition qcore is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true. Proposition x3b is true if proposition x3a is not true. Proposition x2b is true if proposition x2a is not true. Proposition t2 is true if proposition t3 is true. Proposition x3a is true if proposition x3b is not true. Proposition x0a is true if proposition x0b is not true. Proposition s0 is true. Proposition s1 is true. Proposition x1a is true if proposition x1b is not true. Proposition x2a is true if proposition x2b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n4-i4-cred::cred", "rec_id": "hv2-combo-conj_n4-i4", "axis": "combo", "difficulty": "conj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition q is true if proposition t0 is true and proposition wide is true. Proposition x0b is true if proposition x0a is not true. Proposition t1 is true if proposition t2 is true. Proposition x1b is true if proposition x1a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t3 is true if proposition qcore is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true. Proposition x3b is true if proposition x3a is not true. Proposition x2b is true if proposition x2a is not true. Proposition t2 is true if proposition t3 is true. Proposition x3a is true if proposition x3b is not true. Proposition x0a is true if proposition x0b is not true. Proposition s0 is true. Proposition s1 is true. Proposition x1a is true if proposition x1b is not true. Proposition x2a is true if proposition x2b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n4-i4-skept::skept", "rec_id": "hv2-combo-conj_n4-i4", "axis": "combo", "difficulty": "conj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition q is true if proposition t0 is true and proposition wide is true. Proposition x0b is true if proposition x0a is not true. Proposition t1 is true if proposition t2 is true. Proposition x1b is true if proposition x1a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t3 is true if proposition qcore is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true. Proposition x3b is true if proposition x3a is not true. Proposition x2b is true if proposition x2a is not true. Proposition t2 is true if proposition t3 is true. Proposition x3a is true if proposition x3b is not true. Proposition x0a is true if proposition x0b is not true. Proposition s0 is true. Proposition s1 is true. Proposition x1a is true if proposition x1b is not true. Proposition x2a is true if proposition x2b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n4-i4-wfs::wfs", "rec_id": "hv2-combo-conj_n4-i4", "axis": "combo", "difficulty": "conj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition q is true if proposition t0 is true and proposition wide is true. Proposition x0b is true if proposition x0a is not true. Proposition t1 is true if proposition t2 is true. Proposition x1b is true if proposition x1a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t3 is true if proposition qcore is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true. Proposition x3b is true if proposition x3a is not true. Proposition x2b is true if proposition x2a is not true. Proposition t2 is true if proposition t3 is true. Proposition x3a is true if proposition x3b is not true. Proposition x0a is true if proposition x0b is not true. Proposition s0 is true. Proposition s1 is true. Proposition x1a is true if proposition x1b is not true. Proposition x2a is true if proposition x2b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n4-i5-none::none", "rec_id": "hv2-combo-conj_n4-i5", "axis": "combo", "difficulty": "conj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true. Proposition x3b is true if proposition x3a is not true. Proposition x0b is true if proposition x0a is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition x2a is true if proposition x2b is not true. Proposition x2b is true if proposition x2a is not true. Proposition s1 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x1a is true if proposition x1b is not true. Proposition x1b is true if proposition x1a is not true. Proposition s0 is true. Proposition x3a is true if proposition x3b is not true. Proposition t1 is true if proposition qcore is true. Proposition x0a is true if proposition x0b is not true. Proposition t0 is true if proposition t1 is true. Proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n4-i5-closed_world::closed_world", "rec_id": "hv2-combo-conj_n4-i5", "axis": "combo", "difficulty": "conj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true. Proposition x3b is true if proposition x3a is not true. Proposition x0b is true if proposition x0a is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition x2a is true if proposition x2b is not true. Proposition x2b is true if proposition x2a is not true. Proposition s1 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x1a is true if proposition x1b is not true. Proposition x1b is true if proposition x1a is not true. Proposition s0 is true. Proposition x3a is true if proposition x3b is not true. Proposition t1 is true if proposition qcore is true. Proposition x0a is true if proposition x0b is not true. Proposition t0 is true if proposition t1 is true. Proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n4-i5-cred::cred", "rec_id": "hv2-combo-conj_n4-i5", "axis": "combo", "difficulty": "conj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true. Proposition x3b is true if proposition x3a is not true. Proposition x0b is true if proposition x0a is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition x2a is true if proposition x2b is not true. Proposition x2b is true if proposition x2a is not true. Proposition s1 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x1a is true if proposition x1b is not true. Proposition x1b is true if proposition x1a is not true. Proposition s0 is true. Proposition x3a is true if proposition x3b is not true. Proposition t1 is true if proposition qcore is true. Proposition x0a is true if proposition x0b is not true. Proposition t0 is true if proposition t1 is true. Proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n4-i5-skept::skept", "rec_id": "hv2-combo-conj_n4-i5", "axis": "combo", "difficulty": "conj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true. Proposition x3b is true if proposition x3a is not true. Proposition x0b is true if proposition x0a is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition x2a is true if proposition x2b is not true. Proposition x2b is true if proposition x2a is not true. Proposition s1 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x1a is true if proposition x1b is not true. Proposition x1b is true if proposition x1a is not true. Proposition s0 is true. Proposition x3a is true if proposition x3b is not true. Proposition t1 is true if proposition qcore is true. Proposition x0a is true if proposition x0b is not true. Proposition t0 is true if proposition t1 is true. Proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n4-i5-wfs::wfs", "rec_id": "hv2-combo-conj_n4-i5", "axis": "combo", "difficulty": "conj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true. Proposition x3b is true if proposition x3a is not true. Proposition x0b is true if proposition x0a is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition x2a is true if proposition x2b is not true. Proposition x2b is true if proposition x2a is not true. Proposition s1 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x1a is true if proposition x1b is not true. Proposition x1b is true if proposition x1a is not true. Proposition s0 is true. Proposition x3a is true if proposition x3b is not true. Proposition t1 is true if proposition qcore is true. Proposition x0a is true if proposition x0b is not true. Proposition t0 is true if proposition t1 is true. Proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n4-i6-none::none", "rec_id": "hv2-combo-conj_n4-i6", "axis": "combo", "difficulty": "conj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition x2b is true if proposition x2a is not true. Proposition x1a is true if proposition x1b is not true. Proposition x1b is true if proposition x1a is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true. Proposition x2a is true if proposition x2b is not true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition qcore is true. Proposition q is true if proposition t0 is true. Proposition t1 is true if proposition t2 is true. Proposition x3a is true if proposition x3b is not true. Proposition x3b is true if proposition x3a is not true. Proposition t0 is true if proposition t1 is true. Proposition x0a is true if proposition x0b is not true. Proposition x0b is true if proposition x0a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n4-i6-closed_world::closed_world", "rec_id": "hv2-combo-conj_n4-i6", "axis": "combo", "difficulty": "conj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition x2b is true if proposition x2a is not true. Proposition x1a is true if proposition x1b is not true. Proposition x1b is true if proposition x1a is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true. Proposition x2a is true if proposition x2b is not true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition qcore is true. Proposition q is true if proposition t0 is true. Proposition t1 is true if proposition t2 is true. Proposition x3a is true if proposition x3b is not true. Proposition x3b is true if proposition x3a is not true. Proposition t0 is true if proposition t1 is true. Proposition x0a is true if proposition x0b is not true. Proposition x0b is true if proposition x0a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n4-i6-cred::cred", "rec_id": "hv2-combo-conj_n4-i6", "axis": "combo", "difficulty": "conj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition x2b is true if proposition x2a is not true. Proposition x1a is true if proposition x1b is not true. Proposition x1b is true if proposition x1a is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true. Proposition x2a is true if proposition x2b is not true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition qcore is true. Proposition q is true if proposition t0 is true. Proposition t1 is true if proposition t2 is true. Proposition x3a is true if proposition x3b is not true. Proposition x3b is true if proposition x3a is not true. Proposition t0 is true if proposition t1 is true. Proposition x0a is true if proposition x0b is not true. Proposition x0b is true if proposition x0a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n4-i6-skept::skept", "rec_id": "hv2-combo-conj_n4-i6", "axis": "combo", "difficulty": "conj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition x2b is true if proposition x2a is not true. Proposition x1a is true if proposition x1b is not true. Proposition x1b is true if proposition x1a is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true. Proposition x2a is true if proposition x2b is not true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition qcore is true. Proposition q is true if proposition t0 is true. Proposition t1 is true if proposition t2 is true. Proposition x3a is true if proposition x3b is not true. Proposition x3b is true if proposition x3a is not true. Proposition t0 is true if proposition t1 is true. Proposition x0a is true if proposition x0b is not true. Proposition x0b is true if proposition x0a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n4-i6-wfs::wfs", "rec_id": "hv2-combo-conj_n4-i6", "axis": "combo", "difficulty": "conj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition x2b is true if proposition x2a is not true. Proposition x1a is true if proposition x1b is not true. Proposition x1b is true if proposition x1a is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true. Proposition x2a is true if proposition x2b is not true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition qcore is true. Proposition q is true if proposition t0 is true. Proposition t1 is true if proposition t2 is true. Proposition x3a is true if proposition x3b is not true. Proposition x3b is true if proposition x3a is not true. Proposition t0 is true if proposition t1 is true. Proposition x0a is true if proposition x0b is not true. Proposition x0b is true if proposition x0a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n4-i7-none::none", "rec_id": "hv2-combo-conj_n4-i7", "axis": "combo", "difficulty": "conj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition t0 is true if proposition t1 is true. Proposition s1 is true. Proposition x0b is true if proposition x0a is not true. Proposition x1b is true if proposition x1a is not true. Proposition etrue is true. Proposition x3b is true if proposition x3a is not true. Proposition x3a is true if proposition x3b is not true. Proposition x2a is true if proposition x2b is not true. Proposition s0 is true. Proposition s2 is true. Proposition t1 is true if proposition t2 is true. Proposition t3 is true if proposition qcore is true. Proposition x0a is true if proposition x0b is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition s3 is true. Proposition x2b is true if proposition x2a is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t2 is true if proposition t3 is true. Proposition x1a is true if proposition x1b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n4-i7-closed_world::closed_world", "rec_id": "hv2-combo-conj_n4-i7", "axis": "combo", "difficulty": "conj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition t0 is true if proposition t1 is true. Proposition s1 is true. Proposition x0b is true if proposition x0a is not true. Proposition x1b is true if proposition x1a is not true. Proposition etrue is true. Proposition x3b is true if proposition x3a is not true. Proposition x3a is true if proposition x3b is not true. Proposition x2a is true if proposition x2b is not true. Proposition s0 is true. Proposition s2 is true. Proposition t1 is true if proposition t2 is true. Proposition t3 is true if proposition qcore is true. Proposition x0a is true if proposition x0b is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition s3 is true. Proposition x2b is true if proposition x2a is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t2 is true if proposition t3 is true. Proposition x1a is true if proposition x1b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n4-i7-cred::cred", "rec_id": "hv2-combo-conj_n4-i7", "axis": "combo", "difficulty": "conj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition t0 is true if proposition t1 is true. Proposition s1 is true. Proposition x0b is true if proposition x0a is not true. Proposition x1b is true if proposition x1a is not true. Proposition etrue is true. Proposition x3b is true if proposition x3a is not true. Proposition x3a is true if proposition x3b is not true. Proposition x2a is true if proposition x2b is not true. Proposition s0 is true. Proposition s2 is true. Proposition t1 is true if proposition t2 is true. Proposition t3 is true if proposition qcore is true. Proposition x0a is true if proposition x0b is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition s3 is true. Proposition x2b is true if proposition x2a is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t2 is true if proposition t3 is true. Proposition x1a is true if proposition x1b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n4-i7-skept::skept", "rec_id": "hv2-combo-conj_n4-i7", "axis": "combo", "difficulty": "conj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition t0 is true if proposition t1 is true. Proposition s1 is true. Proposition x0b is true if proposition x0a is not true. Proposition x1b is true if proposition x1a is not true. Proposition etrue is true. Proposition x3b is true if proposition x3a is not true. Proposition x3a is true if proposition x3b is not true. Proposition x2a is true if proposition x2b is not true. Proposition s0 is true. Proposition s2 is true. Proposition t1 is true if proposition t2 is true. Proposition t3 is true if proposition qcore is true. Proposition x0a is true if proposition x0b is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition s3 is true. Proposition x2b is true if proposition x2a is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t2 is true if proposition t3 is true. Proposition x1a is true if proposition x1b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n4-i7-wfs::wfs", "rec_id": "hv2-combo-conj_n4-i7", "axis": "combo", "difficulty": "conj_n4", "n_stable_models": 16, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition t0 is true if proposition t1 is true. Proposition s1 is true. Proposition x0b is true if proposition x0a is not true. Proposition x1b is true if proposition x1a is not true. Proposition etrue is true. Proposition x3b is true if proposition x3a is not true. Proposition x3a is true if proposition x3b is not true. Proposition x2a is true if proposition x2b is not true. Proposition s0 is true. Proposition s2 is true. Proposition t1 is true if proposition t2 is true. Proposition t3 is true if proposition qcore is true. Proposition x0a is true if proposition x0b is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition s3 is true. Proposition x2b is true if proposition x2a is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t2 is true if proposition t3 is true. Proposition x1a is true if proposition x1b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n5-i0-none::none", "rec_id": "hv2-combo-conj_n5-i0", "axis": "combo", "difficulty": "conj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition x2b is true if proposition x2a is not true. Proposition t4 is true if proposition t5 is true. Proposition x1a is true if proposition x1b is not true. Proposition s0 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x0a is true if proposition x0b is not true. Proposition t2 is true if proposition t3 is true. Proposition t5 is true if proposition qcore is true. Proposition x3b is true if proposition x3a is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition t1 is true if proposition t2 is true. Proposition x4b is true if proposition x4a is not true. Proposition x2a is true if proposition x2b is not true. Proposition x4a is true if proposition x4b is not true. Proposition x3a is true if proposition x3b is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true. Proposition t3 is true if proposition t4 is true. Proposition x0b is true if proposition x0a is not true. Proposition s1 is true. Proposition t0 is true if proposition t1 is true. Proposition x1b is true if proposition x1a is not true. Proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n5-i0-closed_world::closed_world", "rec_id": "hv2-combo-conj_n5-i0", "axis": "combo", "difficulty": "conj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition x2b is true if proposition x2a is not true. Proposition t4 is true if proposition t5 is true. Proposition x1a is true if proposition x1b is not true. Proposition s0 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x0a is true if proposition x0b is not true. Proposition t2 is true if proposition t3 is true. Proposition t5 is true if proposition qcore is true. Proposition x3b is true if proposition x3a is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition t1 is true if proposition t2 is true. Proposition x4b is true if proposition x4a is not true. Proposition x2a is true if proposition x2b is not true. Proposition x4a is true if proposition x4b is not true. Proposition x3a is true if proposition x3b is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true. Proposition t3 is true if proposition t4 is true. Proposition x0b is true if proposition x0a is not true. Proposition s1 is true. Proposition t0 is true if proposition t1 is true. Proposition x1b is true if proposition x1a is not true. Proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n5-i0-cred::cred", "rec_id": "hv2-combo-conj_n5-i0", "axis": "combo", "difficulty": "conj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition x2b is true if proposition x2a is not true. Proposition t4 is true if proposition t5 is true. Proposition x1a is true if proposition x1b is not true. Proposition s0 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x0a is true if proposition x0b is not true. Proposition t2 is true if proposition t3 is true. Proposition t5 is true if proposition qcore is true. Proposition x3b is true if proposition x3a is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition t1 is true if proposition t2 is true. Proposition x4b is true if proposition x4a is not true. Proposition x2a is true if proposition x2b is not true. Proposition x4a is true if proposition x4b is not true. Proposition x3a is true if proposition x3b is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true. Proposition t3 is true if proposition t4 is true. Proposition x0b is true if proposition x0a is not true. Proposition s1 is true. Proposition t0 is true if proposition t1 is true. Proposition x1b is true if proposition x1a is not true. Proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n5-i0-skept::skept", "rec_id": "hv2-combo-conj_n5-i0", "axis": "combo", "difficulty": "conj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition x2b is true if proposition x2a is not true. Proposition t4 is true if proposition t5 is true. Proposition x1a is true if proposition x1b is not true. Proposition s0 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x0a is true if proposition x0b is not true. Proposition t2 is true if proposition t3 is true. Proposition t5 is true if proposition qcore is true. Proposition x3b is true if proposition x3a is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition t1 is true if proposition t2 is true. Proposition x4b is true if proposition x4a is not true. Proposition x2a is true if proposition x2b is not true. Proposition x4a is true if proposition x4b is not true. Proposition x3a is true if proposition x3b is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true. Proposition t3 is true if proposition t4 is true. Proposition x0b is true if proposition x0a is not true. Proposition s1 is true. Proposition t0 is true if proposition t1 is true. Proposition x1b is true if proposition x1a is not true. Proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n5-i0-wfs::wfs", "rec_id": "hv2-combo-conj_n5-i0", "axis": "combo", "difficulty": "conj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition x2b is true if proposition x2a is not true. Proposition t4 is true if proposition t5 is true. Proposition x1a is true if proposition x1b is not true. Proposition s0 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x0a is true if proposition x0b is not true. Proposition t2 is true if proposition t3 is true. Proposition t5 is true if proposition qcore is true. Proposition x3b is true if proposition x3a is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition t1 is true if proposition t2 is true. Proposition x4b is true if proposition x4a is not true. Proposition x2a is true if proposition x2b is not true. Proposition x4a is true if proposition x4b is not true. Proposition x3a is true if proposition x3b is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true. Proposition t3 is true if proposition t4 is true. Proposition x0b is true if proposition x0a is not true. Proposition s1 is true. Proposition t0 is true if proposition t1 is true. Proposition x1b is true if proposition x1a is not true. Proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n5-i1-none::none", "rec_id": "hv2-combo-conj_n5-i1", "axis": "combo", "difficulty": "conj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition x0a is true if proposition x0b is not true. Proposition x2a is true if proposition x2b is not true. Proposition s2 is true. Proposition q is true if proposition qcore is true and proposition wide is true. Proposition s1 is true. Proposition x2b is true if proposition x2a is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true. Proposition x3b is true if proposition x3a is not true. Proposition x3a is true if proposition x3b is not true. Proposition x4a is true if proposition x4b is not true. Proposition x1b is true if proposition x1a is not true. Proposition x4b is true if proposition x4a is not true. Proposition s0 is true. Proposition x1a is true if proposition x1b is not true. Proposition x0b is true if proposition x0a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n5-i1-closed_world::closed_world", "rec_id": "hv2-combo-conj_n5-i1", "axis": "combo", "difficulty": "conj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition x0a is true if proposition x0b is not true. Proposition x2a is true if proposition x2b is not true. Proposition s2 is true. Proposition q is true if proposition qcore is true and proposition wide is true. Proposition s1 is true. Proposition x2b is true if proposition x2a is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true. Proposition x3b is true if proposition x3a is not true. Proposition x3a is true if proposition x3b is not true. Proposition x4a is true if proposition x4b is not true. Proposition x1b is true if proposition x1a is not true. Proposition x4b is true if proposition x4a is not true. Proposition s0 is true. Proposition x1a is true if proposition x1b is not true. Proposition x0b is true if proposition x0a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n5-i1-cred::cred", "rec_id": "hv2-combo-conj_n5-i1", "axis": "combo", "difficulty": "conj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition x0a is true if proposition x0b is not true. Proposition x2a is true if proposition x2b is not true. Proposition s2 is true. Proposition q is true if proposition qcore is true and proposition wide is true. Proposition s1 is true. Proposition x2b is true if proposition x2a is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true. Proposition x3b is true if proposition x3a is not true. Proposition x3a is true if proposition x3b is not true. Proposition x4a is true if proposition x4b is not true. Proposition x1b is true if proposition x1a is not true. Proposition x4b is true if proposition x4a is not true. Proposition s0 is true. Proposition x1a is true if proposition x1b is not true. Proposition x0b is true if proposition x0a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n5-i1-skept::skept", "rec_id": "hv2-combo-conj_n5-i1", "axis": "combo", "difficulty": "conj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition x0a is true if proposition x0b is not true. Proposition x2a is true if proposition x2b is not true. Proposition s2 is true. Proposition q is true if proposition qcore is true and proposition wide is true. Proposition s1 is true. Proposition x2b is true if proposition x2a is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true. Proposition x3b is true if proposition x3a is not true. Proposition x3a is true if proposition x3b is not true. Proposition x4a is true if proposition x4b is not true. Proposition x1b is true if proposition x1a is not true. Proposition x4b is true if proposition x4a is not true. Proposition s0 is true. Proposition x1a is true if proposition x1b is not true. Proposition x0b is true if proposition x0a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n5-i1-wfs::wfs", "rec_id": "hv2-combo-conj_n5-i1", "axis": "combo", "difficulty": "conj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition x0a is true if proposition x0b is not true. Proposition x2a is true if proposition x2b is not true. Proposition s2 is true. Proposition q is true if proposition qcore is true and proposition wide is true. Proposition s1 is true. Proposition x2b is true if proposition x2a is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true. Proposition x3b is true if proposition x3a is not true. Proposition x3a is true if proposition x3b is not true. Proposition x4a is true if proposition x4b is not true. Proposition x1b is true if proposition x1a is not true. Proposition x4b is true if proposition x4a is not true. Proposition s0 is true. Proposition x1a is true if proposition x1b is not true. Proposition x0b is true if proposition x0a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n5-i2-none::none", "rec_id": "hv2-combo-conj_n5-i2", "axis": "combo", "difficulty": "conj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true. Proposition t3 is true if proposition t4 is true. Proposition x2b is true if proposition x2a is not true. Proposition x1a is true if proposition x1b is not true. Proposition x0a is true if proposition x0b is not true. Proposition t5 is true if proposition qcore is true. Proposition x0b is true if proposition x0a is not true. Proposition t0 is true if proposition t1 is true. Proposition x3b is true if proposition x3a is not true. Proposition x2a is true if proposition x2b is not true. Proposition s0 is true. Proposition x1b is true if proposition x1a is not true. Proposition x3a is true if proposition x3b is not true. Proposition x4a is true if proposition x4b is not true. Proposition t4 is true if proposition t5 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t1 is true if proposition t2 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition s1 is true. Proposition t2 is true if proposition t3 is true. Proposition x4b is true if proposition x4a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n5-i2-closed_world::closed_world", "rec_id": "hv2-combo-conj_n5-i2", "axis": "combo", "difficulty": "conj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true. Proposition t3 is true if proposition t4 is true. Proposition x2b is true if proposition x2a is not true. Proposition x1a is true if proposition x1b is not true. Proposition x0a is true if proposition x0b is not true. Proposition t5 is true if proposition qcore is true. Proposition x0b is true if proposition x0a is not true. Proposition t0 is true if proposition t1 is true. Proposition x3b is true if proposition x3a is not true. Proposition x2a is true if proposition x2b is not true. Proposition s0 is true. Proposition x1b is true if proposition x1a is not true. Proposition x3a is true if proposition x3b is not true. Proposition x4a is true if proposition x4b is not true. Proposition t4 is true if proposition t5 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t1 is true if proposition t2 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition s1 is true. Proposition t2 is true if proposition t3 is true. Proposition x4b is true if proposition x4a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n5-i2-cred::cred", "rec_id": "hv2-combo-conj_n5-i2", "axis": "combo", "difficulty": "conj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true. Proposition t3 is true if proposition t4 is true. Proposition x2b is true if proposition x2a is not true. Proposition x1a is true if proposition x1b is not true. Proposition x0a is true if proposition x0b is not true. Proposition t5 is true if proposition qcore is true. Proposition x0b is true if proposition x0a is not true. Proposition t0 is true if proposition t1 is true. Proposition x3b is true if proposition x3a is not true. Proposition x2a is true if proposition x2b is not true. Proposition s0 is true. Proposition x1b is true if proposition x1a is not true. Proposition x3a is true if proposition x3b is not true. Proposition x4a is true if proposition x4b is not true. Proposition t4 is true if proposition t5 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t1 is true if proposition t2 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition s1 is true. Proposition t2 is true if proposition t3 is true. Proposition x4b is true if proposition x4a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n5-i2-skept::skept", "rec_id": "hv2-combo-conj_n5-i2", "axis": "combo", "difficulty": "conj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true. Proposition t3 is true if proposition t4 is true. Proposition x2b is true if proposition x2a is not true. Proposition x1a is true if proposition x1b is not true. Proposition x0a is true if proposition x0b is not true. Proposition t5 is true if proposition qcore is true. Proposition x0b is true if proposition x0a is not true. Proposition t0 is true if proposition t1 is true. Proposition x3b is true if proposition x3a is not true. Proposition x2a is true if proposition x2b is not true. Proposition s0 is true. Proposition x1b is true if proposition x1a is not true. Proposition x3a is true if proposition x3b is not true. Proposition x4a is true if proposition x4b is not true. Proposition t4 is true if proposition t5 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t1 is true if proposition t2 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition s1 is true. Proposition t2 is true if proposition t3 is true. Proposition x4b is true if proposition x4a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n5-i2-wfs::wfs", "rec_id": "hv2-combo-conj_n5-i2", "axis": "combo", "difficulty": "conj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true. Proposition t3 is true if proposition t4 is true. Proposition x2b is true if proposition x2a is not true. Proposition x1a is true if proposition x1b is not true. Proposition x0a is true if proposition x0b is not true. Proposition t5 is true if proposition qcore is true. Proposition x0b is true if proposition x0a is not true. Proposition t0 is true if proposition t1 is true. Proposition x3b is true if proposition x3a is not true. Proposition x2a is true if proposition x2b is not true. Proposition s0 is true. Proposition x1b is true if proposition x1a is not true. Proposition x3a is true if proposition x3b is not true. Proposition x4a is true if proposition x4b is not true. Proposition t4 is true if proposition t5 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t1 is true if proposition t2 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition s1 is true. Proposition t2 is true if proposition t3 is true. Proposition x4b is true if proposition x4a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n5-i3-none::none", "rec_id": "hv2-combo-conj_n5-i3", "axis": "combo", "difficulty": "conj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition x3a is true if proposition x3b is not true. Proposition t1 is true if proposition t2 is true. Proposition x2b is true if proposition x2a is not true. Proposition t0 is true if proposition t1 is true. Proposition s2 is true. Proposition x3b is true if proposition x3a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition s1 is true. Proposition x4b is true if proposition x4a is not true. Proposition s0 is true. Proposition x1a is true if proposition x1b is not true. Proposition x0b is true if proposition x0a is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true. Proposition x1b is true if proposition x1a is not true. Proposition x0a is true if proposition x0b is not true. Proposition x4a is true if proposition x4b is not true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x2a is true if proposition x2b is not true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition qcore is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n5-i3-closed_world::closed_world", "rec_id": "hv2-combo-conj_n5-i3", "axis": "combo", "difficulty": "conj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition x3a is true if proposition x3b is not true. Proposition t1 is true if proposition t2 is true. Proposition x2b is true if proposition x2a is not true. Proposition t0 is true if proposition t1 is true. Proposition s2 is true. Proposition x3b is true if proposition x3a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition s1 is true. Proposition x4b is true if proposition x4a is not true. Proposition s0 is true. Proposition x1a is true if proposition x1b is not true. Proposition x0b is true if proposition x0a is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true. Proposition x1b is true if proposition x1a is not true. Proposition x0a is true if proposition x0b is not true. Proposition x4a is true if proposition x4b is not true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x2a is true if proposition x2b is not true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition qcore is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n5-i3-cred::cred", "rec_id": "hv2-combo-conj_n5-i3", "axis": "combo", "difficulty": "conj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition x3a is true if proposition x3b is not true. Proposition t1 is true if proposition t2 is true. Proposition x2b is true if proposition x2a is not true. Proposition t0 is true if proposition t1 is true. Proposition s2 is true. Proposition x3b is true if proposition x3a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition s1 is true. Proposition x4b is true if proposition x4a is not true. Proposition s0 is true. Proposition x1a is true if proposition x1b is not true. Proposition x0b is true if proposition x0a is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true. Proposition x1b is true if proposition x1a is not true. Proposition x0a is true if proposition x0b is not true. Proposition x4a is true if proposition x4b is not true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x2a is true if proposition x2b is not true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition qcore is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n5-i3-skept::skept", "rec_id": "hv2-combo-conj_n5-i3", "axis": "combo", "difficulty": "conj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition x3a is true if proposition x3b is not true. Proposition t1 is true if proposition t2 is true. Proposition x2b is true if proposition x2a is not true. Proposition t0 is true if proposition t1 is true. Proposition s2 is true. Proposition x3b is true if proposition x3a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition s1 is true. Proposition x4b is true if proposition x4a is not true. Proposition s0 is true. Proposition x1a is true if proposition x1b is not true. Proposition x0b is true if proposition x0a is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true. Proposition x1b is true if proposition x1a is not true. Proposition x0a is true if proposition x0b is not true. Proposition x4a is true if proposition x4b is not true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x2a is true if proposition x2b is not true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition qcore is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n5-i3-wfs::wfs", "rec_id": "hv2-combo-conj_n5-i3", "axis": "combo", "difficulty": "conj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition x3a is true if proposition x3b is not true. Proposition t1 is true if proposition t2 is true. Proposition x2b is true if proposition x2a is not true. Proposition t0 is true if proposition t1 is true. Proposition s2 is true. Proposition x3b is true if proposition x3a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition s1 is true. Proposition x4b is true if proposition x4a is not true. Proposition s0 is true. Proposition x1a is true if proposition x1b is not true. Proposition x0b is true if proposition x0a is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true. Proposition x1b is true if proposition x1a is not true. Proposition x0a is true if proposition x0b is not true. Proposition x4a is true if proposition x4b is not true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x2a is true if proposition x2b is not true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition qcore is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n5-i4-none::none", "rec_id": "hv2-combo-conj_n5-i4", "axis": "combo", "difficulty": "conj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition x4a is true if proposition x4b is not true. Proposition x2b is true if proposition x2a is not true. Proposition x1b is true if proposition x1a is not true. Proposition t1 is true if proposition qcore is true. Proposition x4b is true if proposition x4a is not true. Proposition x3b is true if proposition x3a is not true. Proposition x3a is true if proposition x3b is not true. Proposition x0a is true if proposition x0b is not true. Proposition s0 is true. Proposition t0 is true if proposition t1 is true. Proposition s1 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x0b is true if proposition x0a is not true. Proposition x1a is true if proposition x1b is not true. Proposition x2a is true if proposition x2b is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n5-i4-closed_world::closed_world", "rec_id": "hv2-combo-conj_n5-i4", "axis": "combo", "difficulty": "conj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition x4a is true if proposition x4b is not true. Proposition x2b is true if proposition x2a is not true. Proposition x1b is true if proposition x1a is not true. Proposition t1 is true if proposition qcore is true. Proposition x4b is true if proposition x4a is not true. Proposition x3b is true if proposition x3a is not true. Proposition x3a is true if proposition x3b is not true. Proposition x0a is true if proposition x0b is not true. Proposition s0 is true. Proposition t0 is true if proposition t1 is true. Proposition s1 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x0b is true if proposition x0a is not true. Proposition x1a is true if proposition x1b is not true. Proposition x2a is true if proposition x2b is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n5-i4-cred::cred", "rec_id": "hv2-combo-conj_n5-i4", "axis": "combo", "difficulty": "conj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition x4a is true if proposition x4b is not true. Proposition x2b is true if proposition x2a is not true. Proposition x1b is true if proposition x1a is not true. Proposition t1 is true if proposition qcore is true. Proposition x4b is true if proposition x4a is not true. Proposition x3b is true if proposition x3a is not true. Proposition x3a is true if proposition x3b is not true. Proposition x0a is true if proposition x0b is not true. Proposition s0 is true. Proposition t0 is true if proposition t1 is true. Proposition s1 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x0b is true if proposition x0a is not true. Proposition x1a is true if proposition x1b is not true. Proposition x2a is true if proposition x2b is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n5-i4-skept::skept", "rec_id": "hv2-combo-conj_n5-i4", "axis": "combo", "difficulty": "conj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition x4a is true if proposition x4b is not true. Proposition x2b is true if proposition x2a is not true. Proposition x1b is true if proposition x1a is not true. Proposition t1 is true if proposition qcore is true. Proposition x4b is true if proposition x4a is not true. Proposition x3b is true if proposition x3a is not true. Proposition x3a is true if proposition x3b is not true. Proposition x0a is true if proposition x0b is not true. Proposition s0 is true. Proposition t0 is true if proposition t1 is true. Proposition s1 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x0b is true if proposition x0a is not true. Proposition x1a is true if proposition x1b is not true. Proposition x2a is true if proposition x2b is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n5-i4-wfs::wfs", "rec_id": "hv2-combo-conj_n5-i4", "axis": "combo", "difficulty": "conj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition x4a is true if proposition x4b is not true. Proposition x2b is true if proposition x2a is not true. Proposition x1b is true if proposition x1a is not true. Proposition t1 is true if proposition qcore is true. Proposition x4b is true if proposition x4a is not true. Proposition x3b is true if proposition x3a is not true. Proposition x3a is true if proposition x3b is not true. Proposition x0a is true if proposition x0b is not true. Proposition s0 is true. Proposition t0 is true if proposition t1 is true. Proposition s1 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x0b is true if proposition x0a is not true. Proposition x1a is true if proposition x1b is not true. Proposition x2a is true if proposition x2b is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n5-i5-none::none", "rec_id": "hv2-combo-conj_n5-i5", "axis": "combo", "difficulty": "conj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition x3b is true if proposition x3a is not true. Proposition x0b is true if proposition x0a is not true. Proposition x3a is true if proposition x3b is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x2a is true if proposition x2b is not true. Proposition s0 is true. Proposition s1 is true. Proposition x0a is true if proposition x0b is not true. Proposition x1b is true if proposition x1a is not true. Proposition x4b is true if proposition x4a is not true. Proposition x4a is true if proposition x4b is not true. Proposition t0 is true if proposition t1 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t1 is true if proposition qcore is true. Proposition x2b is true if proposition x2a is not true. Proposition x1a is true if proposition x1b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n5-i5-closed_world::closed_world", "rec_id": "hv2-combo-conj_n5-i5", "axis": "combo", "difficulty": "conj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition x3b is true if proposition x3a is not true. Proposition x0b is true if proposition x0a is not true. Proposition x3a is true if proposition x3b is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x2a is true if proposition x2b is not true. Proposition s0 is true. Proposition s1 is true. Proposition x0a is true if proposition x0b is not true. Proposition x1b is true if proposition x1a is not true. Proposition x4b is true if proposition x4a is not true. Proposition x4a is true if proposition x4b is not true. Proposition t0 is true if proposition t1 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t1 is true if proposition qcore is true. Proposition x2b is true if proposition x2a is not true. Proposition x1a is true if proposition x1b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n5-i5-cred::cred", "rec_id": "hv2-combo-conj_n5-i5", "axis": "combo", "difficulty": "conj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition x3b is true if proposition x3a is not true. Proposition x0b is true if proposition x0a is not true. Proposition x3a is true if proposition x3b is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x2a is true if proposition x2b is not true. Proposition s0 is true. Proposition s1 is true. Proposition x0a is true if proposition x0b is not true. Proposition x1b is true if proposition x1a is not true. Proposition x4b is true if proposition x4a is not true. Proposition x4a is true if proposition x4b is not true. Proposition t0 is true if proposition t1 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t1 is true if proposition qcore is true. Proposition x2b is true if proposition x2a is not true. Proposition x1a is true if proposition x1b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n5-i5-skept::skept", "rec_id": "hv2-combo-conj_n5-i5", "axis": "combo", "difficulty": "conj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition x3b is true if proposition x3a is not true. Proposition x0b is true if proposition x0a is not true. Proposition x3a is true if proposition x3b is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x2a is true if proposition x2b is not true. Proposition s0 is true. Proposition s1 is true. Proposition x0a is true if proposition x0b is not true. Proposition x1b is true if proposition x1a is not true. Proposition x4b is true if proposition x4a is not true. Proposition x4a is true if proposition x4b is not true. Proposition t0 is true if proposition t1 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t1 is true if proposition qcore is true. Proposition x2b is true if proposition x2a is not true. Proposition x1a is true if proposition x1b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n5-i5-wfs::wfs", "rec_id": "hv2-combo-conj_n5-i5", "axis": "combo", "difficulty": "conj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition x3b is true if proposition x3a is not true. Proposition x0b is true if proposition x0a is not true. Proposition x3a is true if proposition x3b is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x2a is true if proposition x2b is not true. Proposition s0 is true. Proposition s1 is true. Proposition x0a is true if proposition x0b is not true. Proposition x1b is true if proposition x1a is not true. Proposition x4b is true if proposition x4a is not true. Proposition x4a is true if proposition x4b is not true. Proposition t0 is true if proposition t1 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t1 is true if proposition qcore is true. Proposition x2b is true if proposition x2a is not true. Proposition x1a is true if proposition x1b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n5-i6-none::none", "rec_id": "hv2-combo-conj_n5-i6", "axis": "combo", "difficulty": "conj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition etrue is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition x0a is true if proposition x0b is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x4a is true if proposition x4b is not true. Proposition x3b is true if proposition x3a is not true. Proposition t1 is true if proposition t2 is true. Proposition x1a is true if proposition x1b is not true. Proposition x0b is true if proposition x0a is not true. Proposition x4b is true if proposition x4a is not true. Proposition x2b is true if proposition x2a is not true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition qcore is true. Proposition s0 is true. Proposition x2a is true if proposition x2b is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true. Proposition x3a is true if proposition x3b is not true. Proposition t0 is true if proposition t1 is true. Proposition x1b is true if proposition x1a is not true. Proposition s1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n5-i6-closed_world::closed_world", "rec_id": "hv2-combo-conj_n5-i6", "axis": "combo", "difficulty": "conj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition etrue is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition x0a is true if proposition x0b is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x4a is true if proposition x4b is not true. Proposition x3b is true if proposition x3a is not true. Proposition t1 is true if proposition t2 is true. Proposition x1a is true if proposition x1b is not true. Proposition x0b is true if proposition x0a is not true. Proposition x4b is true if proposition x4a is not true. Proposition x2b is true if proposition x2a is not true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition qcore is true. Proposition s0 is true. Proposition x2a is true if proposition x2b is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true. Proposition x3a is true if proposition x3b is not true. Proposition t0 is true if proposition t1 is true. Proposition x1b is true if proposition x1a is not true. Proposition s1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n5-i6-cred::cred", "rec_id": "hv2-combo-conj_n5-i6", "axis": "combo", "difficulty": "conj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition etrue is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition x0a is true if proposition x0b is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x4a is true if proposition x4b is not true. Proposition x3b is true if proposition x3a is not true. Proposition t1 is true if proposition t2 is true. Proposition x1a is true if proposition x1b is not true. Proposition x0b is true if proposition x0a is not true. Proposition x4b is true if proposition x4a is not true. Proposition x2b is true if proposition x2a is not true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition qcore is true. Proposition s0 is true. Proposition x2a is true if proposition x2b is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true. Proposition x3a is true if proposition x3b is not true. Proposition t0 is true if proposition t1 is true. Proposition x1b is true if proposition x1a is not true. Proposition s1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n5-i6-skept::skept", "rec_id": "hv2-combo-conj_n5-i6", "axis": "combo", "difficulty": "conj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition etrue is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition x0a is true if proposition x0b is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x4a is true if proposition x4b is not true. Proposition x3b is true if proposition x3a is not true. Proposition t1 is true if proposition t2 is true. Proposition x1a is true if proposition x1b is not true. Proposition x0b is true if proposition x0a is not true. Proposition x4b is true if proposition x4a is not true. Proposition x2b is true if proposition x2a is not true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition qcore is true. Proposition s0 is true. Proposition x2a is true if proposition x2b is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true. Proposition x3a is true if proposition x3b is not true. Proposition t0 is true if proposition t1 is true. Proposition x1b is true if proposition x1a is not true. Proposition s1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n5-i6-wfs::wfs", "rec_id": "hv2-combo-conj_n5-i6", "axis": "combo", "difficulty": "conj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition etrue is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition x0a is true if proposition x0b is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x4a is true if proposition x4b is not true. Proposition x3b is true if proposition x3a is not true. Proposition t1 is true if proposition t2 is true. Proposition x1a is true if proposition x1b is not true. Proposition x0b is true if proposition x0a is not true. Proposition x4b is true if proposition x4a is not true. Proposition x2b is true if proposition x2a is not true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition qcore is true. Proposition s0 is true. Proposition x2a is true if proposition x2b is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true. Proposition x3a is true if proposition x3b is not true. Proposition t0 is true if proposition t1 is true. Proposition x1b is true if proposition x1a is not true. Proposition s1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n5-i7-none::none", "rec_id": "hv2-combo-conj_n5-i7", "axis": "combo", "difficulty": "conj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition q is true if proposition t0 is true. Proposition x0b is true if proposition x0a is not true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition qcore is true. Proposition x1b is true if proposition x1a is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true. Proposition x2a is true if proposition x2b is not true. Proposition t1 is true if proposition t2 is true. Proposition x4a is true if proposition x4b is not true. Proposition x1a is true if proposition x1b is not true. Proposition x3a is true if proposition x3b is not true. Proposition x4b is true if proposition x4a is not true. Proposition x0a is true if proposition x0b is not true. Proposition x2b is true if proposition x2a is not true. Proposition t0 is true if proposition t1 is true. Proposition x3b is true if proposition x3a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n5-i7-closed_world::closed_world", "rec_id": "hv2-combo-conj_n5-i7", "axis": "combo", "difficulty": "conj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition q is true if proposition t0 is true. Proposition x0b is true if proposition x0a is not true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition qcore is true. Proposition x1b is true if proposition x1a is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true. Proposition x2a is true if proposition x2b is not true. Proposition t1 is true if proposition t2 is true. Proposition x4a is true if proposition x4b is not true. Proposition x1a is true if proposition x1b is not true. Proposition x3a is true if proposition x3b is not true. Proposition x4b is true if proposition x4a is not true. Proposition x0a is true if proposition x0b is not true. Proposition x2b is true if proposition x2a is not true. Proposition t0 is true if proposition t1 is true. Proposition x3b is true if proposition x3a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n5-i7-cred::cred", "rec_id": "hv2-combo-conj_n5-i7", "axis": "combo", "difficulty": "conj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition q is true if proposition t0 is true. Proposition x0b is true if proposition x0a is not true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition qcore is true. Proposition x1b is true if proposition x1a is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true. Proposition x2a is true if proposition x2b is not true. Proposition t1 is true if proposition t2 is true. Proposition x4a is true if proposition x4b is not true. Proposition x1a is true if proposition x1b is not true. Proposition x3a is true if proposition x3b is not true. Proposition x4b is true if proposition x4a is not true. Proposition x0a is true if proposition x0b is not true. Proposition x2b is true if proposition x2a is not true. Proposition t0 is true if proposition t1 is true. Proposition x3b is true if proposition x3a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n5-i7-skept::skept", "rec_id": "hv2-combo-conj_n5-i7", "axis": "combo", "difficulty": "conj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition q is true if proposition t0 is true. Proposition x0b is true if proposition x0a is not true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition qcore is true. Proposition x1b is true if proposition x1a is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true. Proposition x2a is true if proposition x2b is not true. Proposition t1 is true if proposition t2 is true. Proposition x4a is true if proposition x4b is not true. Proposition x1a is true if proposition x1b is not true. Proposition x3a is true if proposition x3b is not true. Proposition x4b is true if proposition x4a is not true. Proposition x0a is true if proposition x0b is not true. Proposition x2b is true if proposition x2a is not true. Proposition t0 is true if proposition t1 is true. Proposition x3b is true if proposition x3a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n5-i7-wfs::wfs", "rec_id": "hv2-combo-conj_n5-i7", "axis": "combo", "difficulty": "conj_n5", "n_stable_models": 32, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition q is true if proposition t0 is true. Proposition x0b is true if proposition x0a is not true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition qcore is true. Proposition x1b is true if proposition x1a is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true. Proposition x2a is true if proposition x2b is not true. Proposition t1 is true if proposition t2 is true. Proposition x4a is true if proposition x4b is not true. Proposition x1a is true if proposition x1b is not true. Proposition x3a is true if proposition x3b is not true. Proposition x4b is true if proposition x4a is not true. Proposition x0a is true if proposition x0b is not true. Proposition x2b is true if proposition x2a is not true. Proposition t0 is true if proposition t1 is true. Proposition x3b is true if proposition x3a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n6-i0-none::none", "rec_id": "hv2-combo-conj_n6-i0", "axis": "combo", "difficulty": "conj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition x1b is true if proposition x1a is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition x3b is true if proposition x3a is not true. Proposition s1 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x0a is true if proposition x0b is not true. Proposition x1a is true if proposition x1b is not true. Proposition x2b is true if proposition x2a is not true. Proposition x4b is true if proposition x4a is not true. Proposition etrue is true. Proposition x3a is true if proposition x3b is not true. Proposition t0 is true if proposition t1 is true. Proposition s0 is true. Proposition t1 is true if proposition qcore is true. Proposition x4a is true if proposition x4b is not true. Proposition x5b is true if proposition x5a is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true and proposition x5a is true. Proposition x0b is true if proposition x0a is not true. Proposition x2a is true if proposition x2b is not true. Proposition x5a is true if proposition x5b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n6-i0-closed_world::closed_world", "rec_id": "hv2-combo-conj_n6-i0", "axis": "combo", "difficulty": "conj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition x1b is true if proposition x1a is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition x3b is true if proposition x3a is not true. Proposition s1 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x0a is true if proposition x0b is not true. Proposition x1a is true if proposition x1b is not true. Proposition x2b is true if proposition x2a is not true. Proposition x4b is true if proposition x4a is not true. Proposition etrue is true. Proposition x3a is true if proposition x3b is not true. Proposition t0 is true if proposition t1 is true. Proposition s0 is true. Proposition t1 is true if proposition qcore is true. Proposition x4a is true if proposition x4b is not true. Proposition x5b is true if proposition x5a is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true and proposition x5a is true. Proposition x0b is true if proposition x0a is not true. Proposition x2a is true if proposition x2b is not true. Proposition x5a is true if proposition x5b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n6-i0-cred::cred", "rec_id": "hv2-combo-conj_n6-i0", "axis": "combo", "difficulty": "conj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition x1b is true if proposition x1a is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition x3b is true if proposition x3a is not true. Proposition s1 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x0a is true if proposition x0b is not true. Proposition x1a is true if proposition x1b is not true. Proposition x2b is true if proposition x2a is not true. Proposition x4b is true if proposition x4a is not true. Proposition etrue is true. Proposition x3a is true if proposition x3b is not true. Proposition t0 is true if proposition t1 is true. Proposition s0 is true. Proposition t1 is true if proposition qcore is true. Proposition x4a is true if proposition x4b is not true. Proposition x5b is true if proposition x5a is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true and proposition x5a is true. Proposition x0b is true if proposition x0a is not true. Proposition x2a is true if proposition x2b is not true. Proposition x5a is true if proposition x5b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n6-i0-skept::skept", "rec_id": "hv2-combo-conj_n6-i0", "axis": "combo", "difficulty": "conj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition x1b is true if proposition x1a is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition x3b is true if proposition x3a is not true. Proposition s1 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x0a is true if proposition x0b is not true. Proposition x1a is true if proposition x1b is not true. Proposition x2b is true if proposition x2a is not true. Proposition x4b is true if proposition x4a is not true. Proposition etrue is true. Proposition x3a is true if proposition x3b is not true. Proposition t0 is true if proposition t1 is true. Proposition s0 is true. Proposition t1 is true if proposition qcore is true. Proposition x4a is true if proposition x4b is not true. Proposition x5b is true if proposition x5a is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true and proposition x5a is true. Proposition x0b is true if proposition x0a is not true. Proposition x2a is true if proposition x2b is not true. Proposition x5a is true if proposition x5b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n6-i0-wfs::wfs", "rec_id": "hv2-combo-conj_n6-i0", "axis": "combo", "difficulty": "conj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition x1b is true if proposition x1a is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition x3b is true if proposition x3a is not true. Proposition s1 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition x0a is true if proposition x0b is not true. Proposition x1a is true if proposition x1b is not true. Proposition x2b is true if proposition x2a is not true. Proposition x4b is true if proposition x4a is not true. Proposition etrue is true. Proposition x3a is true if proposition x3b is not true. Proposition t0 is true if proposition t1 is true. Proposition s0 is true. Proposition t1 is true if proposition qcore is true. Proposition x4a is true if proposition x4b is not true. Proposition x5b is true if proposition x5a is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true and proposition x5a is true. Proposition x0b is true if proposition x0a is not true. Proposition x2a is true if proposition x2b is not true. Proposition x5a is true if proposition x5b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n6-i1-none::none", "rec_id": "hv2-combo-conj_n6-i1", "axis": "combo", "difficulty": "conj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition q is true if proposition t0 is true and proposition wide is true. Proposition x0b is true if proposition x0a is not true. Proposition s2 is true. Proposition x5b is true if proposition x5a is not true. Proposition x2a is true if proposition x2b is not true. Proposition t1 is true if proposition qcore is true. Proposition s0 is true. Proposition x1a is true if proposition x1b is not true. Proposition x3b is true if proposition x3a is not true. Proposition x4b is true if proposition x4a is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true and proposition x5a is true. Proposition x4a is true if proposition x4b is not true. Proposition x3a is true if proposition x3b is not true. Proposition x1b is true if proposition x1a is not true. Proposition t0 is true if proposition t1 is true. Proposition x5a is true if proposition x5b is not true. Proposition x2b is true if proposition x2a is not true. Proposition s1 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition x0a is true if proposition x0b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n6-i1-closed_world::closed_world", "rec_id": "hv2-combo-conj_n6-i1", "axis": "combo", "difficulty": "conj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition q is true if proposition t0 is true and proposition wide is true. Proposition x0b is true if proposition x0a is not true. Proposition s2 is true. Proposition x5b is true if proposition x5a is not true. Proposition x2a is true if proposition x2b is not true. Proposition t1 is true if proposition qcore is true. Proposition s0 is true. Proposition x1a is true if proposition x1b is not true. Proposition x3b is true if proposition x3a is not true. Proposition x4b is true if proposition x4a is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true and proposition x5a is true. Proposition x4a is true if proposition x4b is not true. Proposition x3a is true if proposition x3b is not true. Proposition x1b is true if proposition x1a is not true. Proposition t0 is true if proposition t1 is true. Proposition x5a is true if proposition x5b is not true. Proposition x2b is true if proposition x2a is not true. Proposition s1 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition x0a is true if proposition x0b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n6-i1-cred::cred", "rec_id": "hv2-combo-conj_n6-i1", "axis": "combo", "difficulty": "conj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition q is true if proposition t0 is true and proposition wide is true. Proposition x0b is true if proposition x0a is not true. Proposition s2 is true. Proposition x5b is true if proposition x5a is not true. Proposition x2a is true if proposition x2b is not true. Proposition t1 is true if proposition qcore is true. Proposition s0 is true. Proposition x1a is true if proposition x1b is not true. Proposition x3b is true if proposition x3a is not true. Proposition x4b is true if proposition x4a is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true and proposition x5a is true. Proposition x4a is true if proposition x4b is not true. Proposition x3a is true if proposition x3b is not true. Proposition x1b is true if proposition x1a is not true. Proposition t0 is true if proposition t1 is true. Proposition x5a is true if proposition x5b is not true. Proposition x2b is true if proposition x2a is not true. Proposition s1 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition x0a is true if proposition x0b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n6-i1-skept::skept", "rec_id": "hv2-combo-conj_n6-i1", "axis": "combo", "difficulty": "conj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition q is true if proposition t0 is true and proposition wide is true. Proposition x0b is true if proposition x0a is not true. Proposition s2 is true. Proposition x5b is true if proposition x5a is not true. Proposition x2a is true if proposition x2b is not true. Proposition t1 is true if proposition qcore is true. Proposition s0 is true. Proposition x1a is true if proposition x1b is not true. Proposition x3b is true if proposition x3a is not true. Proposition x4b is true if proposition x4a is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true and proposition x5a is true. Proposition x4a is true if proposition x4b is not true. Proposition x3a is true if proposition x3b is not true. Proposition x1b is true if proposition x1a is not true. Proposition t0 is true if proposition t1 is true. Proposition x5a is true if proposition x5b is not true. Proposition x2b is true if proposition x2a is not true. Proposition s1 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition x0a is true if proposition x0b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n6-i1-wfs::wfs", "rec_id": "hv2-combo-conj_n6-i1", "axis": "combo", "difficulty": "conj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition q is true if proposition t0 is true and proposition wide is true. Proposition x0b is true if proposition x0a is not true. Proposition s2 is true. Proposition x5b is true if proposition x5a is not true. Proposition x2a is true if proposition x2b is not true. Proposition t1 is true if proposition qcore is true. Proposition s0 is true. Proposition x1a is true if proposition x1b is not true. Proposition x3b is true if proposition x3a is not true. Proposition x4b is true if proposition x4a is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true and proposition x5a is true. Proposition x4a is true if proposition x4b is not true. Proposition x3a is true if proposition x3b is not true. Proposition x1b is true if proposition x1a is not true. Proposition t0 is true if proposition t1 is true. Proposition x5a is true if proposition x5b is not true. Proposition x2b is true if proposition x2a is not true. Proposition s1 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition x0a is true if proposition x0b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n6-i2-none::none", "rec_id": "hv2-combo-conj_n6-i2", "axis": "combo", "difficulty": "conj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition wide is true if proposition s0 is true and proposition s1 is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true and proposition x5a is true. Proposition x4a is true if proposition x4b is not true. Proposition x3a is true if proposition x3b is not true. Proposition x2a is true if proposition x2b is not true. Proposition x1b is true if proposition x1a is not true. Proposition q is true if proposition qcore is true and proposition wide is true and proposition etrue is true. Proposition etrue is true. Proposition x3b is true if proposition x3a is not true. Proposition x1a is true if proposition x1b is not true. Proposition x0b is true if proposition x0a is not true. Proposition s1 is true. Proposition s0 is true. Proposition x2b is true if proposition x2a is not true. Proposition x5b is true if proposition x5a is not true. Proposition x0a is true if proposition x0b is not true. Proposition x4b is true if proposition x4a is not true. Proposition x5a is true if proposition x5b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n6-i2-closed_world::closed_world", "rec_id": "hv2-combo-conj_n6-i2", "axis": "combo", "difficulty": "conj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition wide is true if proposition s0 is true and proposition s1 is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true and proposition x5a is true. Proposition x4a is true if proposition x4b is not true. Proposition x3a is true if proposition x3b is not true. Proposition x2a is true if proposition x2b is not true. Proposition x1b is true if proposition x1a is not true. Proposition q is true if proposition qcore is true and proposition wide is true and proposition etrue is true. Proposition etrue is true. Proposition x3b is true if proposition x3a is not true. Proposition x1a is true if proposition x1b is not true. Proposition x0b is true if proposition x0a is not true. Proposition s1 is true. Proposition s0 is true. Proposition x2b is true if proposition x2a is not true. Proposition x5b is true if proposition x5a is not true. Proposition x0a is true if proposition x0b is not true. Proposition x4b is true if proposition x4a is not true. Proposition x5a is true if proposition x5b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n6-i2-cred::cred", "rec_id": "hv2-combo-conj_n6-i2", "axis": "combo", "difficulty": "conj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition wide is true if proposition s0 is true and proposition s1 is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true and proposition x5a is true. Proposition x4a is true if proposition x4b is not true. Proposition x3a is true if proposition x3b is not true. Proposition x2a is true if proposition x2b is not true. Proposition x1b is true if proposition x1a is not true. Proposition q is true if proposition qcore is true and proposition wide is true and proposition etrue is true. Proposition etrue is true. Proposition x3b is true if proposition x3a is not true. Proposition x1a is true if proposition x1b is not true. Proposition x0b is true if proposition x0a is not true. Proposition s1 is true. Proposition s0 is true. Proposition x2b is true if proposition x2a is not true. Proposition x5b is true if proposition x5a is not true. Proposition x0a is true if proposition x0b is not true. Proposition x4b is true if proposition x4a is not true. Proposition x5a is true if proposition x5b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n6-i2-skept::skept", "rec_id": "hv2-combo-conj_n6-i2", "axis": "combo", "difficulty": "conj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition wide is true if proposition s0 is true and proposition s1 is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true and proposition x5a is true. Proposition x4a is true if proposition x4b is not true. Proposition x3a is true if proposition x3b is not true. Proposition x2a is true if proposition x2b is not true. Proposition x1b is true if proposition x1a is not true. Proposition q is true if proposition qcore is true and proposition wide is true and proposition etrue is true. Proposition etrue is true. Proposition x3b is true if proposition x3a is not true. Proposition x1a is true if proposition x1b is not true. Proposition x0b is true if proposition x0a is not true. Proposition s1 is true. Proposition s0 is true. Proposition x2b is true if proposition x2a is not true. Proposition x5b is true if proposition x5a is not true. Proposition x0a is true if proposition x0b is not true. Proposition x4b is true if proposition x4a is not true. Proposition x5a is true if proposition x5b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n6-i2-wfs::wfs", "rec_id": "hv2-combo-conj_n6-i2", "axis": "combo", "difficulty": "conj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition wide is true if proposition s0 is true and proposition s1 is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true and proposition x5a is true. Proposition x4a is true if proposition x4b is not true. Proposition x3a is true if proposition x3b is not true. Proposition x2a is true if proposition x2b is not true. Proposition x1b is true if proposition x1a is not true. Proposition q is true if proposition qcore is true and proposition wide is true and proposition etrue is true. Proposition etrue is true. Proposition x3b is true if proposition x3a is not true. Proposition x1a is true if proposition x1b is not true. Proposition x0b is true if proposition x0a is not true. Proposition s1 is true. Proposition s0 is true. Proposition x2b is true if proposition x2a is not true. Proposition x5b is true if proposition x5a is not true. Proposition x0a is true if proposition x0b is not true. Proposition x4b is true if proposition x4a is not true. Proposition x5a is true if proposition x5b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n6-i3-none::none", "rec_id": "hv2-combo-conj_n6-i3", "axis": "combo", "difficulty": "conj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition x5b is true if proposition x5a is not true. Proposition t1 is true if proposition t2 is true. Proposition x0a is true if proposition x0b is not true. Proposition t0 is true if proposition t1 is true. Proposition x2a is true if proposition x2b is not true. Proposition t2 is true if proposition t3 is true. Proposition x2b is true if proposition x2a is not true. Proposition x4b is true if proposition x4a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition x1b is true if proposition x1a is not true. Proposition x3b is true if proposition x3a is not true. Proposition x4a is true if proposition x4b is not true. Proposition s0 is true. Proposition t3 is true if proposition qcore is true. Proposition s2 is true. Proposition x1a is true if proposition x1b is not true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x0b is true if proposition x0a is not true. Proposition s1 is true. Proposition x5a is true if proposition x5b is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true and proposition x5a is true. Proposition x3a is true if proposition x3b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n6-i3-closed_world::closed_world", "rec_id": "hv2-combo-conj_n6-i3", "axis": "combo", "difficulty": "conj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition x5b is true if proposition x5a is not true. Proposition t1 is true if proposition t2 is true. Proposition x0a is true if proposition x0b is not true. Proposition t0 is true if proposition t1 is true. Proposition x2a is true if proposition x2b is not true. Proposition t2 is true if proposition t3 is true. Proposition x2b is true if proposition x2a is not true. Proposition x4b is true if proposition x4a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition x1b is true if proposition x1a is not true. Proposition x3b is true if proposition x3a is not true. Proposition x4a is true if proposition x4b is not true. Proposition s0 is true. Proposition t3 is true if proposition qcore is true. Proposition s2 is true. Proposition x1a is true if proposition x1b is not true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x0b is true if proposition x0a is not true. Proposition s1 is true. Proposition x5a is true if proposition x5b is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true and proposition x5a is true. Proposition x3a is true if proposition x3b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n6-i3-cred::cred", "rec_id": "hv2-combo-conj_n6-i3", "axis": "combo", "difficulty": "conj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition x5b is true if proposition x5a is not true. Proposition t1 is true if proposition t2 is true. Proposition x0a is true if proposition x0b is not true. Proposition t0 is true if proposition t1 is true. Proposition x2a is true if proposition x2b is not true. Proposition t2 is true if proposition t3 is true. Proposition x2b is true if proposition x2a is not true. Proposition x4b is true if proposition x4a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition x1b is true if proposition x1a is not true. Proposition x3b is true if proposition x3a is not true. Proposition x4a is true if proposition x4b is not true. Proposition s0 is true. Proposition t3 is true if proposition qcore is true. Proposition s2 is true. Proposition x1a is true if proposition x1b is not true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x0b is true if proposition x0a is not true. Proposition s1 is true. Proposition x5a is true if proposition x5b is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true and proposition x5a is true. Proposition x3a is true if proposition x3b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n6-i3-skept::skept", "rec_id": "hv2-combo-conj_n6-i3", "axis": "combo", "difficulty": "conj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition x5b is true if proposition x5a is not true. Proposition t1 is true if proposition t2 is true. Proposition x0a is true if proposition x0b is not true. Proposition t0 is true if proposition t1 is true. Proposition x2a is true if proposition x2b is not true. Proposition t2 is true if proposition t3 is true. Proposition x2b is true if proposition x2a is not true. Proposition x4b is true if proposition x4a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition x1b is true if proposition x1a is not true. Proposition x3b is true if proposition x3a is not true. Proposition x4a is true if proposition x4b is not true. Proposition s0 is true. Proposition t3 is true if proposition qcore is true. Proposition s2 is true. Proposition x1a is true if proposition x1b is not true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x0b is true if proposition x0a is not true. Proposition s1 is true. Proposition x5a is true if proposition x5b is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true and proposition x5a is true. Proposition x3a is true if proposition x3b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n6-i3-wfs::wfs", "rec_id": "hv2-combo-conj_n6-i3", "axis": "combo", "difficulty": "conj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition x5b is true if proposition x5a is not true. Proposition t1 is true if proposition t2 is true. Proposition x0a is true if proposition x0b is not true. Proposition t0 is true if proposition t1 is true. Proposition x2a is true if proposition x2b is not true. Proposition t2 is true if proposition t3 is true. Proposition x2b is true if proposition x2a is not true. Proposition x4b is true if proposition x4a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition x1b is true if proposition x1a is not true. Proposition x3b is true if proposition x3a is not true. Proposition x4a is true if proposition x4b is not true. Proposition s0 is true. Proposition t3 is true if proposition qcore is true. Proposition s2 is true. Proposition x1a is true if proposition x1b is not true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x0b is true if proposition x0a is not true. Proposition s1 is true. Proposition x5a is true if proposition x5b is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true and proposition x5a is true. Proposition x3a is true if proposition x3b is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n6-i4-none::none", "rec_id": "hv2-combo-conj_n6-i4", "axis": "combo", "difficulty": "conj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition x0a is true if proposition x0b is not true. Proposition x3a is true if proposition x3b is not true. Proposition t5 is true if proposition qcore is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x2a is true if proposition x2b is not true. Proposition t0 is true if proposition t1 is true. Proposition t2 is true if proposition t3 is true. Proposition x2b is true if proposition x2a is not true. Proposition s1 is true. Proposition t4 is true if proposition t5 is true. Proposition x1a is true if proposition x1b is not true. Proposition x5b is true if proposition x5a is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true and proposition x5a is true. Proposition t3 is true if proposition t4 is true. Proposition x3b is true if proposition x3a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition x4a is true if proposition x4b is not true. Proposition x1b is true if proposition x1a is not true. Proposition x0b is true if proposition x0a is not true. Proposition t1 is true if proposition t2 is true. Proposition s0 is true. Proposition x4b is true if proposition x4a is not true. Proposition x5a is true if proposition x5b is not true. Proposition s2 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n6-i4-closed_world::closed_world", "rec_id": "hv2-combo-conj_n6-i4", "axis": "combo", "difficulty": "conj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition x0a is true if proposition x0b is not true. Proposition x3a is true if proposition x3b is not true. Proposition t5 is true if proposition qcore is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x2a is true if proposition x2b is not true. Proposition t0 is true if proposition t1 is true. Proposition t2 is true if proposition t3 is true. Proposition x2b is true if proposition x2a is not true. Proposition s1 is true. Proposition t4 is true if proposition t5 is true. Proposition x1a is true if proposition x1b is not true. Proposition x5b is true if proposition x5a is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true and proposition x5a is true. Proposition t3 is true if proposition t4 is true. Proposition x3b is true if proposition x3a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition x4a is true if proposition x4b is not true. Proposition x1b is true if proposition x1a is not true. Proposition x0b is true if proposition x0a is not true. Proposition t1 is true if proposition t2 is true. Proposition s0 is true. Proposition x4b is true if proposition x4a is not true. Proposition x5a is true if proposition x5b is not true. Proposition s2 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n6-i4-cred::cred", "rec_id": "hv2-combo-conj_n6-i4", "axis": "combo", "difficulty": "conj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition x0a is true if proposition x0b is not true. Proposition x3a is true if proposition x3b is not true. Proposition t5 is true if proposition qcore is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x2a is true if proposition x2b is not true. Proposition t0 is true if proposition t1 is true. Proposition t2 is true if proposition t3 is true. Proposition x2b is true if proposition x2a is not true. Proposition s1 is true. Proposition t4 is true if proposition t5 is true. Proposition x1a is true if proposition x1b is not true. Proposition x5b is true if proposition x5a is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true and proposition x5a is true. Proposition t3 is true if proposition t4 is true. Proposition x3b is true if proposition x3a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition x4a is true if proposition x4b is not true. Proposition x1b is true if proposition x1a is not true. Proposition x0b is true if proposition x0a is not true. Proposition t1 is true if proposition t2 is true. Proposition s0 is true. Proposition x4b is true if proposition x4a is not true. Proposition x5a is true if proposition x5b is not true. Proposition s2 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n6-i4-skept::skept", "rec_id": "hv2-combo-conj_n6-i4", "axis": "combo", "difficulty": "conj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition x0a is true if proposition x0b is not true. Proposition x3a is true if proposition x3b is not true. Proposition t5 is true if proposition qcore is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x2a is true if proposition x2b is not true. Proposition t0 is true if proposition t1 is true. Proposition t2 is true if proposition t3 is true. Proposition x2b is true if proposition x2a is not true. Proposition s1 is true. Proposition t4 is true if proposition t5 is true. Proposition x1a is true if proposition x1b is not true. Proposition x5b is true if proposition x5a is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true and proposition x5a is true. Proposition t3 is true if proposition t4 is true. Proposition x3b is true if proposition x3a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition x4a is true if proposition x4b is not true. Proposition x1b is true if proposition x1a is not true. Proposition x0b is true if proposition x0a is not true. Proposition t1 is true if proposition t2 is true. Proposition s0 is true. Proposition x4b is true if proposition x4a is not true. Proposition x5a is true if proposition x5b is not true. Proposition s2 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n6-i4-wfs::wfs", "rec_id": "hv2-combo-conj_n6-i4", "axis": "combo", "difficulty": "conj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition x0a is true if proposition x0b is not true. Proposition x3a is true if proposition x3b is not true. Proposition t5 is true if proposition qcore is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition x2a is true if proposition x2b is not true. Proposition t0 is true if proposition t1 is true. Proposition t2 is true if proposition t3 is true. Proposition x2b is true if proposition x2a is not true. Proposition s1 is true. Proposition t4 is true if proposition t5 is true. Proposition x1a is true if proposition x1b is not true. Proposition x5b is true if proposition x5a is not true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true and proposition x5a is true. Proposition t3 is true if proposition t4 is true. Proposition x3b is true if proposition x3a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition x4a is true if proposition x4b is not true. Proposition x1b is true if proposition x1a is not true. Proposition x0b is true if proposition x0a is not true. Proposition t1 is true if proposition t2 is true. Proposition s0 is true. Proposition x4b is true if proposition x4a is not true. Proposition x5a is true if proposition x5b is not true. Proposition s2 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n6-i5-none::none", "rec_id": "hv2-combo-conj_n6-i5", "axis": "combo", "difficulty": "conj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition x4a is true if proposition x4b is not true. Proposition x2b is true if proposition x2a is not true. Proposition x0b is true if proposition x0a is not true. Proposition x5a is true if proposition x5b is not true. Proposition t3 is true if proposition t4 is true. Proposition t4 is true if proposition t5 is true. Proposition x0a is true if proposition x0b is not true. Proposition etrue is true. Proposition t0 is true if proposition t1 is true. Proposition s1 is true. Proposition x4b is true if proposition x4a is not true. Proposition x2a is true if proposition x2b is not true. Proposition x1b is true if proposition x1a is not true. Proposition t2 is true if proposition t3 is true. Proposition x5b is true if proposition x5a is not true. Proposition x1a is true if proposition x1b is not true. Proposition s0 is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true and proposition x5a is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition x3a is true if proposition x3b is not true. Proposition t1 is true if proposition t2 is true. Proposition x3b is true if proposition x3a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t5 is true if proposition qcore is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n6-i5-closed_world::closed_world", "rec_id": "hv2-combo-conj_n6-i5", "axis": "combo", "difficulty": "conj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition x4a is true if proposition x4b is not true. Proposition x2b is true if proposition x2a is not true. Proposition x0b is true if proposition x0a is not true. Proposition x5a is true if proposition x5b is not true. Proposition t3 is true if proposition t4 is true. Proposition t4 is true if proposition t5 is true. Proposition x0a is true if proposition x0b is not true. Proposition etrue is true. Proposition t0 is true if proposition t1 is true. Proposition s1 is true. Proposition x4b is true if proposition x4a is not true. Proposition x2a is true if proposition x2b is not true. Proposition x1b is true if proposition x1a is not true. Proposition t2 is true if proposition t3 is true. Proposition x5b is true if proposition x5a is not true. Proposition x1a is true if proposition x1b is not true. Proposition s0 is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true and proposition x5a is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition x3a is true if proposition x3b is not true. Proposition t1 is true if proposition t2 is true. Proposition x3b is true if proposition x3a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t5 is true if proposition qcore is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n6-i5-cred::cred", "rec_id": "hv2-combo-conj_n6-i5", "axis": "combo", "difficulty": "conj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition x4a is true if proposition x4b is not true. Proposition x2b is true if proposition x2a is not true. Proposition x0b is true if proposition x0a is not true. Proposition x5a is true if proposition x5b is not true. Proposition t3 is true if proposition t4 is true. Proposition t4 is true if proposition t5 is true. Proposition x0a is true if proposition x0b is not true. Proposition etrue is true. Proposition t0 is true if proposition t1 is true. Proposition s1 is true. Proposition x4b is true if proposition x4a is not true. Proposition x2a is true if proposition x2b is not true. Proposition x1b is true if proposition x1a is not true. Proposition t2 is true if proposition t3 is true. Proposition x5b is true if proposition x5a is not true. Proposition x1a is true if proposition x1b is not true. Proposition s0 is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true and proposition x5a is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition x3a is true if proposition x3b is not true. Proposition t1 is true if proposition t2 is true. Proposition x3b is true if proposition x3a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t5 is true if proposition qcore is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n6-i5-skept::skept", "rec_id": "hv2-combo-conj_n6-i5", "axis": "combo", "difficulty": "conj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition x4a is true if proposition x4b is not true. Proposition x2b is true if proposition x2a is not true. Proposition x0b is true if proposition x0a is not true. Proposition x5a is true if proposition x5b is not true. Proposition t3 is true if proposition t4 is true. Proposition t4 is true if proposition t5 is true. Proposition x0a is true if proposition x0b is not true. Proposition etrue is true. Proposition t0 is true if proposition t1 is true. Proposition s1 is true. Proposition x4b is true if proposition x4a is not true. Proposition x2a is true if proposition x2b is not true. Proposition x1b is true if proposition x1a is not true. Proposition t2 is true if proposition t3 is true. Proposition x5b is true if proposition x5a is not true. Proposition x1a is true if proposition x1b is not true. Proposition s0 is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true and proposition x5a is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition x3a is true if proposition x3b is not true. Proposition t1 is true if proposition t2 is true. Proposition x3b is true if proposition x3a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t5 is true if proposition qcore is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n6-i5-wfs::wfs", "rec_id": "hv2-combo-conj_n6-i5", "axis": "combo", "difficulty": "conj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition x4a is true if proposition x4b is not true. Proposition x2b is true if proposition x2a is not true. Proposition x0b is true if proposition x0a is not true. Proposition x5a is true if proposition x5b is not true. Proposition t3 is true if proposition t4 is true. Proposition t4 is true if proposition t5 is true. Proposition x0a is true if proposition x0b is not true. Proposition etrue is true. Proposition t0 is true if proposition t1 is true. Proposition s1 is true. Proposition x4b is true if proposition x4a is not true. Proposition x2a is true if proposition x2b is not true. Proposition x1b is true if proposition x1a is not true. Proposition t2 is true if proposition t3 is true. Proposition x5b is true if proposition x5a is not true. Proposition x1a is true if proposition x1b is not true. Proposition s0 is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true and proposition x5a is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition x3a is true if proposition x3b is not true. Proposition t1 is true if proposition t2 is true. Proposition x3b is true if proposition x3a is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t5 is true if proposition qcore is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n6-i6-none::none", "rec_id": "hv2-combo-conj_n6-i6", "axis": "combo", "difficulty": "conj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition x5a is true if proposition x5b is not true. Proposition x4a is true if proposition x4b is not true. Proposition x1b is true if proposition x1a is not true. Proposition etrue is true. Proposition t1 is true if proposition qcore is true. Proposition q is true if proposition t0 is true and proposition etrue is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true and proposition x5a is true. Proposition x1a is true if proposition x1b is not true. Proposition x5b is true if proposition x5a is not true. Proposition x2b is true if proposition x2a is not true. Proposition x3a is true if proposition x3b is not true. Proposition t0 is true if proposition t1 is true. Proposition x4b is true if proposition x4a is not true. Proposition x3b is true if proposition x3a is not true. Proposition x2a is true if proposition x2b is not true. Proposition x0a is true if proposition x0b is not true. Proposition x0b is true if proposition x0a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n6-i6-closed_world::closed_world", "rec_id": "hv2-combo-conj_n6-i6", "axis": "combo", "difficulty": "conj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition x5a is true if proposition x5b is not true. Proposition x4a is true if proposition x4b is not true. Proposition x1b is true if proposition x1a is not true. Proposition etrue is true. Proposition t1 is true if proposition qcore is true. Proposition q is true if proposition t0 is true and proposition etrue is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true and proposition x5a is true. Proposition x1a is true if proposition x1b is not true. Proposition x5b is true if proposition x5a is not true. Proposition x2b is true if proposition x2a is not true. Proposition x3a is true if proposition x3b is not true. Proposition t0 is true if proposition t1 is true. Proposition x4b is true if proposition x4a is not true. Proposition x3b is true if proposition x3a is not true. Proposition x2a is true if proposition x2b is not true. Proposition x0a is true if proposition x0b is not true. Proposition x0b is true if proposition x0a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n6-i6-cred::cred", "rec_id": "hv2-combo-conj_n6-i6", "axis": "combo", "difficulty": "conj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition x5a is true if proposition x5b is not true. Proposition x4a is true if proposition x4b is not true. Proposition x1b is true if proposition x1a is not true. Proposition etrue is true. Proposition t1 is true if proposition qcore is true. Proposition q is true if proposition t0 is true and proposition etrue is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true and proposition x5a is true. Proposition x1a is true if proposition x1b is not true. Proposition x5b is true if proposition x5a is not true. Proposition x2b is true if proposition x2a is not true. Proposition x3a is true if proposition x3b is not true. Proposition t0 is true if proposition t1 is true. Proposition x4b is true if proposition x4a is not true. Proposition x3b is true if proposition x3a is not true. Proposition x2a is true if proposition x2b is not true. Proposition x0a is true if proposition x0b is not true. Proposition x0b is true if proposition x0a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n6-i6-skept::skept", "rec_id": "hv2-combo-conj_n6-i6", "axis": "combo", "difficulty": "conj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition x5a is true if proposition x5b is not true. Proposition x4a is true if proposition x4b is not true. Proposition x1b is true if proposition x1a is not true. Proposition etrue is true. Proposition t1 is true if proposition qcore is true. Proposition q is true if proposition t0 is true and proposition etrue is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true and proposition x5a is true. Proposition x1a is true if proposition x1b is not true. Proposition x5b is true if proposition x5a is not true. Proposition x2b is true if proposition x2a is not true. Proposition x3a is true if proposition x3b is not true. Proposition t0 is true if proposition t1 is true. Proposition x4b is true if proposition x4a is not true. Proposition x3b is true if proposition x3a is not true. Proposition x2a is true if proposition x2b is not true. Proposition x0a is true if proposition x0b is not true. Proposition x0b is true if proposition x0a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n6-i6-wfs::wfs", "rec_id": "hv2-combo-conj_n6-i6", "axis": "combo", "difficulty": "conj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition x5a is true if proposition x5b is not true. Proposition x4a is true if proposition x4b is not true. Proposition x1b is true if proposition x1a is not true. Proposition etrue is true. Proposition t1 is true if proposition qcore is true. Proposition q is true if proposition t0 is true and proposition etrue is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true and proposition x5a is true. Proposition x1a is true if proposition x1b is not true. Proposition x5b is true if proposition x5a is not true. Proposition x2b is true if proposition x2a is not true. Proposition x3a is true if proposition x3b is not true. Proposition t0 is true if proposition t1 is true. Proposition x4b is true if proposition x4a is not true. Proposition x3b is true if proposition x3a is not true. Proposition x2a is true if proposition x2b is not true. Proposition x0a is true if proposition x0b is not true. Proposition x0b is true if proposition x0a is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n6-i7-none::none", "rec_id": "hv2-combo-conj_n6-i7", "axis": "combo", "difficulty": "conj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition s2 is true. Proposition x5a is true if proposition x5b is not true. Proposition x3b is true if proposition x3a is not true. Proposition x3a is true if proposition x3b is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition x0a is true if proposition x0b is not true. Proposition x0b is true if proposition x0a is not true. Proposition etrue is true. Proposition s1 is true. Proposition x1a is true if proposition x1b is not true. Proposition t1 is true if proposition t2 is true. Proposition x2a is true if proposition x2b is not true. Proposition x4b is true if proposition x4a is not true. Proposition x2b is true if proposition x2a is not true. Proposition t2 is true if proposition t3 is true. Proposition x4a is true if proposition x4b is not true. Proposition x1b is true if proposition x1a is not true. Proposition x5b is true if proposition x5a is not true. Proposition t3 is true if proposition qcore is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true and proposition x5a is true. Proposition t0 is true if proposition t1 is true. Proposition s0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n6-i7-closed_world::closed_world", "rec_id": "hv2-combo-conj_n6-i7", "axis": "combo", "difficulty": "conj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition s2 is true. Proposition x5a is true if proposition x5b is not true. Proposition x3b is true if proposition x3a is not true. Proposition x3a is true if proposition x3b is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition x0a is true if proposition x0b is not true. Proposition x0b is true if proposition x0a is not true. Proposition etrue is true. Proposition s1 is true. Proposition x1a is true if proposition x1b is not true. Proposition t1 is true if proposition t2 is true. Proposition x2a is true if proposition x2b is not true. Proposition x4b is true if proposition x4a is not true. Proposition x2b is true if proposition x2a is not true. Proposition t2 is true if proposition t3 is true. Proposition x4a is true if proposition x4b is not true. Proposition x1b is true if proposition x1a is not true. Proposition x5b is true if proposition x5a is not true. Proposition t3 is true if proposition qcore is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true and proposition x5a is true. Proposition t0 is true if proposition t1 is true. Proposition s0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n6-i7-cred::cred", "rec_id": "hv2-combo-conj_n6-i7", "axis": "combo", "difficulty": "conj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition s2 is true. Proposition x5a is true if proposition x5b is not true. Proposition x3b is true if proposition x3a is not true. Proposition x3a is true if proposition x3b is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition x0a is true if proposition x0b is not true. Proposition x0b is true if proposition x0a is not true. Proposition etrue is true. Proposition s1 is true. Proposition x1a is true if proposition x1b is not true. Proposition t1 is true if proposition t2 is true. Proposition x2a is true if proposition x2b is not true. Proposition x4b is true if proposition x4a is not true. Proposition x2b is true if proposition x2a is not true. Proposition t2 is true if proposition t3 is true. Proposition x4a is true if proposition x4b is not true. Proposition x1b is true if proposition x1a is not true. Proposition x5b is true if proposition x5a is not true. Proposition t3 is true if proposition qcore is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true and proposition x5a is true. Proposition t0 is true if proposition t1 is true. Proposition s0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n6-i7-skept::skept", "rec_id": "hv2-combo-conj_n6-i7", "axis": "combo", "difficulty": "conj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition s2 is true. Proposition x5a is true if proposition x5b is not true. Proposition x3b is true if proposition x3a is not true. Proposition x3a is true if proposition x3b is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition x0a is true if proposition x0b is not true. Proposition x0b is true if proposition x0a is not true. Proposition etrue is true. Proposition s1 is true. Proposition x1a is true if proposition x1b is not true. Proposition t1 is true if proposition t2 is true. Proposition x2a is true if proposition x2b is not true. Proposition x4b is true if proposition x4a is not true. Proposition x2b is true if proposition x2a is not true. Proposition t2 is true if proposition t3 is true. Proposition x4a is true if proposition x4b is not true. Proposition x1b is true if proposition x1a is not true. Proposition x5b is true if proposition x5a is not true. Proposition t3 is true if proposition qcore is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true and proposition x5a is true. Proposition t0 is true if proposition t1 is true. Proposition s0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-conj_n6-i7-wfs::wfs", "rec_id": "hv2-combo-conj_n6-i7", "axis": "combo", "difficulty": "conj_n6", "n_stable_models": 64, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition s2 is true. Proposition x5a is true if proposition x5b is not true. Proposition x3b is true if proposition x3a is not true. Proposition x3a is true if proposition x3b is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition x0a is true if proposition x0b is not true. Proposition x0b is true if proposition x0a is not true. Proposition etrue is true. Proposition s1 is true. Proposition x1a is true if proposition x1b is not true. Proposition t1 is true if proposition t2 is true. Proposition x2a is true if proposition x2b is not true. Proposition x4b is true if proposition x4a is not true. Proposition x2b is true if proposition x2a is not true. Proposition t2 is true if proposition t3 is true. Proposition x4a is true if proposition x4b is not true. Proposition x1b is true if proposition x1a is not true. Proposition x5b is true if proposition x5a is not true. Proposition t3 is true if proposition qcore is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition qcore is true if proposition x0a is true and proposition x1a is true and proposition x2a is true and proposition x3a is true and proposition x4a is true and proposition x5a is true. Proposition t0 is true if proposition t1 is true. Proposition s0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n2-i0-none::none", "rec_id": "hv2-combo-coupled_n2-i0", "axis": "combo", "difficulty": "coupled_n2", "n_stable_models": 3, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition t1 is true if proposition t2 is true. Proposition s2 is true. Proposition h0 is true if proposition a0 is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition t3 is true if proposition t4 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t4 is true if proposition t5 is true. Proposition t5 is true if proposition qcore is true. Proposition h1 is true if proposition a1 is not true. Proposition etrue is true. Proposition a1 is true if proposition h1 is not true. Proposition s1 is true. Proposition t2 is true if proposition t3 is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition h0 is true. Proposition s0 is true. Proposition s3 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n2-i0-closed_world::closed_world", "rec_id": "hv2-combo-coupled_n2-i0", "axis": "combo", "difficulty": "coupled_n2", "n_stable_models": 3, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition t1 is true if proposition t2 is true. Proposition s2 is true. Proposition h0 is true if proposition a0 is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition t3 is true if proposition t4 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t4 is true if proposition t5 is true. Proposition t5 is true if proposition qcore is true. Proposition h1 is true if proposition a1 is not true. Proposition etrue is true. Proposition a1 is true if proposition h1 is not true. Proposition s1 is true. Proposition t2 is true if proposition t3 is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition h0 is true. Proposition s0 is true. Proposition s3 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n2-i0-cred::cred", "rec_id": "hv2-combo-coupled_n2-i0", "axis": "combo", "difficulty": "coupled_n2", "n_stable_models": 3, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition t1 is true if proposition t2 is true. Proposition s2 is true. Proposition h0 is true if proposition a0 is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition t3 is true if proposition t4 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t4 is true if proposition t5 is true. Proposition t5 is true if proposition qcore is true. Proposition h1 is true if proposition a1 is not true. Proposition etrue is true. Proposition a1 is true if proposition h1 is not true. Proposition s1 is true. Proposition t2 is true if proposition t3 is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition h0 is true. Proposition s0 is true. Proposition s3 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n2-i0-skept::skept", "rec_id": "hv2-combo-coupled_n2-i0", "axis": "combo", "difficulty": "coupled_n2", "n_stable_models": 3, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition t1 is true if proposition t2 is true. Proposition s2 is true. Proposition h0 is true if proposition a0 is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition t3 is true if proposition t4 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t4 is true if proposition t5 is true. Proposition t5 is true if proposition qcore is true. Proposition h1 is true if proposition a1 is not true. Proposition etrue is true. Proposition a1 is true if proposition h1 is not true. Proposition s1 is true. Proposition t2 is true if proposition t3 is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition h0 is true. Proposition s0 is true. Proposition s3 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n2-i0-wfs::wfs", "rec_id": "hv2-combo-coupled_n2-i0", "axis": "combo", "difficulty": "coupled_n2", "n_stable_models": 3, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition t1 is true if proposition t2 is true. Proposition s2 is true. Proposition h0 is true if proposition a0 is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition t3 is true if proposition t4 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t4 is true if proposition t5 is true. Proposition t5 is true if proposition qcore is true. Proposition h1 is true if proposition a1 is not true. Proposition etrue is true. Proposition a1 is true if proposition h1 is not true. Proposition s1 is true. Proposition t2 is true if proposition t3 is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition h0 is true. Proposition s0 is true. Proposition s3 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n2-i1-none::none", "rec_id": "hv2-combo-coupled_n2-i1", "axis": "combo", "difficulty": "coupled_n2", "n_stable_models": 3, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition a1 is true if proposition h1 is not true. Proposition h1 is true if proposition a1 is not true. Proposition q is true if proposition qcore is true and proposition etrue is true. Proposition qcore is true if proposition h0 is true. Proposition etrue is true. Proposition h0 is true if proposition a0 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n2-i1-closed_world::closed_world", "rec_id": "hv2-combo-coupled_n2-i1", "axis": "combo", "difficulty": "coupled_n2", "n_stable_models": 3, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition a1 is true if proposition h1 is not true. Proposition h1 is true if proposition a1 is not true. Proposition q is true if proposition qcore is true and proposition etrue is true. Proposition qcore is true if proposition h0 is true. Proposition etrue is true. Proposition h0 is true if proposition a0 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n2-i1-cred::cred", "rec_id": "hv2-combo-coupled_n2-i1", "axis": "combo", "difficulty": "coupled_n2", "n_stable_models": 3, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition a1 is true if proposition h1 is not true. Proposition h1 is true if proposition a1 is not true. Proposition q is true if proposition qcore is true and proposition etrue is true. Proposition qcore is true if proposition h0 is true. Proposition etrue is true. Proposition h0 is true if proposition a0 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n2-i1-skept::skept", "rec_id": "hv2-combo-coupled_n2-i1", "axis": "combo", "difficulty": "coupled_n2", "n_stable_models": 3, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition a1 is true if proposition h1 is not true. Proposition h1 is true if proposition a1 is not true. Proposition q is true if proposition qcore is true and proposition etrue is true. Proposition qcore is true if proposition h0 is true. Proposition etrue is true. Proposition h0 is true if proposition a0 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n2-i1-wfs::wfs", "rec_id": "hv2-combo-coupled_n2-i1", "axis": "combo", "difficulty": "coupled_n2", "n_stable_models": 3, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition a1 is true if proposition h1 is not true. Proposition h1 is true if proposition a1 is not true. Proposition q is true if proposition qcore is true and proposition etrue is true. Proposition qcore is true if proposition h0 is true. Proposition etrue is true. Proposition h0 is true if proposition a0 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n2-i2-none::none", "rec_id": "hv2-combo-coupled_n2-i2", "axis": "combo", "difficulty": "coupled_n2", "n_stable_models": 3, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition qcore is true if proposition h0 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition a1 is true if proposition h1 is not true. Proposition h0 is true if proposition a0 is not true. Proposition q is true if proposition qcore is true. Proposition h1 is true if proposition a1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n2-i2-closed_world::closed_world", "rec_id": "hv2-combo-coupled_n2-i2", "axis": "combo", "difficulty": "coupled_n2", "n_stable_models": 3, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition qcore is true if proposition h0 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition a1 is true if proposition h1 is not true. Proposition h0 is true if proposition a0 is not true. Proposition q is true if proposition qcore is true. Proposition h1 is true if proposition a1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n2-i2-cred::cred", "rec_id": "hv2-combo-coupled_n2-i2", "axis": "combo", "difficulty": "coupled_n2", "n_stable_models": 3, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition qcore is true if proposition h0 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition a1 is true if proposition h1 is not true. Proposition h0 is true if proposition a0 is not true. Proposition q is true if proposition qcore is true. Proposition h1 is true if proposition a1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n2-i2-skept::skept", "rec_id": "hv2-combo-coupled_n2-i2", "axis": "combo", "difficulty": "coupled_n2", "n_stable_models": 3, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition qcore is true if proposition h0 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition a1 is true if proposition h1 is not true. Proposition h0 is true if proposition a0 is not true. Proposition q is true if proposition qcore is true. Proposition h1 is true if proposition a1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n2-i2-wfs::wfs", "rec_id": "hv2-combo-coupled_n2-i2", "axis": "combo", "difficulty": "coupled_n2", "n_stable_models": 3, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition qcore is true if proposition h0 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition a1 is true if proposition h1 is not true. Proposition h0 is true if proposition a0 is not true. Proposition q is true if proposition qcore is true. Proposition h1 is true if proposition a1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n2-i3-none::none", "rec_id": "hv2-combo-coupled_n2-i3", "axis": "combo", "difficulty": "coupled_n2", "n_stable_models": 3, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition t1 is true if proposition qcore is true. Proposition t0 is true if proposition t1 is true. Proposition h0 is true if proposition a0 is not true. Proposition a1 is true if proposition h1 is not true. Proposition q is true if proposition t0 is true. Proposition h1 is true if proposition a1 is not true. Proposition qcore is true if proposition h0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n2-i3-closed_world::closed_world", "rec_id": "hv2-combo-coupled_n2-i3", "axis": "combo", "difficulty": "coupled_n2", "n_stable_models": 3, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition t1 is true if proposition qcore is true. Proposition t0 is true if proposition t1 is true. Proposition h0 is true if proposition a0 is not true. Proposition a1 is true if proposition h1 is not true. Proposition q is true if proposition t0 is true. Proposition h1 is true if proposition a1 is not true. Proposition qcore is true if proposition h0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n2-i3-cred::cred", "rec_id": "hv2-combo-coupled_n2-i3", "axis": "combo", "difficulty": "coupled_n2", "n_stable_models": 3, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition t1 is true if proposition qcore is true. Proposition t0 is true if proposition t1 is true. Proposition h0 is true if proposition a0 is not true. Proposition a1 is true if proposition h1 is not true. Proposition q is true if proposition t0 is true. Proposition h1 is true if proposition a1 is not true. Proposition qcore is true if proposition h0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n2-i3-skept::skept", "rec_id": "hv2-combo-coupled_n2-i3", "axis": "combo", "difficulty": "coupled_n2", "n_stable_models": 3, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition t1 is true if proposition qcore is true. Proposition t0 is true if proposition t1 is true. Proposition h0 is true if proposition a0 is not true. Proposition a1 is true if proposition h1 is not true. Proposition q is true if proposition t0 is true. Proposition h1 is true if proposition a1 is not true. Proposition qcore is true if proposition h0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n2-i3-wfs::wfs", "rec_id": "hv2-combo-coupled_n2-i3", "axis": "combo", "difficulty": "coupled_n2", "n_stable_models": 3, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition t1 is true if proposition qcore is true. Proposition t0 is true if proposition t1 is true. Proposition h0 is true if proposition a0 is not true. Proposition a1 is true if proposition h1 is not true. Proposition q is true if proposition t0 is true. Proposition h1 is true if proposition a1 is not true. Proposition qcore is true if proposition h0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n2-i4-none::none", "rec_id": "hv2-combo-coupled_n2-i4", "axis": "combo", "difficulty": "coupled_n2", "n_stable_models": 3, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition qcore is true if proposition h0 is true. Proposition q is true if proposition qcore is true and proposition wide is true and proposition etrue is true. Proposition h0 is true if proposition a0 is not true. Proposition s1 is true. Proposition h1 is true if proposition a1 is not true. Proposition s2 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition a1 is true if proposition h1 is not true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition s3 is true. Proposition etrue is true. Proposition s0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n2-i4-closed_world::closed_world", "rec_id": "hv2-combo-coupled_n2-i4", "axis": "combo", "difficulty": "coupled_n2", "n_stable_models": 3, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition qcore is true if proposition h0 is true. Proposition q is true if proposition qcore is true and proposition wide is true and proposition etrue is true. Proposition h0 is true if proposition a0 is not true. Proposition s1 is true. Proposition h1 is true if proposition a1 is not true. Proposition s2 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition a1 is true if proposition h1 is not true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition s3 is true. Proposition etrue is true. Proposition s0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n2-i4-cred::cred", "rec_id": "hv2-combo-coupled_n2-i4", "axis": "combo", "difficulty": "coupled_n2", "n_stable_models": 3, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition qcore is true if proposition h0 is true. Proposition q is true if proposition qcore is true and proposition wide is true and proposition etrue is true. Proposition h0 is true if proposition a0 is not true. Proposition s1 is true. Proposition h1 is true if proposition a1 is not true. Proposition s2 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition a1 is true if proposition h1 is not true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition s3 is true. Proposition etrue is true. Proposition s0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n2-i4-skept::skept", "rec_id": "hv2-combo-coupled_n2-i4", "axis": "combo", "difficulty": "coupled_n2", "n_stable_models": 3, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition qcore is true if proposition h0 is true. Proposition q is true if proposition qcore is true and proposition wide is true and proposition etrue is true. Proposition h0 is true if proposition a0 is not true. Proposition s1 is true. Proposition h1 is true if proposition a1 is not true. Proposition s2 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition a1 is true if proposition h1 is not true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition s3 is true. Proposition etrue is true. Proposition s0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n2-i4-wfs::wfs", "rec_id": "hv2-combo-coupled_n2-i4", "axis": "combo", "difficulty": "coupled_n2", "n_stable_models": 3, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition qcore is true if proposition h0 is true. Proposition q is true if proposition qcore is true and proposition wide is true and proposition etrue is true. Proposition h0 is true if proposition a0 is not true. Proposition s1 is true. Proposition h1 is true if proposition a1 is not true. Proposition s2 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition a1 is true if proposition h1 is not true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition s3 is true. Proposition etrue is true. Proposition s0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n2-i5-none::none", "rec_id": "hv2-combo-coupled_n2-i5", "axis": "combo", "difficulty": "coupled_n2", "n_stable_models": 3, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition h1 is true if proposition a1 is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition a1 is true if proposition h1 is not true. Proposition h0 is true if proposition a0 is not true. Proposition q is true if proposition qcore is true and proposition wide is true. Proposition qcore is true if proposition h0 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition s0 is true. Proposition s1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n2-i5-closed_world::closed_world", "rec_id": "hv2-combo-coupled_n2-i5", "axis": "combo", "difficulty": "coupled_n2", "n_stable_models": 3, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition h1 is true if proposition a1 is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition a1 is true if proposition h1 is not true. Proposition h0 is true if proposition a0 is not true. Proposition q is true if proposition qcore is true and proposition wide is true. Proposition qcore is true if proposition h0 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition s0 is true. Proposition s1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n2-i5-cred::cred", "rec_id": "hv2-combo-coupled_n2-i5", "axis": "combo", "difficulty": "coupled_n2", "n_stable_models": 3, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition h1 is true if proposition a1 is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition a1 is true if proposition h1 is not true. Proposition h0 is true if proposition a0 is not true. Proposition q is true if proposition qcore is true and proposition wide is true. Proposition qcore is true if proposition h0 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition s0 is true. Proposition s1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n2-i5-skept::skept", "rec_id": "hv2-combo-coupled_n2-i5", "axis": "combo", "difficulty": "coupled_n2", "n_stable_models": 3, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition h1 is true if proposition a1 is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition a1 is true if proposition h1 is not true. Proposition h0 is true if proposition a0 is not true. Proposition q is true if proposition qcore is true and proposition wide is true. Proposition qcore is true if proposition h0 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition s0 is true. Proposition s1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n2-i5-wfs::wfs", "rec_id": "hv2-combo-coupled_n2-i5", "axis": "combo", "difficulty": "coupled_n2", "n_stable_models": 3, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition h1 is true if proposition a1 is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition a1 is true if proposition h1 is not true. Proposition h0 is true if proposition a0 is not true. Proposition q is true if proposition qcore is true and proposition wide is true. Proposition qcore is true if proposition h0 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition s0 is true. Proposition s1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n2-i6-none::none", "rec_id": "hv2-combo-coupled_n2-i6", "axis": "combo", "difficulty": "coupled_n2", "n_stable_models": 3, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition t4 is true if proposition t5 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition h0 is true. Proposition t5 is true if proposition qcore is true. Proposition a1 is true if proposition h1 is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition h1 is true if proposition a1 is not true. Proposition s0 is true. Proposition t1 is true if proposition t2 is true. Proposition s1 is true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition t4 is true. Proposition h0 is true if proposition a0 is not true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n2-i6-closed_world::closed_world", "rec_id": "hv2-combo-coupled_n2-i6", "axis": "combo", "difficulty": "coupled_n2", "n_stable_models": 3, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition t4 is true if proposition t5 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition h0 is true. Proposition t5 is true if proposition qcore is true. Proposition a1 is true if proposition h1 is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition h1 is true if proposition a1 is not true. Proposition s0 is true. Proposition t1 is true if proposition t2 is true. Proposition s1 is true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition t4 is true. Proposition h0 is true if proposition a0 is not true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n2-i6-cred::cred", "rec_id": "hv2-combo-coupled_n2-i6", "axis": "combo", "difficulty": "coupled_n2", "n_stable_models": 3, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition t4 is true if proposition t5 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition h0 is true. Proposition t5 is true if proposition qcore is true. Proposition a1 is true if proposition h1 is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition h1 is true if proposition a1 is not true. Proposition s0 is true. Proposition t1 is true if proposition t2 is true. Proposition s1 is true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition t4 is true. Proposition h0 is true if proposition a0 is not true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n2-i6-skept::skept", "rec_id": "hv2-combo-coupled_n2-i6", "axis": "combo", "difficulty": "coupled_n2", "n_stable_models": 3, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition t4 is true if proposition t5 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition h0 is true. Proposition t5 is true if proposition qcore is true. Proposition a1 is true if proposition h1 is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition h1 is true if proposition a1 is not true. Proposition s0 is true. Proposition t1 is true if proposition t2 is true. Proposition s1 is true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition t4 is true. Proposition h0 is true if proposition a0 is not true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n2-i6-wfs::wfs", "rec_id": "hv2-combo-coupled_n2-i6", "axis": "combo", "difficulty": "coupled_n2", "n_stable_models": 3, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition t4 is true if proposition t5 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition h0 is true. Proposition t5 is true if proposition qcore is true. Proposition a1 is true if proposition h1 is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition h1 is true if proposition a1 is not true. Proposition s0 is true. Proposition t1 is true if proposition t2 is true. Proposition s1 is true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition t4 is true. Proposition h0 is true if proposition a0 is not true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n2-i7-none::none", "rec_id": "hv2-combo-coupled_n2-i7", "axis": "combo", "difficulty": "coupled_n2", "n_stable_models": 3, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition a1 is true if proposition h1 is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition t1 is true if proposition qcore is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition t0 is true if proposition t1 is true. Proposition s2 is true. Proposition qcore is true if proposition h0 is true. Proposition h1 is true if proposition a1 is not true. Proposition s0 is true. Proposition h0 is true if proposition a0 is not true. Proposition s1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n2-i7-closed_world::closed_world", "rec_id": "hv2-combo-coupled_n2-i7", "axis": "combo", "difficulty": "coupled_n2", "n_stable_models": 3, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition a1 is true if proposition h1 is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition t1 is true if proposition qcore is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition t0 is true if proposition t1 is true. Proposition s2 is true. Proposition qcore is true if proposition h0 is true. Proposition h1 is true if proposition a1 is not true. Proposition s0 is true. Proposition h0 is true if proposition a0 is not true. Proposition s1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n2-i7-cred::cred", "rec_id": "hv2-combo-coupled_n2-i7", "axis": "combo", "difficulty": "coupled_n2", "n_stable_models": 3, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition a1 is true if proposition h1 is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition t1 is true if proposition qcore is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition t0 is true if proposition t1 is true. Proposition s2 is true. Proposition qcore is true if proposition h0 is true. Proposition h1 is true if proposition a1 is not true. Proposition s0 is true. Proposition h0 is true if proposition a0 is not true. Proposition s1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n2-i7-skept::skept", "rec_id": "hv2-combo-coupled_n2-i7", "axis": "combo", "difficulty": "coupled_n2", "n_stable_models": 3, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition a1 is true if proposition h1 is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition t1 is true if proposition qcore is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition t0 is true if proposition t1 is true. Proposition s2 is true. Proposition qcore is true if proposition h0 is true. Proposition h1 is true if proposition a1 is not true. Proposition s0 is true. Proposition h0 is true if proposition a0 is not true. Proposition s1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n2-i7-wfs::wfs", "rec_id": "hv2-combo-coupled_n2-i7", "axis": "combo", "difficulty": "coupled_n2", "n_stable_models": 3, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition a1 is true if proposition h1 is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition t1 is true if proposition qcore is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition t0 is true if proposition t1 is true. Proposition s2 is true. Proposition qcore is true if proposition h0 is true. Proposition h1 is true if proposition a1 is not true. Proposition s0 is true. Proposition h0 is true if proposition a0 is not true. Proposition s1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n3-i0-none::none", "rec_id": "hv2-combo-coupled_n3-i0", "axis": "combo", "difficulty": "coupled_n3", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition h0 is true if proposition a0 is not true. Proposition s1 is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition h0 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition a2 is true if proposition h2 is not true. Proposition t1 is true if proposition t2 is true. Proposition t2 is true if proposition t3 is true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition h2 is true if proposition a2 is not true. Proposition s0 is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition etrue is true. Proposition t3 is true if proposition qcore is true. Proposition h1 is true if proposition a1 is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n3-i0-closed_world::closed_world", "rec_id": "hv2-combo-coupled_n3-i0", "axis": "combo", "difficulty": "coupled_n3", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition h0 is true if proposition a0 is not true. Proposition s1 is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition h0 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition a2 is true if proposition h2 is not true. Proposition t1 is true if proposition t2 is true. Proposition t2 is true if proposition t3 is true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition h2 is true if proposition a2 is not true. Proposition s0 is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition etrue is true. Proposition t3 is true if proposition qcore is true. Proposition h1 is true if proposition a1 is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n3-i0-cred::cred", "rec_id": "hv2-combo-coupled_n3-i0", "axis": "combo", "difficulty": "coupled_n3", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition h0 is true if proposition a0 is not true. Proposition s1 is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition h0 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition a2 is true if proposition h2 is not true. Proposition t1 is true if proposition t2 is true. Proposition t2 is true if proposition t3 is true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition h2 is true if proposition a2 is not true. Proposition s0 is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition etrue is true. Proposition t3 is true if proposition qcore is true. Proposition h1 is true if proposition a1 is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n3-i0-skept::skept", "rec_id": "hv2-combo-coupled_n3-i0", "axis": "combo", "difficulty": "coupled_n3", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition h0 is true if proposition a0 is not true. Proposition s1 is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition h0 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition a2 is true if proposition h2 is not true. Proposition t1 is true if proposition t2 is true. Proposition t2 is true if proposition t3 is true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition h2 is true if proposition a2 is not true. Proposition s0 is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition etrue is true. Proposition t3 is true if proposition qcore is true. Proposition h1 is true if proposition a1 is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n3-i0-wfs::wfs", "rec_id": "hv2-combo-coupled_n3-i0", "axis": "combo", "difficulty": "coupled_n3", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition h0 is true if proposition a0 is not true. Proposition s1 is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition h0 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition a2 is true if proposition h2 is not true. Proposition t1 is true if proposition t2 is true. Proposition t2 is true if proposition t3 is true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition h2 is true if proposition a2 is not true. Proposition s0 is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition etrue is true. Proposition t3 is true if proposition qcore is true. Proposition h1 is true if proposition a1 is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n3-i1-none::none", "rec_id": "hv2-combo-coupled_n3-i1", "axis": "combo", "difficulty": "coupled_n3", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition s1 is true. Proposition h0 is true if proposition a0 is not true. Proposition qcore is true if proposition h0 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition h1 is true if proposition a1 is not true. Proposition s0 is true. Proposition s2 is true. Proposition t1 is true if proposition qcore is true. Proposition h2 is true if proposition a2 is not true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition t0 is true if proposition t1 is true. Proposition a2 is true if proposition h2 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n3-i1-closed_world::closed_world", "rec_id": "hv2-combo-coupled_n3-i1", "axis": "combo", "difficulty": "coupled_n3", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition s1 is true. Proposition h0 is true if proposition a0 is not true. Proposition qcore is true if proposition h0 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition h1 is true if proposition a1 is not true. Proposition s0 is true. Proposition s2 is true. Proposition t1 is true if proposition qcore is true. Proposition h2 is true if proposition a2 is not true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition t0 is true if proposition t1 is true. Proposition a2 is true if proposition h2 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n3-i1-cred::cred", "rec_id": "hv2-combo-coupled_n3-i1", "axis": "combo", "difficulty": "coupled_n3", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition s1 is true. Proposition h0 is true if proposition a0 is not true. Proposition qcore is true if proposition h0 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition h1 is true if proposition a1 is not true. Proposition s0 is true. Proposition s2 is true. Proposition t1 is true if proposition qcore is true. Proposition h2 is true if proposition a2 is not true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition t0 is true if proposition t1 is true. Proposition a2 is true if proposition h2 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n3-i1-skept::skept", "rec_id": "hv2-combo-coupled_n3-i1", "axis": "combo", "difficulty": "coupled_n3", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition s1 is true. Proposition h0 is true if proposition a0 is not true. Proposition qcore is true if proposition h0 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition h1 is true if proposition a1 is not true. Proposition s0 is true. Proposition s2 is true. Proposition t1 is true if proposition qcore is true. Proposition h2 is true if proposition a2 is not true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition t0 is true if proposition t1 is true. Proposition a2 is true if proposition h2 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n3-i1-wfs::wfs", "rec_id": "hv2-combo-coupled_n3-i1", "axis": "combo", "difficulty": "coupled_n3", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition s1 is true. Proposition h0 is true if proposition a0 is not true. Proposition qcore is true if proposition h0 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition h1 is true if proposition a1 is not true. Proposition s0 is true. Proposition s2 is true. Proposition t1 is true if proposition qcore is true. Proposition h2 is true if proposition a2 is not true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition t0 is true if proposition t1 is true. Proposition a2 is true if proposition h2 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n3-i2-none::none", "rec_id": "hv2-combo-coupled_n3-i2", "axis": "combo", "difficulty": "coupled_n3", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition a2 is true if proposition h2 is not true. Proposition t4 is true if proposition t5 is true. Proposition t2 is true if proposition t3 is true. Proposition h2 is true if proposition a2 is not true. Proposition s1 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition h1 is true if proposition a1 is not true. Proposition t1 is true if proposition t2 is true. Proposition t5 is true if proposition qcore is true. Proposition s0 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t3 is true if proposition t4 is true. Proposition qcore is true if proposition h0 is true. Proposition t0 is true if proposition t1 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition h0 is true if proposition a0 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n3-i2-closed_world::closed_world", "rec_id": "hv2-combo-coupled_n3-i2", "axis": "combo", "difficulty": "coupled_n3", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition a2 is true if proposition h2 is not true. Proposition t4 is true if proposition t5 is true. Proposition t2 is true if proposition t3 is true. Proposition h2 is true if proposition a2 is not true. Proposition s1 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition h1 is true if proposition a1 is not true. Proposition t1 is true if proposition t2 is true. Proposition t5 is true if proposition qcore is true. Proposition s0 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t3 is true if proposition t4 is true. Proposition qcore is true if proposition h0 is true. Proposition t0 is true if proposition t1 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition h0 is true if proposition a0 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n3-i2-cred::cred", "rec_id": "hv2-combo-coupled_n3-i2", "axis": "combo", "difficulty": "coupled_n3", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition a2 is true if proposition h2 is not true. Proposition t4 is true if proposition t5 is true. Proposition t2 is true if proposition t3 is true. Proposition h2 is true if proposition a2 is not true. Proposition s1 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition h1 is true if proposition a1 is not true. Proposition t1 is true if proposition t2 is true. Proposition t5 is true if proposition qcore is true. Proposition s0 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t3 is true if proposition t4 is true. Proposition qcore is true if proposition h0 is true. Proposition t0 is true if proposition t1 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition h0 is true if proposition a0 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n3-i2-skept::skept", "rec_id": "hv2-combo-coupled_n3-i2", "axis": "combo", "difficulty": "coupled_n3", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition a2 is true if proposition h2 is not true. Proposition t4 is true if proposition t5 is true. Proposition t2 is true if proposition t3 is true. Proposition h2 is true if proposition a2 is not true. Proposition s1 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition h1 is true if proposition a1 is not true. Proposition t1 is true if proposition t2 is true. Proposition t5 is true if proposition qcore is true. Proposition s0 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t3 is true if proposition t4 is true. Proposition qcore is true if proposition h0 is true. Proposition t0 is true if proposition t1 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition h0 is true if proposition a0 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n3-i2-wfs::wfs", "rec_id": "hv2-combo-coupled_n3-i2", "axis": "combo", "difficulty": "coupled_n3", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition a2 is true if proposition h2 is not true. Proposition t4 is true if proposition t5 is true. Proposition t2 is true if proposition t3 is true. Proposition h2 is true if proposition a2 is not true. Proposition s1 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition h1 is true if proposition a1 is not true. Proposition t1 is true if proposition t2 is true. Proposition t5 is true if proposition qcore is true. Proposition s0 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t3 is true if proposition t4 is true. Proposition qcore is true if proposition h0 is true. Proposition t0 is true if proposition t1 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition h0 is true if proposition a0 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n3-i3-none::none", "rec_id": "hv2-combo-coupled_n3-i3", "axis": "combo", "difficulty": "coupled_n3", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition h2 is true if proposition a2 is not true. Proposition s2 is true. Proposition s3 is true. Proposition qcore is true if proposition h0 is true. Proposition t2 is true if proposition t3 is true. Proposition t0 is true if proposition t1 is true. Proposition s0 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition h1 is true if proposition a1 is not true. Proposition s1 is true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition h0 is true if proposition a0 is not true. Proposition t1 is true if proposition t2 is true. Proposition a2 is true if proposition h2 is not true. Proposition t3 is true if proposition qcore is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n3-i3-closed_world::closed_world", "rec_id": "hv2-combo-coupled_n3-i3", "axis": "combo", "difficulty": "coupled_n3", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition h2 is true if proposition a2 is not true. Proposition s2 is true. Proposition s3 is true. Proposition qcore is true if proposition h0 is true. Proposition t2 is true if proposition t3 is true. Proposition t0 is true if proposition t1 is true. Proposition s0 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition h1 is true if proposition a1 is not true. Proposition s1 is true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition h0 is true if proposition a0 is not true. Proposition t1 is true if proposition t2 is true. Proposition a2 is true if proposition h2 is not true. Proposition t3 is true if proposition qcore is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n3-i3-cred::cred", "rec_id": "hv2-combo-coupled_n3-i3", "axis": "combo", "difficulty": "coupled_n3", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition h2 is true if proposition a2 is not true. Proposition s2 is true. Proposition s3 is true. Proposition qcore is true if proposition h0 is true. Proposition t2 is true if proposition t3 is true. Proposition t0 is true if proposition t1 is true. Proposition s0 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition h1 is true if proposition a1 is not true. Proposition s1 is true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition h0 is true if proposition a0 is not true. Proposition t1 is true if proposition t2 is true. Proposition a2 is true if proposition h2 is not true. Proposition t3 is true if proposition qcore is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n3-i3-skept::skept", "rec_id": "hv2-combo-coupled_n3-i3", "axis": "combo", "difficulty": "coupled_n3", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition h2 is true if proposition a2 is not true. Proposition s2 is true. Proposition s3 is true. Proposition qcore is true if proposition h0 is true. Proposition t2 is true if proposition t3 is true. Proposition t0 is true if proposition t1 is true. Proposition s0 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition h1 is true if proposition a1 is not true. Proposition s1 is true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition h0 is true if proposition a0 is not true. Proposition t1 is true if proposition t2 is true. Proposition a2 is true if proposition h2 is not true. Proposition t3 is true if proposition qcore is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n3-i3-wfs::wfs", "rec_id": "hv2-combo-coupled_n3-i3", "axis": "combo", "difficulty": "coupled_n3", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition h2 is true if proposition a2 is not true. Proposition s2 is true. Proposition s3 is true. Proposition qcore is true if proposition h0 is true. Proposition t2 is true if proposition t3 is true. Proposition t0 is true if proposition t1 is true. Proposition s0 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition h1 is true if proposition a1 is not true. Proposition s1 is true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition h0 is true if proposition a0 is not true. Proposition t1 is true if proposition t2 is true. Proposition a2 is true if proposition h2 is not true. Proposition t3 is true if proposition qcore is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n3-i4-none::none", "rec_id": "hv2-combo-coupled_n3-i4", "axis": "combo", "difficulty": "coupled_n3", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition h1 is true if proposition a1 is not true. Proposition qcore is true if proposition h0 is true. Proposition h2 is true if proposition a2 is not true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition s2 is true. Proposition s0 is true. Proposition s1 is true. Proposition a2 is true if proposition h2 is not true. Proposition etrue is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition h0 is true if proposition a0 is not true. Proposition s3 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition q is true if proposition qcore is true and proposition wide is true and proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n3-i4-closed_world::closed_world", "rec_id": "hv2-combo-coupled_n3-i4", "axis": "combo", "difficulty": "coupled_n3", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition h1 is true if proposition a1 is not true. Proposition qcore is true if proposition h0 is true. Proposition h2 is true if proposition a2 is not true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition s2 is true. Proposition s0 is true. Proposition s1 is true. Proposition a2 is true if proposition h2 is not true. Proposition etrue is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition h0 is true if proposition a0 is not true. Proposition s3 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition q is true if proposition qcore is true and proposition wide is true and proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n3-i4-cred::cred", "rec_id": "hv2-combo-coupled_n3-i4", "axis": "combo", "difficulty": "coupled_n3", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition h1 is true if proposition a1 is not true. Proposition qcore is true if proposition h0 is true. Proposition h2 is true if proposition a2 is not true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition s2 is true. Proposition s0 is true. Proposition s1 is true. Proposition a2 is true if proposition h2 is not true. Proposition etrue is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition h0 is true if proposition a0 is not true. Proposition s3 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition q is true if proposition qcore is true and proposition wide is true and proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n3-i4-skept::skept", "rec_id": "hv2-combo-coupled_n3-i4", "axis": "combo", "difficulty": "coupled_n3", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition h1 is true if proposition a1 is not true. Proposition qcore is true if proposition h0 is true. Proposition h2 is true if proposition a2 is not true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition s2 is true. Proposition s0 is true. Proposition s1 is true. Proposition a2 is true if proposition h2 is not true. Proposition etrue is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition h0 is true if proposition a0 is not true. Proposition s3 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition q is true if proposition qcore is true and proposition wide is true and proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n3-i4-wfs::wfs", "rec_id": "hv2-combo-coupled_n3-i4", "axis": "combo", "difficulty": "coupled_n3", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition h1 is true if proposition a1 is not true. Proposition qcore is true if proposition h0 is true. Proposition h2 is true if proposition a2 is not true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition s2 is true. Proposition s0 is true. Proposition s1 is true. Proposition a2 is true if proposition h2 is not true. Proposition etrue is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition h0 is true if proposition a0 is not true. Proposition s3 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition q is true if proposition qcore is true and proposition wide is true and proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n3-i5-none::none", "rec_id": "hv2-combo-coupled_n3-i5", "axis": "combo", "difficulty": "coupled_n3", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition qcore is true if proposition h0 is true. Proposition t1 is true if proposition t2 is true. Proposition t3 is true if proposition qcore is true. Proposition t0 is true if proposition t1 is true. Proposition h1 is true if proposition a1 is not true. Proposition h2 is true if proposition a2 is not true. Proposition a2 is true if proposition h2 is not true. Proposition h0 is true if proposition a0 is not true. Proposition t2 is true if proposition t3 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition q is true if proposition t0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n3-i5-closed_world::closed_world", "rec_id": "hv2-combo-coupled_n3-i5", "axis": "combo", "difficulty": "coupled_n3", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition qcore is true if proposition h0 is true. Proposition t1 is true if proposition t2 is true. Proposition t3 is true if proposition qcore is true. Proposition t0 is true if proposition t1 is true. Proposition h1 is true if proposition a1 is not true. Proposition h2 is true if proposition a2 is not true. Proposition a2 is true if proposition h2 is not true. Proposition h0 is true if proposition a0 is not true. Proposition t2 is true if proposition t3 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition q is true if proposition t0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n3-i5-cred::cred", "rec_id": "hv2-combo-coupled_n3-i5", "axis": "combo", "difficulty": "coupled_n3", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition qcore is true if proposition h0 is true. Proposition t1 is true if proposition t2 is true. Proposition t3 is true if proposition qcore is true. Proposition t0 is true if proposition t1 is true. Proposition h1 is true if proposition a1 is not true. Proposition h2 is true if proposition a2 is not true. Proposition a2 is true if proposition h2 is not true. Proposition h0 is true if proposition a0 is not true. Proposition t2 is true if proposition t3 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition q is true if proposition t0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n3-i5-skept::skept", "rec_id": "hv2-combo-coupled_n3-i5", "axis": "combo", "difficulty": "coupled_n3", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition qcore is true if proposition h0 is true. Proposition t1 is true if proposition t2 is true. Proposition t3 is true if proposition qcore is true. Proposition t0 is true if proposition t1 is true. Proposition h1 is true if proposition a1 is not true. Proposition h2 is true if proposition a2 is not true. Proposition a2 is true if proposition h2 is not true. Proposition h0 is true if proposition a0 is not true. Proposition t2 is true if proposition t3 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition q is true if proposition t0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n3-i5-wfs::wfs", "rec_id": "hv2-combo-coupled_n3-i5", "axis": "combo", "difficulty": "coupled_n3", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition qcore is true if proposition h0 is true. Proposition t1 is true if proposition t2 is true. Proposition t3 is true if proposition qcore is true. Proposition t0 is true if proposition t1 is true. Proposition h1 is true if proposition a1 is not true. Proposition h2 is true if proposition a2 is not true. Proposition a2 is true if proposition h2 is not true. Proposition h0 is true if proposition a0 is not true. Proposition t2 is true if proposition t3 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition q is true if proposition t0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n3-i6-none::none", "rec_id": "hv2-combo-coupled_n3-i6", "axis": "combo", "difficulty": "coupled_n3", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition t2 is true if proposition t3 is true. Proposition t4 is true if proposition t5 is true. Proposition s2 is true. Proposition t3 is true if proposition t4 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition s1 is true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition h0 is true if proposition a0 is not true. Proposition t5 is true if proposition qcore is true. Proposition a2 is true if proposition h2 is not true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition h2 is true if proposition a2 is not true. Proposition t1 is true if proposition t2 is true. Proposition s0 is true. Proposition etrue is true. Proposition h1 is true if proposition a1 is not true. Proposition qcore is true if proposition h0 is true. Proposition t0 is true if proposition t1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n3-i6-closed_world::closed_world", "rec_id": "hv2-combo-coupled_n3-i6", "axis": "combo", "difficulty": "coupled_n3", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition t2 is true if proposition t3 is true. Proposition t4 is true if proposition t5 is true. Proposition s2 is true. Proposition t3 is true if proposition t4 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition s1 is true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition h0 is true if proposition a0 is not true. Proposition t5 is true if proposition qcore is true. Proposition a2 is true if proposition h2 is not true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition h2 is true if proposition a2 is not true. Proposition t1 is true if proposition t2 is true. Proposition s0 is true. Proposition etrue is true. Proposition h1 is true if proposition a1 is not true. Proposition qcore is true if proposition h0 is true. Proposition t0 is true if proposition t1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n3-i6-cred::cred", "rec_id": "hv2-combo-coupled_n3-i6", "axis": "combo", "difficulty": "coupled_n3", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition t2 is true if proposition t3 is true. Proposition t4 is true if proposition t5 is true. Proposition s2 is true. Proposition t3 is true if proposition t4 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition s1 is true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition h0 is true if proposition a0 is not true. Proposition t5 is true if proposition qcore is true. Proposition a2 is true if proposition h2 is not true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition h2 is true if proposition a2 is not true. Proposition t1 is true if proposition t2 is true. Proposition s0 is true. Proposition etrue is true. Proposition h1 is true if proposition a1 is not true. Proposition qcore is true if proposition h0 is true. Proposition t0 is true if proposition t1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n3-i6-skept::skept", "rec_id": "hv2-combo-coupled_n3-i6", "axis": "combo", "difficulty": "coupled_n3", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition t2 is true if proposition t3 is true. Proposition t4 is true if proposition t5 is true. Proposition s2 is true. Proposition t3 is true if proposition t4 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition s1 is true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition h0 is true if proposition a0 is not true. Proposition t5 is true if proposition qcore is true. Proposition a2 is true if proposition h2 is not true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition h2 is true if proposition a2 is not true. Proposition t1 is true if proposition t2 is true. Proposition s0 is true. Proposition etrue is true. Proposition h1 is true if proposition a1 is not true. Proposition qcore is true if proposition h0 is true. Proposition t0 is true if proposition t1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n3-i6-wfs::wfs", "rec_id": "hv2-combo-coupled_n3-i6", "axis": "combo", "difficulty": "coupled_n3", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition t2 is true if proposition t3 is true. Proposition t4 is true if proposition t5 is true. Proposition s2 is true. Proposition t3 is true if proposition t4 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition s1 is true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition h0 is true if proposition a0 is not true. Proposition t5 is true if proposition qcore is true. Proposition a2 is true if proposition h2 is not true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition h2 is true if proposition a2 is not true. Proposition t1 is true if proposition t2 is true. Proposition s0 is true. Proposition etrue is true. Proposition h1 is true if proposition a1 is not true. Proposition qcore is true if proposition h0 is true. Proposition t0 is true if proposition t1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n3-i7-none::none", "rec_id": "hv2-combo-coupled_n3-i7", "axis": "combo", "difficulty": "coupled_n3", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition h0 is true if proposition a0 is not true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition h1 is true if proposition a1 is not true. Proposition s0 is true. Proposition h2 is true if proposition a2 is not true. Proposition s1 is true. Proposition t1 is true if proposition qcore is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition etrue is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition a2 is true if proposition h2 is not true. Proposition s2 is true. Proposition t0 is true if proposition t1 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition qcore is true if proposition h0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n3-i7-closed_world::closed_world", "rec_id": "hv2-combo-coupled_n3-i7", "axis": "combo", "difficulty": "coupled_n3", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition h0 is true if proposition a0 is not true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition h1 is true if proposition a1 is not true. Proposition s0 is true. Proposition h2 is true if proposition a2 is not true. Proposition s1 is true. Proposition t1 is true if proposition qcore is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition etrue is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition a2 is true if proposition h2 is not true. Proposition s2 is true. Proposition t0 is true if proposition t1 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition qcore is true if proposition h0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n3-i7-cred::cred", "rec_id": "hv2-combo-coupled_n3-i7", "axis": "combo", "difficulty": "coupled_n3", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition h0 is true if proposition a0 is not true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition h1 is true if proposition a1 is not true. Proposition s0 is true. Proposition h2 is true if proposition a2 is not true. Proposition s1 is true. Proposition t1 is true if proposition qcore is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition etrue is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition a2 is true if proposition h2 is not true. Proposition s2 is true. Proposition t0 is true if proposition t1 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition qcore is true if proposition h0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n3-i7-skept::skept", "rec_id": "hv2-combo-coupled_n3-i7", "axis": "combo", "difficulty": "coupled_n3", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition h0 is true if proposition a0 is not true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition h1 is true if proposition a1 is not true. Proposition s0 is true. Proposition h2 is true if proposition a2 is not true. Proposition s1 is true. Proposition t1 is true if proposition qcore is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition etrue is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition a2 is true if proposition h2 is not true. Proposition s2 is true. Proposition t0 is true if proposition t1 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition qcore is true if proposition h0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n3-i7-wfs::wfs", "rec_id": "hv2-combo-coupled_n3-i7", "axis": "combo", "difficulty": "coupled_n3", "n_stable_models": 4, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition h0 is true if proposition a0 is not true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition h1 is true if proposition a1 is not true. Proposition s0 is true. Proposition h2 is true if proposition a2 is not true. Proposition s1 is true. Proposition t1 is true if proposition qcore is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition etrue is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition a2 is true if proposition h2 is not true. Proposition s2 is true. Proposition t0 is true if proposition t1 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition qcore is true if proposition h0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n4-i0-none::none", "rec_id": "hv2-combo-coupled_n4-i0", "axis": "combo", "difficulty": "coupled_n4", "n_stable_models": 5, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition qcore is true if proposition h0 is true. Proposition t0 is true if proposition t1 is true. Proposition s2 is true. Proposition h1 is true if proposition a1 is not true. Proposition t5 is true if proposition qcore is true. Proposition t2 is true if proposition t3 is true. Proposition h2 is true if proposition a2 is not true. Proposition h3 is true if proposition a3 is not true. Proposition s0 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition t4 is true if proposition t5 is true. Proposition h0 is true if proposition a0 is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t3 is true if proposition t4 is true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition s3 is true. Proposition s1 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition t1 is true if proposition t2 is true. Proposition a3 is true if proposition h3 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n4-i0-closed_world::closed_world", "rec_id": "hv2-combo-coupled_n4-i0", "axis": "combo", "difficulty": "coupled_n4", "n_stable_models": 5, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition qcore is true if proposition h0 is true. Proposition t0 is true if proposition t1 is true. Proposition s2 is true. Proposition h1 is true if proposition a1 is not true. Proposition t5 is true if proposition qcore is true. Proposition t2 is true if proposition t3 is true. Proposition h2 is true if proposition a2 is not true. Proposition h3 is true if proposition a3 is not true. Proposition s0 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition t4 is true if proposition t5 is true. Proposition h0 is true if proposition a0 is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t3 is true if proposition t4 is true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition s3 is true. Proposition s1 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition t1 is true if proposition t2 is true. Proposition a3 is true if proposition h3 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n4-i0-cred::cred", "rec_id": "hv2-combo-coupled_n4-i0", "axis": "combo", "difficulty": "coupled_n4", "n_stable_models": 5, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition qcore is true if proposition h0 is true. Proposition t0 is true if proposition t1 is true. Proposition s2 is true. Proposition h1 is true if proposition a1 is not true. Proposition t5 is true if proposition qcore is true. Proposition t2 is true if proposition t3 is true. Proposition h2 is true if proposition a2 is not true. Proposition h3 is true if proposition a3 is not true. Proposition s0 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition t4 is true if proposition t5 is true. Proposition h0 is true if proposition a0 is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t3 is true if proposition t4 is true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition s3 is true. Proposition s1 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition t1 is true if proposition t2 is true. Proposition a3 is true if proposition h3 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n4-i0-skept::skept", "rec_id": "hv2-combo-coupled_n4-i0", "axis": "combo", "difficulty": "coupled_n4", "n_stable_models": 5, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition qcore is true if proposition h0 is true. Proposition t0 is true if proposition t1 is true. Proposition s2 is true. Proposition h1 is true if proposition a1 is not true. Proposition t5 is true if proposition qcore is true. Proposition t2 is true if proposition t3 is true. Proposition h2 is true if proposition a2 is not true. Proposition h3 is true if proposition a3 is not true. Proposition s0 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition t4 is true if proposition t5 is true. Proposition h0 is true if proposition a0 is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t3 is true if proposition t4 is true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition s3 is true. Proposition s1 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition t1 is true if proposition t2 is true. Proposition a3 is true if proposition h3 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n4-i0-wfs::wfs", "rec_id": "hv2-combo-coupled_n4-i0", "axis": "combo", "difficulty": "coupled_n4", "n_stable_models": 5, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition qcore is true if proposition h0 is true. Proposition t0 is true if proposition t1 is true. Proposition s2 is true. Proposition h1 is true if proposition a1 is not true. Proposition t5 is true if proposition qcore is true. Proposition t2 is true if proposition t3 is true. Proposition h2 is true if proposition a2 is not true. Proposition h3 is true if proposition a3 is not true. Proposition s0 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition t4 is true if proposition t5 is true. Proposition h0 is true if proposition a0 is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t3 is true if proposition t4 is true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition s3 is true. Proposition s1 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition t1 is true if proposition t2 is true. Proposition a3 is true if proposition h3 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n4-i1-none::none", "rec_id": "hv2-combo-coupled_n4-i1", "axis": "combo", "difficulty": "coupled_n4", "n_stable_models": 5, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s2 is true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition h1 is true if proposition a1 is not true. Proposition qcore is true if proposition h0 is true. Proposition s1 is true. Proposition h3 is true if proposition a3 is not true. Proposition a3 is true if proposition h3 is not true. Proposition h2 is true if proposition a2 is not true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition q is true if proposition qcore is true and proposition wide is true. Proposition s0 is true. Proposition s3 is true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition h0 is true if proposition a0 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n4-i1-closed_world::closed_world", "rec_id": "hv2-combo-coupled_n4-i1", "axis": "combo", "difficulty": "coupled_n4", "n_stable_models": 5, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s2 is true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition h1 is true if proposition a1 is not true. Proposition qcore is true if proposition h0 is true. Proposition s1 is true. Proposition h3 is true if proposition a3 is not true. Proposition a3 is true if proposition h3 is not true. Proposition h2 is true if proposition a2 is not true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition q is true if proposition qcore is true and proposition wide is true. Proposition s0 is true. Proposition s3 is true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition h0 is true if proposition a0 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n4-i1-cred::cred", "rec_id": "hv2-combo-coupled_n4-i1", "axis": "combo", "difficulty": "coupled_n4", "n_stable_models": 5, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s2 is true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition h1 is true if proposition a1 is not true. Proposition qcore is true if proposition h0 is true. Proposition s1 is true. Proposition h3 is true if proposition a3 is not true. Proposition a3 is true if proposition h3 is not true. Proposition h2 is true if proposition a2 is not true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition q is true if proposition qcore is true and proposition wide is true. Proposition s0 is true. Proposition s3 is true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition h0 is true if proposition a0 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n4-i1-skept::skept", "rec_id": "hv2-combo-coupled_n4-i1", "axis": "combo", "difficulty": "coupled_n4", "n_stable_models": 5, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s2 is true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition h1 is true if proposition a1 is not true. Proposition qcore is true if proposition h0 is true. Proposition s1 is true. Proposition h3 is true if proposition a3 is not true. Proposition a3 is true if proposition h3 is not true. Proposition h2 is true if proposition a2 is not true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition q is true if proposition qcore is true and proposition wide is true. Proposition s0 is true. Proposition s3 is true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition h0 is true if proposition a0 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n4-i1-wfs::wfs", "rec_id": "hv2-combo-coupled_n4-i1", "axis": "combo", "difficulty": "coupled_n4", "n_stable_models": 5, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s2 is true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition h1 is true if proposition a1 is not true. Proposition qcore is true if proposition h0 is true. Proposition s1 is true. Proposition h3 is true if proposition a3 is not true. Proposition a3 is true if proposition h3 is not true. Proposition h2 is true if proposition a2 is not true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition q is true if proposition qcore is true and proposition wide is true. Proposition s0 is true. Proposition s3 is true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition h0 is true if proposition a0 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n4-i2-none::none", "rec_id": "hv2-combo-coupled_n4-i2", "axis": "combo", "difficulty": "coupled_n4", "n_stable_models": 5, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition a3 is true if proposition h3 is not true. Proposition h1 is true if proposition a1 is not true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition t1 is true if proposition qcore is true. Proposition h3 is true if proposition a3 is not true. Proposition h2 is true if proposition a2 is not true. Proposition s0 is true. Proposition qcore is true if proposition h0 is true. Proposition s1 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition t0 is true if proposition t1 is true. Proposition h0 is true if proposition a0 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n4-i2-closed_world::closed_world", "rec_id": "hv2-combo-coupled_n4-i2", "axis": "combo", "difficulty": "coupled_n4", "n_stable_models": 5, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition a3 is true if proposition h3 is not true. Proposition h1 is true if proposition a1 is not true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition t1 is true if proposition qcore is true. Proposition h3 is true if proposition a3 is not true. Proposition h2 is true if proposition a2 is not true. Proposition s0 is true. Proposition qcore is true if proposition h0 is true. Proposition s1 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition t0 is true if proposition t1 is true. Proposition h0 is true if proposition a0 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n4-i2-cred::cred", "rec_id": "hv2-combo-coupled_n4-i2", "axis": "combo", "difficulty": "coupled_n4", "n_stable_models": 5, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition a3 is true if proposition h3 is not true. Proposition h1 is true if proposition a1 is not true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition t1 is true if proposition qcore is true. Proposition h3 is true if proposition a3 is not true. Proposition h2 is true if proposition a2 is not true. Proposition s0 is true. Proposition qcore is true if proposition h0 is true. Proposition s1 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition t0 is true if proposition t1 is true. Proposition h0 is true if proposition a0 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n4-i2-skept::skept", "rec_id": "hv2-combo-coupled_n4-i2", "axis": "combo", "difficulty": "coupled_n4", "n_stable_models": 5, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition a3 is true if proposition h3 is not true. Proposition h1 is true if proposition a1 is not true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition t1 is true if proposition qcore is true. Proposition h3 is true if proposition a3 is not true. Proposition h2 is true if proposition a2 is not true. Proposition s0 is true. Proposition qcore is true if proposition h0 is true. Proposition s1 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition t0 is true if proposition t1 is true. Proposition h0 is true if proposition a0 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n4-i2-wfs::wfs", "rec_id": "hv2-combo-coupled_n4-i2", "axis": "combo", "difficulty": "coupled_n4", "n_stable_models": 5, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition a3 is true if proposition h3 is not true. Proposition h1 is true if proposition a1 is not true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition t1 is true if proposition qcore is true. Proposition h3 is true if proposition a3 is not true. Proposition h2 is true if proposition a2 is not true. Proposition s0 is true. Proposition qcore is true if proposition h0 is true. Proposition s1 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition t0 is true if proposition t1 is true. Proposition h0 is true if proposition a0 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n4-i3-none::none", "rec_id": "hv2-combo-coupled_n4-i3", "axis": "combo", "difficulty": "coupled_n4", "n_stable_models": 5, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition s1 is true. Proposition h3 is true if proposition a3 is not true. Proposition qcore is true if proposition h0 is true. Proposition a3 is true if proposition h3 is not true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition q is true if proposition qcore is true and proposition wide is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition s0 is true. Proposition h0 is true if proposition a0 is not true. Proposition h2 is true if proposition a2 is not true. Proposition h1 is true if proposition a1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n4-i3-closed_world::closed_world", "rec_id": "hv2-combo-coupled_n4-i3", "axis": "combo", "difficulty": "coupled_n4", "n_stable_models": 5, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition s1 is true. Proposition h3 is true if proposition a3 is not true. Proposition qcore is true if proposition h0 is true. Proposition a3 is true if proposition h3 is not true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition q is true if proposition qcore is true and proposition wide is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition s0 is true. Proposition h0 is true if proposition a0 is not true. Proposition h2 is true if proposition a2 is not true. Proposition h1 is true if proposition a1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n4-i3-cred::cred", "rec_id": "hv2-combo-coupled_n4-i3", "axis": "combo", "difficulty": "coupled_n4", "n_stable_models": 5, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition s1 is true. Proposition h3 is true if proposition a3 is not true. Proposition qcore is true if proposition h0 is true. Proposition a3 is true if proposition h3 is not true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition q is true if proposition qcore is true and proposition wide is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition s0 is true. Proposition h0 is true if proposition a0 is not true. Proposition h2 is true if proposition a2 is not true. Proposition h1 is true if proposition a1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n4-i3-skept::skept", "rec_id": "hv2-combo-coupled_n4-i3", "axis": "combo", "difficulty": "coupled_n4", "n_stable_models": 5, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition s1 is true. Proposition h3 is true if proposition a3 is not true. Proposition qcore is true if proposition h0 is true. Proposition a3 is true if proposition h3 is not true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition q is true if proposition qcore is true and proposition wide is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition s0 is true. Proposition h0 is true if proposition a0 is not true. Proposition h2 is true if proposition a2 is not true. Proposition h1 is true if proposition a1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n4-i3-wfs::wfs", "rec_id": "hv2-combo-coupled_n4-i3", "axis": "combo", "difficulty": "coupled_n4", "n_stable_models": 5, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition s1 is true. Proposition h3 is true if proposition a3 is not true. Proposition qcore is true if proposition h0 is true. Proposition a3 is true if proposition h3 is not true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition q is true if proposition qcore is true and proposition wide is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition s0 is true. Proposition h0 is true if proposition a0 is not true. Proposition h2 is true if proposition a2 is not true. Proposition h1 is true if proposition a1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n4-i4-none::none", "rec_id": "hv2-combo-coupled_n4-i4", "axis": "combo", "difficulty": "coupled_n4", "n_stable_models": 5, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition q is true if proposition t0 is true and proposition wide is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition t1 is true if proposition t2 is true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t3 is true if proposition qcore is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition h0 is true. Proposition a3 is true if proposition h3 is not true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition t2 is true if proposition t3 is true. Proposition h3 is true if proposition a3 is not true. Proposition h0 is true if proposition a0 is not true. Proposition s0 is true. Proposition s1 is true. Proposition h1 is true if proposition a1 is not true. Proposition h2 is true if proposition a2 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n4-i4-closed_world::closed_world", "rec_id": "hv2-combo-coupled_n4-i4", "axis": "combo", "difficulty": "coupled_n4", "n_stable_models": 5, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition q is true if proposition t0 is true and proposition wide is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition t1 is true if proposition t2 is true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t3 is true if proposition qcore is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition h0 is true. Proposition a3 is true if proposition h3 is not true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition t2 is true if proposition t3 is true. Proposition h3 is true if proposition a3 is not true. Proposition h0 is true if proposition a0 is not true. Proposition s0 is true. Proposition s1 is true. Proposition h1 is true if proposition a1 is not true. Proposition h2 is true if proposition a2 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n4-i4-cred::cred", "rec_id": "hv2-combo-coupled_n4-i4", "axis": "combo", "difficulty": "coupled_n4", "n_stable_models": 5, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition q is true if proposition t0 is true and proposition wide is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition t1 is true if proposition t2 is true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t3 is true if proposition qcore is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition h0 is true. Proposition a3 is true if proposition h3 is not true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition t2 is true if proposition t3 is true. Proposition h3 is true if proposition a3 is not true. Proposition h0 is true if proposition a0 is not true. Proposition s0 is true. Proposition s1 is true. Proposition h1 is true if proposition a1 is not true. Proposition h2 is true if proposition a2 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n4-i4-skept::skept", "rec_id": "hv2-combo-coupled_n4-i4", "axis": "combo", "difficulty": "coupled_n4", "n_stable_models": 5, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition q is true if proposition t0 is true and proposition wide is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition t1 is true if proposition t2 is true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t3 is true if proposition qcore is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition h0 is true. Proposition a3 is true if proposition h3 is not true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition t2 is true if proposition t3 is true. Proposition h3 is true if proposition a3 is not true. Proposition h0 is true if proposition a0 is not true. Proposition s0 is true. Proposition s1 is true. Proposition h1 is true if proposition a1 is not true. Proposition h2 is true if proposition a2 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n4-i4-wfs::wfs", "rec_id": "hv2-combo-coupled_n4-i4", "axis": "combo", "difficulty": "coupled_n4", "n_stable_models": 5, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition q is true if proposition t0 is true and proposition wide is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition t1 is true if proposition t2 is true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t3 is true if proposition qcore is true. Proposition t0 is true if proposition t1 is true. Proposition qcore is true if proposition h0 is true. Proposition a3 is true if proposition h3 is not true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition t2 is true if proposition t3 is true. Proposition h3 is true if proposition a3 is not true. Proposition h0 is true if proposition a0 is not true. Proposition s0 is true. Proposition s1 is true. Proposition h1 is true if proposition a1 is not true. Proposition h2 is true if proposition a2 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n4-i5-none::none", "rec_id": "hv2-combo-coupled_n4-i5", "axis": "combo", "difficulty": "coupled_n4", "n_stable_models": 5, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition qcore is true if proposition h0 is true. Proposition a3 is true if proposition h3 is not true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition h2 is true if proposition a2 is not true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition s1 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition h1 is true if proposition a1 is not true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition s0 is true. Proposition h3 is true if proposition a3 is not true. Proposition t1 is true if proposition qcore is true. Proposition h0 is true if proposition a0 is not true. Proposition t0 is true if proposition t1 is true. Proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n4-i5-closed_world::closed_world", "rec_id": "hv2-combo-coupled_n4-i5", "axis": "combo", "difficulty": "coupled_n4", "n_stable_models": 5, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition qcore is true if proposition h0 is true. Proposition a3 is true if proposition h3 is not true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition h2 is true if proposition a2 is not true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition s1 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition h1 is true if proposition a1 is not true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition s0 is true. Proposition h3 is true if proposition a3 is not true. Proposition t1 is true if proposition qcore is true. Proposition h0 is true if proposition a0 is not true. Proposition t0 is true if proposition t1 is true. Proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n4-i5-cred::cred", "rec_id": "hv2-combo-coupled_n4-i5", "axis": "combo", "difficulty": "coupled_n4", "n_stable_models": 5, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition qcore is true if proposition h0 is true. Proposition a3 is true if proposition h3 is not true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition h2 is true if proposition a2 is not true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition s1 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition h1 is true if proposition a1 is not true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition s0 is true. Proposition h3 is true if proposition a3 is not true. Proposition t1 is true if proposition qcore is true. Proposition h0 is true if proposition a0 is not true. Proposition t0 is true if proposition t1 is true. Proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n4-i5-skept::skept", "rec_id": "hv2-combo-coupled_n4-i5", "axis": "combo", "difficulty": "coupled_n4", "n_stable_models": 5, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition qcore is true if proposition h0 is true. Proposition a3 is true if proposition h3 is not true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition h2 is true if proposition a2 is not true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition s1 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition h1 is true if proposition a1 is not true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition s0 is true. Proposition h3 is true if proposition a3 is not true. Proposition t1 is true if proposition qcore is true. Proposition h0 is true if proposition a0 is not true. Proposition t0 is true if proposition t1 is true. Proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n4-i5-wfs::wfs", "rec_id": "hv2-combo-coupled_n4-i5", "axis": "combo", "difficulty": "coupled_n4", "n_stable_models": 5, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition qcore is true if proposition h0 is true. Proposition a3 is true if proposition h3 is not true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition h2 is true if proposition a2 is not true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition s1 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition h1 is true if proposition a1 is not true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition s0 is true. Proposition h3 is true if proposition a3 is not true. Proposition t1 is true if proposition qcore is true. Proposition h0 is true if proposition a0 is not true. Proposition t0 is true if proposition t1 is true. Proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n4-i6-none::none", "rec_id": "hv2-combo-coupled_n4-i6", "axis": "combo", "difficulty": "coupled_n4", "n_stable_models": 5, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition h1 is true if proposition a1 is not true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition qcore is true if proposition h0 is true. Proposition h2 is true if proposition a2 is not true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition qcore is true. Proposition q is true if proposition t0 is true. Proposition t1 is true if proposition t2 is true. Proposition h3 is true if proposition a3 is not true. Proposition a3 is true if proposition h3 is not true. Proposition t0 is true if proposition t1 is true. Proposition h0 is true if proposition a0 is not true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n4-i6-closed_world::closed_world", "rec_id": "hv2-combo-coupled_n4-i6", "axis": "combo", "difficulty": "coupled_n4", "n_stable_models": 5, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition h1 is true if proposition a1 is not true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition qcore is true if proposition h0 is true. Proposition h2 is true if proposition a2 is not true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition qcore is true. Proposition q is true if proposition t0 is true. Proposition t1 is true if proposition t2 is true. Proposition h3 is true if proposition a3 is not true. Proposition a3 is true if proposition h3 is not true. Proposition t0 is true if proposition t1 is true. Proposition h0 is true if proposition a0 is not true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n4-i6-cred::cred", "rec_id": "hv2-combo-coupled_n4-i6", "axis": "combo", "difficulty": "coupled_n4", "n_stable_models": 5, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition h1 is true if proposition a1 is not true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition qcore is true if proposition h0 is true. Proposition h2 is true if proposition a2 is not true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition qcore is true. Proposition q is true if proposition t0 is true. Proposition t1 is true if proposition t2 is true. Proposition h3 is true if proposition a3 is not true. Proposition a3 is true if proposition h3 is not true. Proposition t0 is true if proposition t1 is true. Proposition h0 is true if proposition a0 is not true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n4-i6-skept::skept", "rec_id": "hv2-combo-coupled_n4-i6", "axis": "combo", "difficulty": "coupled_n4", "n_stable_models": 5, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition h1 is true if proposition a1 is not true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition qcore is true if proposition h0 is true. Proposition h2 is true if proposition a2 is not true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition qcore is true. Proposition q is true if proposition t0 is true. Proposition t1 is true if proposition t2 is true. Proposition h3 is true if proposition a3 is not true. Proposition a3 is true if proposition h3 is not true. Proposition t0 is true if proposition t1 is true. Proposition h0 is true if proposition a0 is not true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n4-i6-wfs::wfs", "rec_id": "hv2-combo-coupled_n4-i6", "axis": "combo", "difficulty": "coupled_n4", "n_stable_models": 5, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition h1 is true if proposition a1 is not true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition qcore is true if proposition h0 is true. Proposition h2 is true if proposition a2 is not true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition qcore is true. Proposition q is true if proposition t0 is true. Proposition t1 is true if proposition t2 is true. Proposition h3 is true if proposition a3 is not true. Proposition a3 is true if proposition h3 is not true. Proposition t0 is true if proposition t1 is true. Proposition h0 is true if proposition a0 is not true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n4-i7-none::none", "rec_id": "hv2-combo-coupled_n4-i7", "axis": "combo", "difficulty": "coupled_n4", "n_stable_models": 5, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition t0 is true if proposition t1 is true. Proposition s1 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition etrue is true. Proposition a3 is true if proposition h3 is not true. Proposition h3 is true if proposition a3 is not true. Proposition h2 is true if proposition a2 is not true. Proposition s0 is true. Proposition s2 is true. Proposition t1 is true if proposition t2 is true. Proposition t3 is true if proposition qcore is true. Proposition h0 is true if proposition a0 is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition s3 is true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition qcore is true if proposition h0 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t2 is true if proposition t3 is true. Proposition h1 is true if proposition a1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n4-i7-closed_world::closed_world", "rec_id": "hv2-combo-coupled_n4-i7", "axis": "combo", "difficulty": "coupled_n4", "n_stable_models": 5, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition t0 is true if proposition t1 is true. Proposition s1 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition etrue is true. Proposition a3 is true if proposition h3 is not true. Proposition h3 is true if proposition a3 is not true. Proposition h2 is true if proposition a2 is not true. Proposition s0 is true. Proposition s2 is true. Proposition t1 is true if proposition t2 is true. Proposition t3 is true if proposition qcore is true. Proposition h0 is true if proposition a0 is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition s3 is true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition qcore is true if proposition h0 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t2 is true if proposition t3 is true. Proposition h1 is true if proposition a1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n4-i7-cred::cred", "rec_id": "hv2-combo-coupled_n4-i7", "axis": "combo", "difficulty": "coupled_n4", "n_stable_models": 5, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition t0 is true if proposition t1 is true. Proposition s1 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition etrue is true. Proposition a3 is true if proposition h3 is not true. Proposition h3 is true if proposition a3 is not true. Proposition h2 is true if proposition a2 is not true. Proposition s0 is true. Proposition s2 is true. Proposition t1 is true if proposition t2 is true. Proposition t3 is true if proposition qcore is true. Proposition h0 is true if proposition a0 is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition s3 is true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition qcore is true if proposition h0 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t2 is true if proposition t3 is true. Proposition h1 is true if proposition a1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n4-i7-skept::skept", "rec_id": "hv2-combo-coupled_n4-i7", "axis": "combo", "difficulty": "coupled_n4", "n_stable_models": 5, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition t0 is true if proposition t1 is true. Proposition s1 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition etrue is true. Proposition a3 is true if proposition h3 is not true. Proposition h3 is true if proposition a3 is not true. Proposition h2 is true if proposition a2 is not true. Proposition s0 is true. Proposition s2 is true. Proposition t1 is true if proposition t2 is true. Proposition t3 is true if proposition qcore is true. Proposition h0 is true if proposition a0 is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition s3 is true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition qcore is true if proposition h0 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t2 is true if proposition t3 is true. Proposition h1 is true if proposition a1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n4-i7-wfs::wfs", "rec_id": "hv2-combo-coupled_n4-i7", "axis": "combo", "difficulty": "coupled_n4", "n_stable_models": 5, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition t0 is true if proposition t1 is true. Proposition s1 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition etrue is true. Proposition a3 is true if proposition h3 is not true. Proposition h3 is true if proposition a3 is not true. Proposition h2 is true if proposition a2 is not true. Proposition s0 is true. Proposition s2 is true. Proposition t1 is true if proposition t2 is true. Proposition t3 is true if proposition qcore is true. Proposition h0 is true if proposition a0 is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition s3 is true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition qcore is true if proposition h0 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t2 is true if proposition t3 is true. Proposition h1 is true if proposition a1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n5-i0-none::none", "rec_id": "hv2-combo-coupled_n5-i0", "axis": "combo", "difficulty": "coupled_n5", "n_stable_models": 6, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition t4 is true if proposition t5 is true. Proposition h1 is true if proposition a1 is not true. Proposition s0 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition h0 is true if proposition a0 is not true. Proposition t2 is true if proposition t3 is true. Proposition t5 is true if proposition qcore is true. Proposition a3 is true if proposition h3 is not true and proposition h4 is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition t1 is true if proposition t2 is true. Proposition a4 is true if proposition h4 is not true. Proposition h2 is true if proposition a2 is not true. Proposition h4 is true if proposition a4 is not true. Proposition h3 is true if proposition a3 is not true. Proposition qcore is true if proposition h0 is true. Proposition t3 is true if proposition t4 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition s1 is true. Proposition t0 is true if proposition t1 is true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n5-i0-closed_world::closed_world", "rec_id": "hv2-combo-coupled_n5-i0", "axis": "combo", "difficulty": "coupled_n5", "n_stable_models": 6, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition t4 is true if proposition t5 is true. Proposition h1 is true if proposition a1 is not true. Proposition s0 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition h0 is true if proposition a0 is not true. Proposition t2 is true if proposition t3 is true. Proposition t5 is true if proposition qcore is true. Proposition a3 is true if proposition h3 is not true and proposition h4 is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition t1 is true if proposition t2 is true. Proposition a4 is true if proposition h4 is not true. Proposition h2 is true if proposition a2 is not true. Proposition h4 is true if proposition a4 is not true. Proposition h3 is true if proposition a3 is not true. Proposition qcore is true if proposition h0 is true. Proposition t3 is true if proposition t4 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition s1 is true. Proposition t0 is true if proposition t1 is true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n5-i0-cred::cred", "rec_id": "hv2-combo-coupled_n5-i0", "axis": "combo", "difficulty": "coupled_n5", "n_stable_models": 6, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition t4 is true if proposition t5 is true. Proposition h1 is true if proposition a1 is not true. Proposition s0 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition h0 is true if proposition a0 is not true. Proposition t2 is true if proposition t3 is true. Proposition t5 is true if proposition qcore is true. Proposition a3 is true if proposition h3 is not true and proposition h4 is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition t1 is true if proposition t2 is true. Proposition a4 is true if proposition h4 is not true. Proposition h2 is true if proposition a2 is not true. Proposition h4 is true if proposition a4 is not true. Proposition h3 is true if proposition a3 is not true. Proposition qcore is true if proposition h0 is true. Proposition t3 is true if proposition t4 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition s1 is true. Proposition t0 is true if proposition t1 is true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n5-i0-skept::skept", "rec_id": "hv2-combo-coupled_n5-i0", "axis": "combo", "difficulty": "coupled_n5", "n_stable_models": 6, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition t4 is true if proposition t5 is true. Proposition h1 is true if proposition a1 is not true. Proposition s0 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition h0 is true if proposition a0 is not true. Proposition t2 is true if proposition t3 is true. Proposition t5 is true if proposition qcore is true. Proposition a3 is true if proposition h3 is not true and proposition h4 is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition t1 is true if proposition t2 is true. Proposition a4 is true if proposition h4 is not true. Proposition h2 is true if proposition a2 is not true. Proposition h4 is true if proposition a4 is not true. Proposition h3 is true if proposition a3 is not true. Proposition qcore is true if proposition h0 is true. Proposition t3 is true if proposition t4 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition s1 is true. Proposition t0 is true if proposition t1 is true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n5-i0-wfs::wfs", "rec_id": "hv2-combo-coupled_n5-i0", "axis": "combo", "difficulty": "coupled_n5", "n_stable_models": 6, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition t4 is true if proposition t5 is true. Proposition h1 is true if proposition a1 is not true. Proposition s0 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition h0 is true if proposition a0 is not true. Proposition t2 is true if proposition t3 is true. Proposition t5 is true if proposition qcore is true. Proposition a3 is true if proposition h3 is not true and proposition h4 is not true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition t1 is true if proposition t2 is true. Proposition a4 is true if proposition h4 is not true. Proposition h2 is true if proposition a2 is not true. Proposition h4 is true if proposition a4 is not true. Proposition h3 is true if proposition a3 is not true. Proposition qcore is true if proposition h0 is true. Proposition t3 is true if proposition t4 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition s1 is true. Proposition t0 is true if proposition t1 is true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n5-i1-none::none", "rec_id": "hv2-combo-coupled_n5-i1", "axis": "combo", "difficulty": "coupled_n5", "n_stable_models": 6, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition h0 is true if proposition a0 is not true. Proposition h2 is true if proposition a2 is not true. Proposition s2 is true. Proposition q is true if proposition qcore is true and proposition wide is true. Proposition s1 is true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition qcore is true if proposition h0 is true. Proposition a3 is true if proposition h3 is not true and proposition h4 is not true. Proposition h3 is true if proposition a3 is not true. Proposition h4 is true if proposition a4 is not true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition a4 is true if proposition h4 is not true. Proposition s0 is true. Proposition h1 is true if proposition a1 is not true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n5-i1-closed_world::closed_world", "rec_id": "hv2-combo-coupled_n5-i1", "axis": "combo", "difficulty": "coupled_n5", "n_stable_models": 6, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition h0 is true if proposition a0 is not true. Proposition h2 is true if proposition a2 is not true. Proposition s2 is true. Proposition q is true if proposition qcore is true and proposition wide is true. Proposition s1 is true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition qcore is true if proposition h0 is true. Proposition a3 is true if proposition h3 is not true and proposition h4 is not true. Proposition h3 is true if proposition a3 is not true. Proposition h4 is true if proposition a4 is not true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition a4 is true if proposition h4 is not true. Proposition s0 is true. Proposition h1 is true if proposition a1 is not true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n5-i1-cred::cred", "rec_id": "hv2-combo-coupled_n5-i1", "axis": "combo", "difficulty": "coupled_n5", "n_stable_models": 6, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition h0 is true if proposition a0 is not true. Proposition h2 is true if proposition a2 is not true. Proposition s2 is true. Proposition q is true if proposition qcore is true and proposition wide is true. Proposition s1 is true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition qcore is true if proposition h0 is true. Proposition a3 is true if proposition h3 is not true and proposition h4 is not true. Proposition h3 is true if proposition a3 is not true. Proposition h4 is true if proposition a4 is not true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition a4 is true if proposition h4 is not true. Proposition s0 is true. Proposition h1 is true if proposition a1 is not true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n5-i1-skept::skept", "rec_id": "hv2-combo-coupled_n5-i1", "axis": "combo", "difficulty": "coupled_n5", "n_stable_models": 6, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition h0 is true if proposition a0 is not true. Proposition h2 is true if proposition a2 is not true. Proposition s2 is true. Proposition q is true if proposition qcore is true and proposition wide is true. Proposition s1 is true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition qcore is true if proposition h0 is true. Proposition a3 is true if proposition h3 is not true and proposition h4 is not true. Proposition h3 is true if proposition a3 is not true. Proposition h4 is true if proposition a4 is not true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition a4 is true if proposition h4 is not true. Proposition s0 is true. Proposition h1 is true if proposition a1 is not true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n5-i1-wfs::wfs", "rec_id": "hv2-combo-coupled_n5-i1", "axis": "combo", "difficulty": "coupled_n5", "n_stable_models": 6, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition h0 is true if proposition a0 is not true. Proposition h2 is true if proposition a2 is not true. Proposition s2 is true. Proposition q is true if proposition qcore is true and proposition wide is true. Proposition s1 is true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition qcore is true if proposition h0 is true. Proposition a3 is true if proposition h3 is not true and proposition h4 is not true. Proposition h3 is true if proposition a3 is not true. Proposition h4 is true if proposition a4 is not true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition a4 is true if proposition h4 is not true. Proposition s0 is true. Proposition h1 is true if proposition a1 is not true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n5-i2-none::none", "rec_id": "hv2-combo-coupled_n5-i2", "axis": "combo", "difficulty": "coupled_n5", "n_stable_models": 6, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition qcore is true if proposition h0 is true. Proposition t3 is true if proposition t4 is true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition h1 is true if proposition a1 is not true. Proposition h0 is true if proposition a0 is not true. Proposition t5 is true if proposition qcore is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition t0 is true if proposition t1 is true. Proposition a3 is true if proposition h3 is not true and proposition h4 is not true. Proposition h2 is true if proposition a2 is not true. Proposition s0 is true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition h3 is true if proposition a3 is not true. Proposition h4 is true if proposition a4 is not true. Proposition t4 is true if proposition t5 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t1 is true if proposition t2 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition s1 is true. Proposition t2 is true if proposition t3 is true. Proposition a4 is true if proposition h4 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n5-i2-closed_world::closed_world", "rec_id": "hv2-combo-coupled_n5-i2", "axis": "combo", "difficulty": "coupled_n5", "n_stable_models": 6, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition qcore is true if proposition h0 is true. Proposition t3 is true if proposition t4 is true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition h1 is true if proposition a1 is not true. Proposition h0 is true if proposition a0 is not true. Proposition t5 is true if proposition qcore is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition t0 is true if proposition t1 is true. Proposition a3 is true if proposition h3 is not true and proposition h4 is not true. Proposition h2 is true if proposition a2 is not true. Proposition s0 is true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition h3 is true if proposition a3 is not true. Proposition h4 is true if proposition a4 is not true. Proposition t4 is true if proposition t5 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t1 is true if proposition t2 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition s1 is true. Proposition t2 is true if proposition t3 is true. Proposition a4 is true if proposition h4 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n5-i2-cred::cred", "rec_id": "hv2-combo-coupled_n5-i2", "axis": "combo", "difficulty": "coupled_n5", "n_stable_models": 6, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition qcore is true if proposition h0 is true. Proposition t3 is true if proposition t4 is true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition h1 is true if proposition a1 is not true. Proposition h0 is true if proposition a0 is not true. Proposition t5 is true if proposition qcore is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition t0 is true if proposition t1 is true. Proposition a3 is true if proposition h3 is not true and proposition h4 is not true. Proposition h2 is true if proposition a2 is not true. Proposition s0 is true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition h3 is true if proposition a3 is not true. Proposition h4 is true if proposition a4 is not true. Proposition t4 is true if proposition t5 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t1 is true if proposition t2 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition s1 is true. Proposition t2 is true if proposition t3 is true. Proposition a4 is true if proposition h4 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n5-i2-skept::skept", "rec_id": "hv2-combo-coupled_n5-i2", "axis": "combo", "difficulty": "coupled_n5", "n_stable_models": 6, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition qcore is true if proposition h0 is true. Proposition t3 is true if proposition t4 is true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition h1 is true if proposition a1 is not true. Proposition h0 is true if proposition a0 is not true. Proposition t5 is true if proposition qcore is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition t0 is true if proposition t1 is true. Proposition a3 is true if proposition h3 is not true and proposition h4 is not true. Proposition h2 is true if proposition a2 is not true. Proposition s0 is true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition h3 is true if proposition a3 is not true. Proposition h4 is true if proposition a4 is not true. Proposition t4 is true if proposition t5 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t1 is true if proposition t2 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition s1 is true. Proposition t2 is true if proposition t3 is true. Proposition a4 is true if proposition h4 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n5-i2-wfs::wfs", "rec_id": "hv2-combo-coupled_n5-i2", "axis": "combo", "difficulty": "coupled_n5", "n_stable_models": 6, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition qcore is true if proposition h0 is true. Proposition t3 is true if proposition t4 is true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition h1 is true if proposition a1 is not true. Proposition h0 is true if proposition a0 is not true. Proposition t5 is true if proposition qcore is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition t0 is true if proposition t1 is true. Proposition a3 is true if proposition h3 is not true and proposition h4 is not true. Proposition h2 is true if proposition a2 is not true. Proposition s0 is true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition h3 is true if proposition a3 is not true. Proposition h4 is true if proposition a4 is not true. Proposition t4 is true if proposition t5 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t1 is true if proposition t2 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition s1 is true. Proposition t2 is true if proposition t3 is true. Proposition a4 is true if proposition h4 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n5-i3-none::none", "rec_id": "hv2-combo-coupled_n5-i3", "axis": "combo", "difficulty": "coupled_n5", "n_stable_models": 6, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition h3 is true if proposition a3 is not true. Proposition t1 is true if proposition t2 is true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition t0 is true if proposition t1 is true. Proposition s2 is true. Proposition a3 is true if proposition h3 is not true and proposition h4 is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition s1 is true. Proposition a4 is true if proposition h4 is not true. Proposition s0 is true. Proposition h1 is true if proposition a1 is not true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition qcore is true if proposition h0 is true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition h0 is true if proposition a0 is not true. Proposition h4 is true if proposition a4 is not true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition h2 is true if proposition a2 is not true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition qcore is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n5-i3-closed_world::closed_world", "rec_id": "hv2-combo-coupled_n5-i3", "axis": "combo", "difficulty": "coupled_n5", "n_stable_models": 6, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition h3 is true if proposition a3 is not true. Proposition t1 is true if proposition t2 is true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition t0 is true if proposition t1 is true. Proposition s2 is true. Proposition a3 is true if proposition h3 is not true and proposition h4 is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition s1 is true. Proposition a4 is true if proposition h4 is not true. Proposition s0 is true. Proposition h1 is true if proposition a1 is not true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition qcore is true if proposition h0 is true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition h0 is true if proposition a0 is not true. Proposition h4 is true if proposition a4 is not true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition h2 is true if proposition a2 is not true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition qcore is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n5-i3-cred::cred", "rec_id": "hv2-combo-coupled_n5-i3", "axis": "combo", "difficulty": "coupled_n5", "n_stable_models": 6, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition h3 is true if proposition a3 is not true. Proposition t1 is true if proposition t2 is true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition t0 is true if proposition t1 is true. Proposition s2 is true. Proposition a3 is true if proposition h3 is not true and proposition h4 is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition s1 is true. Proposition a4 is true if proposition h4 is not true. Proposition s0 is true. Proposition h1 is true if proposition a1 is not true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition qcore is true if proposition h0 is true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition h0 is true if proposition a0 is not true. Proposition h4 is true if proposition a4 is not true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition h2 is true if proposition a2 is not true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition qcore is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n5-i3-skept::skept", "rec_id": "hv2-combo-coupled_n5-i3", "axis": "combo", "difficulty": "coupled_n5", "n_stable_models": 6, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition h3 is true if proposition a3 is not true. Proposition t1 is true if proposition t2 is true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition t0 is true if proposition t1 is true. Proposition s2 is true. Proposition a3 is true if proposition h3 is not true and proposition h4 is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition s1 is true. Proposition a4 is true if proposition h4 is not true. Proposition s0 is true. Proposition h1 is true if proposition a1 is not true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition qcore is true if proposition h0 is true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition h0 is true if proposition a0 is not true. Proposition h4 is true if proposition a4 is not true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition h2 is true if proposition a2 is not true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition qcore is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n5-i3-wfs::wfs", "rec_id": "hv2-combo-coupled_n5-i3", "axis": "combo", "difficulty": "coupled_n5", "n_stable_models": 6, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition h3 is true if proposition a3 is not true. Proposition t1 is true if proposition t2 is true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition t0 is true if proposition t1 is true. Proposition s2 is true. Proposition a3 is true if proposition h3 is not true and proposition h4 is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true and proposition s2 is true. Proposition s1 is true. Proposition a4 is true if proposition h4 is not true. Proposition s0 is true. Proposition h1 is true if proposition a1 is not true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition qcore is true if proposition h0 is true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition h0 is true if proposition a0 is not true. Proposition h4 is true if proposition a4 is not true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition h2 is true if proposition a2 is not true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition qcore is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n5-i4-none::none", "rec_id": "hv2-combo-coupled_n5-i4", "axis": "combo", "difficulty": "coupled_n5", "n_stable_models": 6, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition h4 is true if proposition a4 is not true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition t1 is true if proposition qcore is true. Proposition a4 is true if proposition h4 is not true. Proposition a3 is true if proposition h3 is not true and proposition h4 is not true. Proposition h3 is true if proposition a3 is not true. Proposition h0 is true if proposition a0 is not true. Proposition s0 is true. Proposition t0 is true if proposition t1 is true. Proposition s1 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition h1 is true if proposition a1 is not true. Proposition h2 is true if proposition a2 is not true. Proposition qcore is true if proposition h0 is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n5-i4-closed_world::closed_world", "rec_id": "hv2-combo-coupled_n5-i4", "axis": "combo", "difficulty": "coupled_n5", "n_stable_models": 6, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition h4 is true if proposition a4 is not true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition t1 is true if proposition qcore is true. Proposition a4 is true if proposition h4 is not true. Proposition a3 is true if proposition h3 is not true and proposition h4 is not true. Proposition h3 is true if proposition a3 is not true. Proposition h0 is true if proposition a0 is not true. Proposition s0 is true. Proposition t0 is true if proposition t1 is true. Proposition s1 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition h1 is true if proposition a1 is not true. Proposition h2 is true if proposition a2 is not true. Proposition qcore is true if proposition h0 is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n5-i4-cred::cred", "rec_id": "hv2-combo-coupled_n5-i4", "axis": "combo", "difficulty": "coupled_n5", "n_stable_models": 6, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition h4 is true if proposition a4 is not true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition t1 is true if proposition qcore is true. Proposition a4 is true if proposition h4 is not true. Proposition a3 is true if proposition h3 is not true and proposition h4 is not true. Proposition h3 is true if proposition a3 is not true. Proposition h0 is true if proposition a0 is not true. Proposition s0 is true. Proposition t0 is true if proposition t1 is true. Proposition s1 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition h1 is true if proposition a1 is not true. Proposition h2 is true if proposition a2 is not true. Proposition qcore is true if proposition h0 is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n5-i4-skept::skept", "rec_id": "hv2-combo-coupled_n5-i4", "axis": "combo", "difficulty": "coupled_n5", "n_stable_models": 6, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition h4 is true if proposition a4 is not true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition t1 is true if proposition qcore is true. Proposition a4 is true if proposition h4 is not true. Proposition a3 is true if proposition h3 is not true and proposition h4 is not true. Proposition h3 is true if proposition a3 is not true. Proposition h0 is true if proposition a0 is not true. Proposition s0 is true. Proposition t0 is true if proposition t1 is true. Proposition s1 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition h1 is true if proposition a1 is not true. Proposition h2 is true if proposition a2 is not true. Proposition qcore is true if proposition h0 is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n5-i4-wfs::wfs", "rec_id": "hv2-combo-coupled_n5-i4", "axis": "combo", "difficulty": "coupled_n5", "n_stable_models": 6, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition h4 is true if proposition a4 is not true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition t1 is true if proposition qcore is true. Proposition a4 is true if proposition h4 is not true. Proposition a3 is true if proposition h3 is not true and proposition h4 is not true. Proposition h3 is true if proposition a3 is not true. Proposition h0 is true if proposition a0 is not true. Proposition s0 is true. Proposition t0 is true if proposition t1 is true. Proposition s1 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition h1 is true if proposition a1 is not true. Proposition h2 is true if proposition a2 is not true. Proposition qcore is true if proposition h0 is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n5-i5-none::none", "rec_id": "hv2-combo-coupled_n5-i5", "axis": "combo", "difficulty": "coupled_n5", "n_stable_models": 6, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition a3 is true if proposition h3 is not true and proposition h4 is not true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition h3 is true if proposition a3 is not true. Proposition qcore is true if proposition h0 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition h2 is true if proposition a2 is not true. Proposition s0 is true. Proposition s1 is true. Proposition h0 is true if proposition a0 is not true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition a4 is true if proposition h4 is not true. Proposition h4 is true if proposition a4 is not true. Proposition t0 is true if proposition t1 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t1 is true if proposition qcore is true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition h1 is true if proposition a1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n5-i5-closed_world::closed_world", "rec_id": "hv2-combo-coupled_n5-i5", "axis": "combo", "difficulty": "coupled_n5", "n_stable_models": 6, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition a3 is true if proposition h3 is not true and proposition h4 is not true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition h3 is true if proposition a3 is not true. Proposition qcore is true if proposition h0 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition h2 is true if proposition a2 is not true. Proposition s0 is true. Proposition s1 is true. Proposition h0 is true if proposition a0 is not true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition a4 is true if proposition h4 is not true. Proposition h4 is true if proposition a4 is not true. Proposition t0 is true if proposition t1 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t1 is true if proposition qcore is true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition h1 is true if proposition a1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n5-i5-cred::cred", "rec_id": "hv2-combo-coupled_n5-i5", "axis": "combo", "difficulty": "coupled_n5", "n_stable_models": 6, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition a3 is true if proposition h3 is not true and proposition h4 is not true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition h3 is true if proposition a3 is not true. Proposition qcore is true if proposition h0 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition h2 is true if proposition a2 is not true. Proposition s0 is true. Proposition s1 is true. Proposition h0 is true if proposition a0 is not true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition a4 is true if proposition h4 is not true. Proposition h4 is true if proposition a4 is not true. Proposition t0 is true if proposition t1 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t1 is true if proposition qcore is true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition h1 is true if proposition a1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n5-i5-skept::skept", "rec_id": "hv2-combo-coupled_n5-i5", "axis": "combo", "difficulty": "coupled_n5", "n_stable_models": 6, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition a3 is true if proposition h3 is not true and proposition h4 is not true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition h3 is true if proposition a3 is not true. Proposition qcore is true if proposition h0 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition h2 is true if proposition a2 is not true. Proposition s0 is true. Proposition s1 is true. Proposition h0 is true if proposition a0 is not true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition a4 is true if proposition h4 is not true. Proposition h4 is true if proposition a4 is not true. Proposition t0 is true if proposition t1 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t1 is true if proposition qcore is true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition h1 is true if proposition a1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n5-i5-wfs::wfs", "rec_id": "hv2-combo-coupled_n5-i5", "axis": "combo", "difficulty": "coupled_n5", "n_stable_models": 6, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition a3 is true if proposition h3 is not true and proposition h4 is not true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition h3 is true if proposition a3 is not true. Proposition qcore is true if proposition h0 is true. Proposition q is true if proposition t0 is true and proposition wide is true. Proposition h2 is true if proposition a2 is not true. Proposition s0 is true. Proposition s1 is true. Proposition h0 is true if proposition a0 is not true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition a4 is true if proposition h4 is not true. Proposition h4 is true if proposition a4 is not true. Proposition t0 is true if proposition t1 is true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition t1 is true if proposition qcore is true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition h1 is true if proposition a1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n5-i6-none::none", "rec_id": "hv2-combo-coupled_n5-i6", "axis": "combo", "difficulty": "coupled_n5", "n_stable_models": 6, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition etrue is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition h0 is true if proposition a0 is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition h4 is true if proposition a4 is not true. Proposition a3 is true if proposition h3 is not true and proposition h4 is not true. Proposition t1 is true if proposition t2 is true. Proposition h1 is true if proposition a1 is not true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition a4 is true if proposition h4 is not true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition qcore is true. Proposition s0 is true. Proposition h2 is true if proposition a2 is not true. Proposition qcore is true if proposition h0 is true. Proposition h3 is true if proposition a3 is not true. Proposition t0 is true if proposition t1 is true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition s1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n5-i6-closed_world::closed_world", "rec_id": "hv2-combo-coupled_n5-i6", "axis": "combo", "difficulty": "coupled_n5", "n_stable_models": 6, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition etrue is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition h0 is true if proposition a0 is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition h4 is true if proposition a4 is not true. Proposition a3 is true if proposition h3 is not true and proposition h4 is not true. Proposition t1 is true if proposition t2 is true. Proposition h1 is true if proposition a1 is not true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition a4 is true if proposition h4 is not true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition qcore is true. Proposition s0 is true. Proposition h2 is true if proposition a2 is not true. Proposition qcore is true if proposition h0 is true. Proposition h3 is true if proposition a3 is not true. Proposition t0 is true if proposition t1 is true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition s1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n5-i6-cred::cred", "rec_id": "hv2-combo-coupled_n5-i6", "axis": "combo", "difficulty": "coupled_n5", "n_stable_models": 6, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition etrue is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition h0 is true if proposition a0 is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition h4 is true if proposition a4 is not true. Proposition a3 is true if proposition h3 is not true and proposition h4 is not true. Proposition t1 is true if proposition t2 is true. Proposition h1 is true if proposition a1 is not true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition a4 is true if proposition h4 is not true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition qcore is true. Proposition s0 is true. Proposition h2 is true if proposition a2 is not true. Proposition qcore is true if proposition h0 is true. Proposition h3 is true if proposition a3 is not true. Proposition t0 is true if proposition t1 is true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition s1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n5-i6-skept::skept", "rec_id": "hv2-combo-coupled_n5-i6", "axis": "combo", "difficulty": "coupled_n5", "n_stable_models": 6, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition etrue is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition h0 is true if proposition a0 is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition h4 is true if proposition a4 is not true. Proposition a3 is true if proposition h3 is not true and proposition h4 is not true. Proposition t1 is true if proposition t2 is true. Proposition h1 is true if proposition a1 is not true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition a4 is true if proposition h4 is not true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition qcore is true. Proposition s0 is true. Proposition h2 is true if proposition a2 is not true. Proposition qcore is true if proposition h0 is true. Proposition h3 is true if proposition a3 is not true. Proposition t0 is true if proposition t1 is true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition s1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n5-i6-wfs::wfs", "rec_id": "hv2-combo-coupled_n5-i6", "axis": "combo", "difficulty": "coupled_n5", "n_stable_models": 6, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition etrue is true. Proposition q is true if proposition t0 is true and proposition wide is true and proposition etrue is true. Proposition h0 is true if proposition a0 is not true. Proposition wide is true if proposition s0 is true and proposition s1 is true. Proposition h4 is true if proposition a4 is not true. Proposition a3 is true if proposition h3 is not true and proposition h4 is not true. Proposition t1 is true if proposition t2 is true. Proposition h1 is true if proposition a1 is not true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition a4 is true if proposition h4 is not true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition qcore is true. Proposition s0 is true. Proposition h2 is true if proposition a2 is not true. Proposition qcore is true if proposition h0 is true. Proposition h3 is true if proposition a3 is not true. Proposition t0 is true if proposition t1 is true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition s1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n5-i7-none::none", "rec_id": "hv2-combo-coupled_n5-i7", "axis": "combo", "difficulty": "coupled_n5", "n_stable_models": 6, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition q is true if proposition t0 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition qcore is true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition qcore is true if proposition h0 is true. Proposition h2 is true if proposition a2 is not true. Proposition t1 is true if proposition t2 is true. Proposition h4 is true if proposition a4 is not true. Proposition h1 is true if proposition a1 is not true. Proposition h3 is true if proposition a3 is not true. Proposition a4 is true if proposition h4 is not true. Proposition h0 is true if proposition a0 is not true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition t0 is true if proposition t1 is true. Proposition a3 is true if proposition h3 is not true and proposition h4 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n5-i7-closed_world::closed_world", "rec_id": "hv2-combo-coupled_n5-i7", "axis": "combo", "difficulty": "coupled_n5", "n_stable_models": 6, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition q is true if proposition t0 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition qcore is true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition qcore is true if proposition h0 is true. Proposition h2 is true if proposition a2 is not true. Proposition t1 is true if proposition t2 is true. Proposition h4 is true if proposition a4 is not true. Proposition h1 is true if proposition a1 is not true. Proposition h3 is true if proposition a3 is not true. Proposition a4 is true if proposition h4 is not true. Proposition h0 is true if proposition a0 is not true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition t0 is true if proposition t1 is true. Proposition a3 is true if proposition h3 is not true and proposition h4 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n5-i7-cred::cred", "rec_id": "hv2-combo-coupled_n5-i7", "axis": "combo", "difficulty": "coupled_n5", "n_stable_models": 6, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition q is true if proposition t0 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition qcore is true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition qcore is true if proposition h0 is true. Proposition h2 is true if proposition a2 is not true. Proposition t1 is true if proposition t2 is true. Proposition h4 is true if proposition a4 is not true. Proposition h1 is true if proposition a1 is not true. Proposition h3 is true if proposition a3 is not true. Proposition a4 is true if proposition h4 is not true. Proposition h0 is true if proposition a0 is not true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition t0 is true if proposition t1 is true. Proposition a3 is true if proposition h3 is not true and proposition h4 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n5-i7-skept::skept", "rec_id": "hv2-combo-coupled_n5-i7", "axis": "combo", "difficulty": "coupled_n5", "n_stable_models": 6, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition q is true if proposition t0 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition qcore is true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition qcore is true if proposition h0 is true. Proposition h2 is true if proposition a2 is not true. Proposition t1 is true if proposition t2 is true. Proposition h4 is true if proposition a4 is not true. Proposition h1 is true if proposition a1 is not true. Proposition h3 is true if proposition a3 is not true. Proposition a4 is true if proposition h4 is not true. Proposition h0 is true if proposition a0 is not true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition t0 is true if proposition t1 is true. Proposition a3 is true if proposition h3 is not true and proposition h4 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-combo-coupled_n5-i7-wfs::wfs", "rec_id": "hv2-combo-coupled_n5-i7", "axis": "combo", "difficulty": "coupled_n5", "n_stable_models": 6, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition q is true if proposition t0 is true. Proposition a0 is true if proposition h0 is not true and proposition h1 is not true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition qcore is true. Proposition a1 is true if proposition h1 is not true and proposition h2 is not true. Proposition qcore is true if proposition h0 is true. Proposition h2 is true if proposition a2 is not true. Proposition t1 is true if proposition t2 is true. Proposition h4 is true if proposition a4 is not true. Proposition h1 is true if proposition a1 is not true. Proposition h3 is true if proposition a3 is not true. Proposition a4 is true if proposition h4 is not true. Proposition h0 is true if proposition a0 is not true. Proposition a2 is true if proposition h2 is not true and proposition h3 is not true. Proposition t0 is true if proposition t1 is true. Proposition a3 is true if proposition h3 is not true and proposition h4 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k4-i0-none::none", "rec_id": "hv2-cyclen-even_one_sided_k4-i0", "axis": "cyclen", "difficulty": "even_one_sided_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition s0 is true if proposition g0_0 is true and proposition g0_1 is true and proposition g0_2 is true. Proposition cq is true if proposition x0 is true. Proposition t6 is true if proposition t7 is true. Proposition g0_0 is true. Proposition btrue is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition q is true if proposition t0 is true. Proposition p0 is true if proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition t4 is true. Proposition t1 is true if proposition t2 is true. Proposition t0 is true if proposition t1 is true. Proposition t4 is true if proposition t5 is true. Proposition t5 is true if proposition t6 is true and proposition cq is true. Proposition x2 is true if proposition x3 is not true. Proposition x0 is true if proposition x1 is not true. Proposition x1 is true if proposition x2 is not true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true. Proposition g2_0 is true. Proposition g1_1 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition t7 is true if proposition btrue is true and proposition wide is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s3 is true. Proposition s3 is true if proposition g3_0 is true. Proposition g3_0 is true. Proposition g0_1 is true. Proposition g1_0 is true. Proposition g0_2 is true. Proposition x3 is true if proposition x0 is not true. Proposition g2_1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k4-i0-closed_world::closed_world", "rec_id": "hv2-cyclen-even_one_sided_k4-i0", "axis": "cyclen", "difficulty": "even_one_sided_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition s0 is true if proposition g0_0 is true and proposition g0_1 is true and proposition g0_2 is true. Proposition cq is true if proposition x0 is true. Proposition t6 is true if proposition t7 is true. Proposition g0_0 is true. Proposition btrue is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition q is true if proposition t0 is true. Proposition p0 is true if proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition t4 is true. Proposition t1 is true if proposition t2 is true. Proposition t0 is true if proposition t1 is true. Proposition t4 is true if proposition t5 is true. Proposition t5 is true if proposition t6 is true and proposition cq is true. Proposition x2 is true if proposition x3 is not true. Proposition x0 is true if proposition x1 is not true. Proposition x1 is true if proposition x2 is not true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true. Proposition g2_0 is true. Proposition g1_1 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition t7 is true if proposition btrue is true and proposition wide is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s3 is true. Proposition s3 is true if proposition g3_0 is true. Proposition g3_0 is true. Proposition g0_1 is true. Proposition g1_0 is true. Proposition g0_2 is true. Proposition x3 is true if proposition x0 is not true. Proposition g2_1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k4-i0-cred::cred", "rec_id": "hv2-cyclen-even_one_sided_k4-i0", "axis": "cyclen", "difficulty": "even_one_sided_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition s0 is true if proposition g0_0 is true and proposition g0_1 is true and proposition g0_2 is true. Proposition cq is true if proposition x0 is true. Proposition t6 is true if proposition t7 is true. Proposition g0_0 is true. Proposition btrue is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition q is true if proposition t0 is true. Proposition p0 is true if proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition t4 is true. Proposition t1 is true if proposition t2 is true. Proposition t0 is true if proposition t1 is true. Proposition t4 is true if proposition t5 is true. Proposition t5 is true if proposition t6 is true and proposition cq is true. Proposition x2 is true if proposition x3 is not true. Proposition x0 is true if proposition x1 is not true. Proposition x1 is true if proposition x2 is not true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true. Proposition g2_0 is true. Proposition g1_1 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition t7 is true if proposition btrue is true and proposition wide is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s3 is true. Proposition s3 is true if proposition g3_0 is true. Proposition g3_0 is true. Proposition g0_1 is true. Proposition g1_0 is true. Proposition g0_2 is true. Proposition x3 is true if proposition x0 is not true. Proposition g2_1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k4-i0-skept::skept", "rec_id": "hv2-cyclen-even_one_sided_k4-i0", "axis": "cyclen", "difficulty": "even_one_sided_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition s0 is true if proposition g0_0 is true and proposition g0_1 is true and proposition g0_2 is true. Proposition cq is true if proposition x0 is true. Proposition t6 is true if proposition t7 is true. Proposition g0_0 is true. Proposition btrue is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition q is true if proposition t0 is true. Proposition p0 is true if proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition t4 is true. Proposition t1 is true if proposition t2 is true. Proposition t0 is true if proposition t1 is true. Proposition t4 is true if proposition t5 is true. Proposition t5 is true if proposition t6 is true and proposition cq is true. Proposition x2 is true if proposition x3 is not true. Proposition x0 is true if proposition x1 is not true. Proposition x1 is true if proposition x2 is not true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true. Proposition g2_0 is true. Proposition g1_1 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition t7 is true if proposition btrue is true and proposition wide is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s3 is true. Proposition s3 is true if proposition g3_0 is true. Proposition g3_0 is true. Proposition g0_1 is true. Proposition g1_0 is true. Proposition g0_2 is true. Proposition x3 is true if proposition x0 is not true. Proposition g2_1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k4-i0-wfs::wfs", "rec_id": "hv2-cyclen-even_one_sided_k4-i0", "axis": "cyclen", "difficulty": "even_one_sided_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition s0 is true if proposition g0_0 is true and proposition g0_1 is true and proposition g0_2 is true. Proposition cq is true if proposition x0 is true. Proposition t6 is true if proposition t7 is true. Proposition g0_0 is true. Proposition btrue is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition q is true if proposition t0 is true. Proposition p0 is true if proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition t4 is true. Proposition t1 is true if proposition t2 is true. Proposition t0 is true if proposition t1 is true. Proposition t4 is true if proposition t5 is true. Proposition t5 is true if proposition t6 is true and proposition cq is true. Proposition x2 is true if proposition x3 is not true. Proposition x0 is true if proposition x1 is not true. Proposition x1 is true if proposition x2 is not true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true. Proposition g2_0 is true. Proposition g1_1 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition t7 is true if proposition btrue is true and proposition wide is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s3 is true. Proposition s3 is true if proposition g3_0 is true. Proposition g3_0 is true. Proposition g0_1 is true. Proposition g1_0 is true. Proposition g0_2 is true. Proposition x3 is true if proposition x0 is not true. Proposition g2_1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k4-i1-none::none", "rec_id": "hv2-cyclen-even_one_sided_k4-i1", "axis": "cyclen", "difficulty": "even_one_sided_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition t3 is true if proposition t4 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition g0_0 is true. Proposition t1 is true if proposition t2 is true and proposition cq is true. Proposition q is true if proposition t0 is true. Proposition t4 is true if proposition t5 is true. Proposition x3 is true if proposition etrue is true and proposition x0 is not true. Proposition g3_1 is true. Proposition t0 is true if proposition t1 is true. Proposition g0_1 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition cq is true if proposition x0 is true. Proposition g3_0 is true. Proposition g2_0 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g2_1 is true. Proposition t2 is true if proposition t3 is true. Proposition g1_1 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition btrue is true. Proposition g1_0 is true. Proposition t5 is true if proposition t6 is true and proposition wide is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition t6 is true if proposition t7 is true. Proposition t7 is true if proposition btrue is true. Proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k4-i1-closed_world::closed_world", "rec_id": "hv2-cyclen-even_one_sided_k4-i1", "axis": "cyclen", "difficulty": "even_one_sided_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition t3 is true if proposition t4 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition g0_0 is true. Proposition t1 is true if proposition t2 is true and proposition cq is true. Proposition q is true if proposition t0 is true. Proposition t4 is true if proposition t5 is true. Proposition x3 is true if proposition etrue is true and proposition x0 is not true. Proposition g3_1 is true. Proposition t0 is true if proposition t1 is true. Proposition g0_1 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition cq is true if proposition x0 is true. Proposition g3_0 is true. Proposition g2_0 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g2_1 is true. Proposition t2 is true if proposition t3 is true. Proposition g1_1 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition btrue is true. Proposition g1_0 is true. Proposition t5 is true if proposition t6 is true and proposition wide is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition t6 is true if proposition t7 is true. Proposition t7 is true if proposition btrue is true. Proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k4-i1-cred::cred", "rec_id": "hv2-cyclen-even_one_sided_k4-i1", "axis": "cyclen", "difficulty": "even_one_sided_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition t3 is true if proposition t4 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition g0_0 is true. Proposition t1 is true if proposition t2 is true and proposition cq is true. Proposition q is true if proposition t0 is true. Proposition t4 is true if proposition t5 is true. Proposition x3 is true if proposition etrue is true and proposition x0 is not true. Proposition g3_1 is true. Proposition t0 is true if proposition t1 is true. Proposition g0_1 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition cq is true if proposition x0 is true. Proposition g3_0 is true. Proposition g2_0 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g2_1 is true. Proposition t2 is true if proposition t3 is true. Proposition g1_1 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition btrue is true. Proposition g1_0 is true. Proposition t5 is true if proposition t6 is true and proposition wide is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition t6 is true if proposition t7 is true. Proposition t7 is true if proposition btrue is true. Proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k4-i1-skept::skept", "rec_id": "hv2-cyclen-even_one_sided_k4-i1", "axis": "cyclen", "difficulty": "even_one_sided_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition t3 is true if proposition t4 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition g0_0 is true. Proposition t1 is true if proposition t2 is true and proposition cq is true. Proposition q is true if proposition t0 is true. Proposition t4 is true if proposition t5 is true. Proposition x3 is true if proposition etrue is true and proposition x0 is not true. Proposition g3_1 is true. Proposition t0 is true if proposition t1 is true. Proposition g0_1 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition cq is true if proposition x0 is true. Proposition g3_0 is true. Proposition g2_0 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g2_1 is true. Proposition t2 is true if proposition t3 is true. Proposition g1_1 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition btrue is true. Proposition g1_0 is true. Proposition t5 is true if proposition t6 is true and proposition wide is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition t6 is true if proposition t7 is true. Proposition t7 is true if proposition btrue is true. Proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k4-i1-wfs::wfs", "rec_id": "hv2-cyclen-even_one_sided_k4-i1", "axis": "cyclen", "difficulty": "even_one_sided_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition t3 is true if proposition t4 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition g0_0 is true. Proposition t1 is true if proposition t2 is true and proposition cq is true. Proposition q is true if proposition t0 is true. Proposition t4 is true if proposition t5 is true. Proposition x3 is true if proposition etrue is true and proposition x0 is not true. Proposition g3_1 is true. Proposition t0 is true if proposition t1 is true. Proposition g0_1 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition cq is true if proposition x0 is true. Proposition g3_0 is true. Proposition g2_0 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g2_1 is true. Proposition t2 is true if proposition t3 is true. Proposition g1_1 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition btrue is true. Proposition g1_0 is true. Proposition t5 is true if proposition t6 is true and proposition wide is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition t6 is true if proposition t7 is true. Proposition t7 is true if proposition btrue is true. Proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k4-i2-none::none", "rec_id": "hv2-cyclen-even_one_sided_k4-i2", "axis": "cyclen", "difficulty": "even_one_sided_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition x1 is true if proposition x2 is not true. Proposition t5 is true if proposition t6 is true and proposition cq is true. Proposition g0_1 is true. Proposition g2_1 is true. Proposition x3 is true if proposition x0 is not true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition btrue is true. Proposition x2 is true if proposition x3 is not true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true and proposition g0_2 is true. Proposition t1 is true if proposition t2 is true. Proposition t4 is true if proposition t5 is true. Proposition t3 is true if proposition t4 is true. Proposition g2_0 is true. Proposition g0_0 is true. Proposition cq is true if proposition x0 is true. Proposition g3_1 is true. Proposition t6 is true if proposition t7 is true. Proposition t0 is true if proposition t1 is true. Proposition x0 is true if proposition x1 is not true. Proposition q is true if proposition t0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t2 is true if proposition t3 is true. Proposition g3_0 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition s1 is true if proposition g1_0 is true. Proposition g1_0 is true. Proposition g0_2 is true. Proposition t7 is true if proposition btrue is true and proposition wide is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k4-i2-closed_world::closed_world", "rec_id": "hv2-cyclen-even_one_sided_k4-i2", "axis": "cyclen", "difficulty": "even_one_sided_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition x1 is true if proposition x2 is not true. Proposition t5 is true if proposition t6 is true and proposition cq is true. Proposition g0_1 is true. Proposition g2_1 is true. Proposition x3 is true if proposition x0 is not true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition btrue is true. Proposition x2 is true if proposition x3 is not true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true and proposition g0_2 is true. Proposition t1 is true if proposition t2 is true. Proposition t4 is true if proposition t5 is true. Proposition t3 is true if proposition t4 is true. Proposition g2_0 is true. Proposition g0_0 is true. Proposition cq is true if proposition x0 is true. Proposition g3_1 is true. Proposition t6 is true if proposition t7 is true. Proposition t0 is true if proposition t1 is true. Proposition x0 is true if proposition x1 is not true. Proposition q is true if proposition t0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t2 is true if proposition t3 is true. Proposition g3_0 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition s1 is true if proposition g1_0 is true. Proposition g1_0 is true. Proposition g0_2 is true. Proposition t7 is true if proposition btrue is true and proposition wide is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k4-i2-cred::cred", "rec_id": "hv2-cyclen-even_one_sided_k4-i2", "axis": "cyclen", "difficulty": "even_one_sided_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition x1 is true if proposition x2 is not true. Proposition t5 is true if proposition t6 is true and proposition cq is true. Proposition g0_1 is true. Proposition g2_1 is true. Proposition x3 is true if proposition x0 is not true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition btrue is true. Proposition x2 is true if proposition x3 is not true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true and proposition g0_2 is true. Proposition t1 is true if proposition t2 is true. Proposition t4 is true if proposition t5 is true. Proposition t3 is true if proposition t4 is true. Proposition g2_0 is true. Proposition g0_0 is true. Proposition cq is true if proposition x0 is true. Proposition g3_1 is true. Proposition t6 is true if proposition t7 is true. Proposition t0 is true if proposition t1 is true. Proposition x0 is true if proposition x1 is not true. Proposition q is true if proposition t0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t2 is true if proposition t3 is true. Proposition g3_0 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition s1 is true if proposition g1_0 is true. Proposition g1_0 is true. Proposition g0_2 is true. Proposition t7 is true if proposition btrue is true and proposition wide is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k4-i2-skept::skept", "rec_id": "hv2-cyclen-even_one_sided_k4-i2", "axis": "cyclen", "difficulty": "even_one_sided_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition x1 is true if proposition x2 is not true. Proposition t5 is true if proposition t6 is true and proposition cq is true. Proposition g0_1 is true. Proposition g2_1 is true. Proposition x3 is true if proposition x0 is not true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition btrue is true. Proposition x2 is true if proposition x3 is not true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true and proposition g0_2 is true. Proposition t1 is true if proposition t2 is true. Proposition t4 is true if proposition t5 is true. Proposition t3 is true if proposition t4 is true. Proposition g2_0 is true. Proposition g0_0 is true. Proposition cq is true if proposition x0 is true. Proposition g3_1 is true. Proposition t6 is true if proposition t7 is true. Proposition t0 is true if proposition t1 is true. Proposition x0 is true if proposition x1 is not true. Proposition q is true if proposition t0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t2 is true if proposition t3 is true. Proposition g3_0 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition s1 is true if proposition g1_0 is true. Proposition g1_0 is true. Proposition g0_2 is true. Proposition t7 is true if proposition btrue is true and proposition wide is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k4-i2-wfs::wfs", "rec_id": "hv2-cyclen-even_one_sided_k4-i2", "axis": "cyclen", "difficulty": "even_one_sided_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition x1 is true if proposition x2 is not true. Proposition t5 is true if proposition t6 is true and proposition cq is true. Proposition g0_1 is true. Proposition g2_1 is true. Proposition x3 is true if proposition x0 is not true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition btrue is true. Proposition x2 is true if proposition x3 is not true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true and proposition g0_2 is true. Proposition t1 is true if proposition t2 is true. Proposition t4 is true if proposition t5 is true. Proposition t3 is true if proposition t4 is true. Proposition g2_0 is true. Proposition g0_0 is true. Proposition cq is true if proposition x0 is true. Proposition g3_1 is true. Proposition t6 is true if proposition t7 is true. Proposition t0 is true if proposition t1 is true. Proposition x0 is true if proposition x1 is not true. Proposition q is true if proposition t0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t2 is true if proposition t3 is true. Proposition g3_0 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition s1 is true if proposition g1_0 is true. Proposition g1_0 is true. Proposition g0_2 is true. Proposition t7 is true if proposition btrue is true and proposition wide is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k4-i3-none::none", "rec_id": "hv2-cyclen-even_one_sided_k4-i3", "axis": "cyclen", "difficulty": "even_one_sided_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition g3_0 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true and proposition g1_3 is true. Proposition cq is true if proposition x0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition t1 is true if proposition t2 is true. Proposition btrue is true. Proposition t6 is true if proposition t7 is true and proposition cq is true. Proposition t2 is true if proposition t3 is true. Proposition p0 is true if proposition s1 is true and proposition s2 is true. Proposition g1_3 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition p1 is true if proposition s3 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition t0 is true if proposition t1 is true and proposition wide is true. Proposition t3 is true if proposition t4 is true. Proposition g0_0 is true. Proposition t7 is true if proposition btrue is true. Proposition g1_1 is true. Proposition t4 is true if proposition t5 is true. Proposition g3_1 is true. Proposition g1_0 is true. Proposition s0 is true if proposition g0_0 is true. Proposition s2 is true if proposition g2_0 is true. Proposition g1_2 is true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition etrue is true. Proposition t5 is true if proposition t6 is true. Proposition p3 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true. Proposition g2_0 is true. Proposition q is true if proposition t0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true and proposition p3 is true. Proposition x3 is true if proposition etrue is true and proposition x0 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k4-i3-closed_world::closed_world", "rec_id": "hv2-cyclen-even_one_sided_k4-i3", "axis": "cyclen", "difficulty": "even_one_sided_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition g3_0 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true and proposition g1_3 is true. Proposition cq is true if proposition x0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition t1 is true if proposition t2 is true. Proposition btrue is true. Proposition t6 is true if proposition t7 is true and proposition cq is true. Proposition t2 is true if proposition t3 is true. Proposition p0 is true if proposition s1 is true and proposition s2 is true. Proposition g1_3 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition p1 is true if proposition s3 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition t0 is true if proposition t1 is true and proposition wide is true. Proposition t3 is true if proposition t4 is true. Proposition g0_0 is true. Proposition t7 is true if proposition btrue is true. Proposition g1_1 is true. Proposition t4 is true if proposition t5 is true. Proposition g3_1 is true. Proposition g1_0 is true. Proposition s0 is true if proposition g0_0 is true. Proposition s2 is true if proposition g2_0 is true. Proposition g1_2 is true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition etrue is true. Proposition t5 is true if proposition t6 is true. Proposition p3 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true. Proposition g2_0 is true. Proposition q is true if proposition t0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true and proposition p3 is true. Proposition x3 is true if proposition etrue is true and proposition x0 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k4-i3-cred::cred", "rec_id": "hv2-cyclen-even_one_sided_k4-i3", "axis": "cyclen", "difficulty": "even_one_sided_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition g3_0 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true and proposition g1_3 is true. Proposition cq is true if proposition x0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition t1 is true if proposition t2 is true. Proposition btrue is true. Proposition t6 is true if proposition t7 is true and proposition cq is true. Proposition t2 is true if proposition t3 is true. Proposition p0 is true if proposition s1 is true and proposition s2 is true. Proposition g1_3 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition p1 is true if proposition s3 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition t0 is true if proposition t1 is true and proposition wide is true. Proposition t3 is true if proposition t4 is true. Proposition g0_0 is true. Proposition t7 is true if proposition btrue is true. Proposition g1_1 is true. Proposition t4 is true if proposition t5 is true. Proposition g3_1 is true. Proposition g1_0 is true. Proposition s0 is true if proposition g0_0 is true. Proposition s2 is true if proposition g2_0 is true. Proposition g1_2 is true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition etrue is true. Proposition t5 is true if proposition t6 is true. Proposition p3 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true. Proposition g2_0 is true. Proposition q is true if proposition t0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true and proposition p3 is true. Proposition x3 is true if proposition etrue is true and proposition x0 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k4-i3-skept::skept", "rec_id": "hv2-cyclen-even_one_sided_k4-i3", "axis": "cyclen", "difficulty": "even_one_sided_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition g3_0 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true and proposition g1_3 is true. Proposition cq is true if proposition x0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition t1 is true if proposition t2 is true. Proposition btrue is true. Proposition t6 is true if proposition t7 is true and proposition cq is true. Proposition t2 is true if proposition t3 is true. Proposition p0 is true if proposition s1 is true and proposition s2 is true. Proposition g1_3 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition p1 is true if proposition s3 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition t0 is true if proposition t1 is true and proposition wide is true. Proposition t3 is true if proposition t4 is true. Proposition g0_0 is true. Proposition t7 is true if proposition btrue is true. Proposition g1_1 is true. Proposition t4 is true if proposition t5 is true. Proposition g3_1 is true. Proposition g1_0 is true. Proposition s0 is true if proposition g0_0 is true. Proposition s2 is true if proposition g2_0 is true. Proposition g1_2 is true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition etrue is true. Proposition t5 is true if proposition t6 is true. Proposition p3 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true. Proposition g2_0 is true. Proposition q is true if proposition t0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true and proposition p3 is true. Proposition x3 is true if proposition etrue is true and proposition x0 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k4-i3-wfs::wfs", "rec_id": "hv2-cyclen-even_one_sided_k4-i3", "axis": "cyclen", "difficulty": "even_one_sided_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition g3_0 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true and proposition g1_3 is true. Proposition cq is true if proposition x0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition t1 is true if proposition t2 is true. Proposition btrue is true. Proposition t6 is true if proposition t7 is true and proposition cq is true. Proposition t2 is true if proposition t3 is true. Proposition p0 is true if proposition s1 is true and proposition s2 is true. Proposition g1_3 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition p1 is true if proposition s3 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition t0 is true if proposition t1 is true and proposition wide is true. Proposition t3 is true if proposition t4 is true. Proposition g0_0 is true. Proposition t7 is true if proposition btrue is true. Proposition g1_1 is true. Proposition t4 is true if proposition t5 is true. Proposition g3_1 is true. Proposition g1_0 is true. Proposition s0 is true if proposition g0_0 is true. Proposition s2 is true if proposition g2_0 is true. Proposition g1_2 is true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition etrue is true. Proposition t5 is true if proposition t6 is true. Proposition p3 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true. Proposition g2_0 is true. Proposition q is true if proposition t0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true and proposition p3 is true. Proposition x3 is true if proposition etrue is true and proposition x0 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k4-i4-none::none", "rec_id": "hv2-cyclen-even_one_sided_k4-i4", "axis": "cyclen", "difficulty": "even_one_sided_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition t1 is true if proposition t2 is true. Proposition t5 is true if proposition t6 is true. Proposition g1_2 is true. Proposition q is true if proposition t0 is true. Proposition btrue is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true. Proposition g1_1 is true. Proposition t7 is true if proposition btrue is true. Proposition etrue is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition t6 is true if proposition t7 is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true. Proposition x3 is true if proposition etrue is true and proposition x0 is not true. Proposition g2_1 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition t4 is true if proposition t5 is true and proposition wide is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g2_0 is true. Proposition g1_0 is true. Proposition g0_1 is true. Proposition g3_0 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition t0 is true if proposition t1 is true and proposition cq is true. Proposition s3 is true if proposition g3_0 is true. Proposition t3 is true if proposition t4 is true. Proposition g0_0 is true. Proposition cq is true if proposition x0 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t2 is true if proposition t3 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k4-i4-closed_world::closed_world", "rec_id": "hv2-cyclen-even_one_sided_k4-i4", "axis": "cyclen", "difficulty": "even_one_sided_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition t1 is true if proposition t2 is true. Proposition t5 is true if proposition t6 is true. Proposition g1_2 is true. Proposition q is true if proposition t0 is true. Proposition btrue is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true. Proposition g1_1 is true. Proposition t7 is true if proposition btrue is true. Proposition etrue is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition t6 is true if proposition t7 is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true. Proposition x3 is true if proposition etrue is true and proposition x0 is not true. Proposition g2_1 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition t4 is true if proposition t5 is true and proposition wide is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g2_0 is true. Proposition g1_0 is true. Proposition g0_1 is true. Proposition g3_0 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition t0 is true if proposition t1 is true and proposition cq is true. Proposition s3 is true if proposition g3_0 is true. Proposition t3 is true if proposition t4 is true. Proposition g0_0 is true. Proposition cq is true if proposition x0 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t2 is true if proposition t3 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k4-i4-cred::cred", "rec_id": "hv2-cyclen-even_one_sided_k4-i4", "axis": "cyclen", "difficulty": "even_one_sided_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition t1 is true if proposition t2 is true. Proposition t5 is true if proposition t6 is true. Proposition g1_2 is true. Proposition q is true if proposition t0 is true. Proposition btrue is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true. Proposition g1_1 is true. Proposition t7 is true if proposition btrue is true. Proposition etrue is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition t6 is true if proposition t7 is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true. Proposition x3 is true if proposition etrue is true and proposition x0 is not true. Proposition g2_1 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition t4 is true if proposition t5 is true and proposition wide is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g2_0 is true. Proposition g1_0 is true. Proposition g0_1 is true. Proposition g3_0 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition t0 is true if proposition t1 is true and proposition cq is true. Proposition s3 is true if proposition g3_0 is true. Proposition t3 is true if proposition t4 is true. Proposition g0_0 is true. Proposition cq is true if proposition x0 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t2 is true if proposition t3 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k4-i4-skept::skept", "rec_id": "hv2-cyclen-even_one_sided_k4-i4", "axis": "cyclen", "difficulty": "even_one_sided_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition t1 is true if proposition t2 is true. Proposition t5 is true if proposition t6 is true. Proposition g1_2 is true. Proposition q is true if proposition t0 is true. Proposition btrue is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true. Proposition g1_1 is true. Proposition t7 is true if proposition btrue is true. Proposition etrue is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition t6 is true if proposition t7 is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true. Proposition x3 is true if proposition etrue is true and proposition x0 is not true. Proposition g2_1 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition t4 is true if proposition t5 is true and proposition wide is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g2_0 is true. Proposition g1_0 is true. Proposition g0_1 is true. Proposition g3_0 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition t0 is true if proposition t1 is true and proposition cq is true. Proposition s3 is true if proposition g3_0 is true. Proposition t3 is true if proposition t4 is true. Proposition g0_0 is true. Proposition cq is true if proposition x0 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t2 is true if proposition t3 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k4-i4-wfs::wfs", "rec_id": "hv2-cyclen-even_one_sided_k4-i4", "axis": "cyclen", "difficulty": "even_one_sided_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition t1 is true if proposition t2 is true. Proposition t5 is true if proposition t6 is true. Proposition g1_2 is true. Proposition q is true if proposition t0 is true. Proposition btrue is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true. Proposition g1_1 is true. Proposition t7 is true if proposition btrue is true. Proposition etrue is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition t6 is true if proposition t7 is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true. Proposition x3 is true if proposition etrue is true and proposition x0 is not true. Proposition g2_1 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition t4 is true if proposition t5 is true and proposition wide is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g2_0 is true. Proposition g1_0 is true. Proposition g0_1 is true. Proposition g3_0 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition t0 is true if proposition t1 is true and proposition cq is true. Proposition s3 is true if proposition g3_0 is true. Proposition t3 is true if proposition t4 is true. Proposition g0_0 is true. Proposition cq is true if proposition x0 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t2 is true if proposition t3 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k4-i5-none::none", "rec_id": "hv2-cyclen-even_one_sided_k4-i5", "axis": "cyclen", "difficulty": "even_one_sided_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition g1_1 is true. Proposition g1_2 is true. Proposition p3 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g3_1 is true. Proposition g3_0 is true. Proposition p2 is true if proposition s1 is true and proposition s3 is true. Proposition x1 is true if proposition x2 is not true. Proposition x2 is true if proposition x3 is not true. Proposition g1_0 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true. Proposition t2 is true if proposition t3 is true and proposition wide is true. Proposition t5 is true if proposition t6 is true. Proposition g2_0 is true. Proposition x3 is true if proposition x0 is not true. Proposition s0 is true if proposition g0_0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition g2_1 is true. Proposition q is true if proposition t0 is true. Proposition g0_0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true and proposition p3 is true. Proposition btrue is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t1 is true if proposition t2 is true and proposition cq is true. Proposition t0 is true if proposition t1 is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t7 is true if proposition btrue is true. Proposition t3 is true if proposition t4 is true. Proposition x0 is true if proposition x1 is not true. Proposition t6 is true if proposition t7 is true. Proposition cq is true if proposition x0 is true. Proposition t4 is true if proposition t5 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k4-i5-closed_world::closed_world", "rec_id": "hv2-cyclen-even_one_sided_k4-i5", "axis": "cyclen", "difficulty": "even_one_sided_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition g1_1 is true. Proposition g1_2 is true. Proposition p3 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g3_1 is true. Proposition g3_0 is true. Proposition p2 is true if proposition s1 is true and proposition s3 is true. Proposition x1 is true if proposition x2 is not true. Proposition x2 is true if proposition x3 is not true. Proposition g1_0 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true. Proposition t2 is true if proposition t3 is true and proposition wide is true. Proposition t5 is true if proposition t6 is true. Proposition g2_0 is true. Proposition x3 is true if proposition x0 is not true. Proposition s0 is true if proposition g0_0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition g2_1 is true. Proposition q is true if proposition t0 is true. Proposition g0_0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true and proposition p3 is true. Proposition btrue is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t1 is true if proposition t2 is true and proposition cq is true. Proposition t0 is true if proposition t1 is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t7 is true if proposition btrue is true. Proposition t3 is true if proposition t4 is true. Proposition x0 is true if proposition x1 is not true. Proposition t6 is true if proposition t7 is true. Proposition cq is true if proposition x0 is true. Proposition t4 is true if proposition t5 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k4-i5-cred::cred", "rec_id": "hv2-cyclen-even_one_sided_k4-i5", "axis": "cyclen", "difficulty": "even_one_sided_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition g1_1 is true. Proposition g1_2 is true. Proposition p3 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g3_1 is true. Proposition g3_0 is true. Proposition p2 is true if proposition s1 is true and proposition s3 is true. Proposition x1 is true if proposition x2 is not true. Proposition x2 is true if proposition x3 is not true. Proposition g1_0 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true. Proposition t2 is true if proposition t3 is true and proposition wide is true. Proposition t5 is true if proposition t6 is true. Proposition g2_0 is true. Proposition x3 is true if proposition x0 is not true. Proposition s0 is true if proposition g0_0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition g2_1 is true. Proposition q is true if proposition t0 is true. Proposition g0_0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true and proposition p3 is true. Proposition btrue is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t1 is true if proposition t2 is true and proposition cq is true. Proposition t0 is true if proposition t1 is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t7 is true if proposition btrue is true. Proposition t3 is true if proposition t4 is true. Proposition x0 is true if proposition x1 is not true. Proposition t6 is true if proposition t7 is true. Proposition cq is true if proposition x0 is true. Proposition t4 is true if proposition t5 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k4-i5-skept::skept", "rec_id": "hv2-cyclen-even_one_sided_k4-i5", "axis": "cyclen", "difficulty": "even_one_sided_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition g1_1 is true. Proposition g1_2 is true. Proposition p3 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g3_1 is true. Proposition g3_0 is true. Proposition p2 is true if proposition s1 is true and proposition s3 is true. Proposition x1 is true if proposition x2 is not true. Proposition x2 is true if proposition x3 is not true. Proposition g1_0 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true. Proposition t2 is true if proposition t3 is true and proposition wide is true. Proposition t5 is true if proposition t6 is true. Proposition g2_0 is true. Proposition x3 is true if proposition x0 is not true. Proposition s0 is true if proposition g0_0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition g2_1 is true. Proposition q is true if proposition t0 is true. Proposition g0_0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true and proposition p3 is true. Proposition btrue is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t1 is true if proposition t2 is true and proposition cq is true. Proposition t0 is true if proposition t1 is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t7 is true if proposition btrue is true. Proposition t3 is true if proposition t4 is true. Proposition x0 is true if proposition x1 is not true. Proposition t6 is true if proposition t7 is true. Proposition cq is true if proposition x0 is true. Proposition t4 is true if proposition t5 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k4-i5-wfs::wfs", "rec_id": "hv2-cyclen-even_one_sided_k4-i5", "axis": "cyclen", "difficulty": "even_one_sided_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition g1_1 is true. Proposition g1_2 is true. Proposition p3 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g3_1 is true. Proposition g3_0 is true. Proposition p2 is true if proposition s1 is true and proposition s3 is true. Proposition x1 is true if proposition x2 is not true. Proposition x2 is true if proposition x3 is not true. Proposition g1_0 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true. Proposition t2 is true if proposition t3 is true and proposition wide is true. Proposition t5 is true if proposition t6 is true. Proposition g2_0 is true. Proposition x3 is true if proposition x0 is not true. Proposition s0 is true if proposition g0_0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition g2_1 is true. Proposition q is true if proposition t0 is true. Proposition g0_0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true and proposition p3 is true. Proposition btrue is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t1 is true if proposition t2 is true and proposition cq is true. Proposition t0 is true if proposition t1 is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t7 is true if proposition btrue is true. Proposition t3 is true if proposition t4 is true. Proposition x0 is true if proposition x1 is not true. Proposition t6 is true if proposition t7 is true. Proposition cq is true if proposition x0 is true. Proposition t4 is true if proposition t5 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k4-i6-none::none", "rec_id": "hv2-cyclen-even_one_sided_k4-i6", "axis": "cyclen", "difficulty": "even_one_sided_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition t1 is true if proposition t2 is true and proposition cq is true. Proposition t7 is true if proposition btrue is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition t6 is true if proposition t7 is true and proposition wide is true. Proposition g1_1 is true. Proposition cq is true if proposition x0 is true. Proposition g3_0 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition x0 is true if proposition x1 is not true. Proposition g2_1 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition t2 is true if proposition t3 is true. Proposition g3_1 is true. Proposition t4 is true if proposition t5 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t5 is true if proposition t6 is true. Proposition btrue is true. Proposition x1 is true if proposition x2 is not true. Proposition q is true if proposition t0 is true. Proposition g0_1 is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition x3 is true if proposition x0 is not true. Proposition t0 is true if proposition t1 is true. Proposition g2_0 is true. Proposition g0_0 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition t3 is true if proposition t4 is true. Proposition g1_0 is true. Proposition x2 is true if proposition x3 is not true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k4-i6-closed_world::closed_world", "rec_id": "hv2-cyclen-even_one_sided_k4-i6", "axis": "cyclen", "difficulty": "even_one_sided_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition t1 is true if proposition t2 is true and proposition cq is true. Proposition t7 is true if proposition btrue is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition t6 is true if proposition t7 is true and proposition wide is true. Proposition g1_1 is true. Proposition cq is true if proposition x0 is true. Proposition g3_0 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition x0 is true if proposition x1 is not true. Proposition g2_1 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition t2 is true if proposition t3 is true. Proposition g3_1 is true. Proposition t4 is true if proposition t5 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t5 is true if proposition t6 is true. Proposition btrue is true. Proposition x1 is true if proposition x2 is not true. Proposition q is true if proposition t0 is true. Proposition g0_1 is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition x3 is true if proposition x0 is not true. Proposition t0 is true if proposition t1 is true. Proposition g2_0 is true. Proposition g0_0 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition t3 is true if proposition t4 is true. Proposition g1_0 is true. Proposition x2 is true if proposition x3 is not true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k4-i6-cred::cred", "rec_id": "hv2-cyclen-even_one_sided_k4-i6", "axis": "cyclen", "difficulty": "even_one_sided_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition t1 is true if proposition t2 is true and proposition cq is true. Proposition t7 is true if proposition btrue is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition t6 is true if proposition t7 is true and proposition wide is true. Proposition g1_1 is true. Proposition cq is true if proposition x0 is true. Proposition g3_0 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition x0 is true if proposition x1 is not true. Proposition g2_1 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition t2 is true if proposition t3 is true. Proposition g3_1 is true. Proposition t4 is true if proposition t5 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t5 is true if proposition t6 is true. Proposition btrue is true. Proposition x1 is true if proposition x2 is not true. Proposition q is true if proposition t0 is true. Proposition g0_1 is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition x3 is true if proposition x0 is not true. Proposition t0 is true if proposition t1 is true. Proposition g2_0 is true. Proposition g0_0 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition t3 is true if proposition t4 is true. Proposition g1_0 is true. Proposition x2 is true if proposition x3 is not true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k4-i6-skept::skept", "rec_id": "hv2-cyclen-even_one_sided_k4-i6", "axis": "cyclen", "difficulty": "even_one_sided_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition t1 is true if proposition t2 is true and proposition cq is true. Proposition t7 is true if proposition btrue is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition t6 is true if proposition t7 is true and proposition wide is true. Proposition g1_1 is true. Proposition cq is true if proposition x0 is true. Proposition g3_0 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition x0 is true if proposition x1 is not true. Proposition g2_1 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition t2 is true if proposition t3 is true. Proposition g3_1 is true. Proposition t4 is true if proposition t5 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t5 is true if proposition t6 is true. Proposition btrue is true. Proposition x1 is true if proposition x2 is not true. Proposition q is true if proposition t0 is true. Proposition g0_1 is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition x3 is true if proposition x0 is not true. Proposition t0 is true if proposition t1 is true. Proposition g2_0 is true. Proposition g0_0 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition t3 is true if proposition t4 is true. Proposition g1_0 is true. Proposition x2 is true if proposition x3 is not true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k4-i6-wfs::wfs", "rec_id": "hv2-cyclen-even_one_sided_k4-i6", "axis": "cyclen", "difficulty": "even_one_sided_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition t1 is true if proposition t2 is true and proposition cq is true. Proposition t7 is true if proposition btrue is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition t6 is true if proposition t7 is true and proposition wide is true. Proposition g1_1 is true. Proposition cq is true if proposition x0 is true. Proposition g3_0 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition x0 is true if proposition x1 is not true. Proposition g2_1 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition t2 is true if proposition t3 is true. Proposition g3_1 is true. Proposition t4 is true if proposition t5 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t5 is true if proposition t6 is true. Proposition btrue is true. Proposition x1 is true if proposition x2 is not true. Proposition q is true if proposition t0 is true. Proposition g0_1 is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition x3 is true if proposition x0 is not true. Proposition t0 is true if proposition t1 is true. Proposition g2_0 is true. Proposition g0_0 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition t3 is true if proposition t4 is true. Proposition g1_0 is true. Proposition x2 is true if proposition x3 is not true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k4-i7-none::none", "rec_id": "hv2-cyclen-even_one_sided_k4-i7", "axis": "cyclen", "difficulty": "even_one_sided_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition g0_0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true. Proposition s2 is true if proposition g2_0 is true. Proposition g1_1 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition etrue is true. Proposition x3 is true if proposition etrue is true and proposition x0 is not true. Proposition g3_1 is true. Proposition t3 is true if proposition t4 is true and proposition wide is true. Proposition t6 is true if proposition t7 is true. Proposition p1 is true if proposition s2 is true and proposition s3 is true. Proposition t2 is true if proposition t3 is true. Proposition g0_1 is true. Proposition t4 is true if proposition t5 is true. Proposition cq is true if proposition x0 is true. Proposition t7 is true if proposition btrue is true and proposition cq is true. Proposition g3_2 is true. Proposition g3_0 is true. Proposition t1 is true if proposition t2 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition t0 is true if proposition t1 is true. Proposition t5 is true if proposition t6 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true and proposition g3_2 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition q is true if proposition t0 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition g1_0 is true. Proposition g2_0 is true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s3 is true. Proposition btrue is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k4-i7-closed_world::closed_world", "rec_id": "hv2-cyclen-even_one_sided_k4-i7", "axis": "cyclen", "difficulty": "even_one_sided_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition g0_0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true. Proposition s2 is true if proposition g2_0 is true. Proposition g1_1 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition etrue is true. Proposition x3 is true if proposition etrue is true and proposition x0 is not true. Proposition g3_1 is true. Proposition t3 is true if proposition t4 is true and proposition wide is true. Proposition t6 is true if proposition t7 is true. Proposition p1 is true if proposition s2 is true and proposition s3 is true. Proposition t2 is true if proposition t3 is true. Proposition g0_1 is true. Proposition t4 is true if proposition t5 is true. Proposition cq is true if proposition x0 is true. Proposition t7 is true if proposition btrue is true and proposition cq is true. Proposition g3_2 is true. Proposition g3_0 is true. Proposition t1 is true if proposition t2 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition t0 is true if proposition t1 is true. Proposition t5 is true if proposition t6 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true and proposition g3_2 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition q is true if proposition t0 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition g1_0 is true. Proposition g2_0 is true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s3 is true. Proposition btrue is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k4-i7-cred::cred", "rec_id": "hv2-cyclen-even_one_sided_k4-i7", "axis": "cyclen", "difficulty": "even_one_sided_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition g0_0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true. Proposition s2 is true if proposition g2_0 is true. Proposition g1_1 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition etrue is true. Proposition x3 is true if proposition etrue is true and proposition x0 is not true. Proposition g3_1 is true. Proposition t3 is true if proposition t4 is true and proposition wide is true. Proposition t6 is true if proposition t7 is true. Proposition p1 is true if proposition s2 is true and proposition s3 is true. Proposition t2 is true if proposition t3 is true. Proposition g0_1 is true. Proposition t4 is true if proposition t5 is true. Proposition cq is true if proposition x0 is true. Proposition t7 is true if proposition btrue is true and proposition cq is true. Proposition g3_2 is true. Proposition g3_0 is true. Proposition t1 is true if proposition t2 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition t0 is true if proposition t1 is true. Proposition t5 is true if proposition t6 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true and proposition g3_2 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition q is true if proposition t0 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition g1_0 is true. Proposition g2_0 is true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s3 is true. Proposition btrue is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k4-i7-skept::skept", "rec_id": "hv2-cyclen-even_one_sided_k4-i7", "axis": "cyclen", "difficulty": "even_one_sided_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition g0_0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true. Proposition s2 is true if proposition g2_0 is true. Proposition g1_1 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition etrue is true. Proposition x3 is true if proposition etrue is true and proposition x0 is not true. Proposition g3_1 is true. Proposition t3 is true if proposition t4 is true and proposition wide is true. Proposition t6 is true if proposition t7 is true. Proposition p1 is true if proposition s2 is true and proposition s3 is true. Proposition t2 is true if proposition t3 is true. Proposition g0_1 is true. Proposition t4 is true if proposition t5 is true. Proposition cq is true if proposition x0 is true. Proposition t7 is true if proposition btrue is true and proposition cq is true. Proposition g3_2 is true. Proposition g3_0 is true. Proposition t1 is true if proposition t2 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition t0 is true if proposition t1 is true. Proposition t5 is true if proposition t6 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true and proposition g3_2 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition q is true if proposition t0 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition g1_0 is true. Proposition g2_0 is true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s3 is true. Proposition btrue is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k4-i7-wfs::wfs", "rec_id": "hv2-cyclen-even_one_sided_k4-i7", "axis": "cyclen", "difficulty": "even_one_sided_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition g0_0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true. Proposition s2 is true if proposition g2_0 is true. Proposition g1_1 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition etrue is true. Proposition x3 is true if proposition etrue is true and proposition x0 is not true. Proposition g3_1 is true. Proposition t3 is true if proposition t4 is true and proposition wide is true. Proposition t6 is true if proposition t7 is true. Proposition p1 is true if proposition s2 is true and proposition s3 is true. Proposition t2 is true if proposition t3 is true. Proposition g0_1 is true. Proposition t4 is true if proposition t5 is true. Proposition cq is true if proposition x0 is true. Proposition t7 is true if proposition btrue is true and proposition cq is true. Proposition g3_2 is true. Proposition g3_0 is true. Proposition t1 is true if proposition t2 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition t0 is true if proposition t1 is true. Proposition t5 is true if proposition t6 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true and proposition g3_2 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition q is true if proposition t0 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition g1_0 is true. Proposition g2_0 is true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s3 is true. Proposition btrue is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k4-i0-none::none", "rec_id": "hv2-cyclen-odd_k4-i0", "axis": "cyclen", "difficulty": "odd_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition s0 is true if proposition g0_0 is true and proposition g0_1 is true and proposition g0_2 is true. Proposition cq is true if proposition x0 is true. Proposition t6 is true if proposition t7 is true. Proposition g0_0 is true. Proposition btrue is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition q is true if proposition t0 is true. Proposition p0 is true if proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition t4 is true. Proposition t1 is true if proposition t2 is true. Proposition t0 is true if proposition t1 is true. Proposition t4 is true if proposition t5 is true. Proposition t5 is true if proposition t6 is true and proposition cq is true. Proposition x2 is true if proposition x3 is not true. Proposition x0 is true if proposition x1 is not true. Proposition x1 is true if proposition x2 is not true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true. Proposition g2_0 is true. Proposition g1_1 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition t7 is true if proposition btrue is true and proposition wide is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s3 is true. Proposition s3 is true if proposition g3_0 is true. Proposition g3_0 is true. Proposition g0_1 is true. Proposition g1_0 is true. Proposition g0_2 is true. Proposition x3 is true if proposition x0 is not true. Proposition g2_1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k4-i0-closed_world::closed_world", "rec_id": "hv2-cyclen-odd_k4-i0", "axis": "cyclen", "difficulty": "odd_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition s0 is true if proposition g0_0 is true and proposition g0_1 is true and proposition g0_2 is true. Proposition cq is true if proposition x0 is true. Proposition t6 is true if proposition t7 is true. Proposition g0_0 is true. Proposition btrue is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition q is true if proposition t0 is true. Proposition p0 is true if proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition t4 is true. Proposition t1 is true if proposition t2 is true. Proposition t0 is true if proposition t1 is true. Proposition t4 is true if proposition t5 is true. Proposition t5 is true if proposition t6 is true and proposition cq is true. Proposition x2 is true if proposition x3 is not true. Proposition x0 is true if proposition x1 is not true. Proposition x1 is true if proposition x2 is not true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true. Proposition g2_0 is true. Proposition g1_1 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition t7 is true if proposition btrue is true and proposition wide is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s3 is true. Proposition s3 is true if proposition g3_0 is true. Proposition g3_0 is true. Proposition g0_1 is true. Proposition g1_0 is true. Proposition g0_2 is true. Proposition x3 is true if proposition x0 is not true. Proposition g2_1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k4-i0-cred::cred", "rec_id": "hv2-cyclen-odd_k4-i0", "axis": "cyclen", "difficulty": "odd_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition s0 is true if proposition g0_0 is true and proposition g0_1 is true and proposition g0_2 is true. Proposition cq is true if proposition x0 is true. Proposition t6 is true if proposition t7 is true. Proposition g0_0 is true. Proposition btrue is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition q is true if proposition t0 is true. Proposition p0 is true if proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition t4 is true. Proposition t1 is true if proposition t2 is true. Proposition t0 is true if proposition t1 is true. Proposition t4 is true if proposition t5 is true. Proposition t5 is true if proposition t6 is true and proposition cq is true. Proposition x2 is true if proposition x3 is not true. Proposition x0 is true if proposition x1 is not true. Proposition x1 is true if proposition x2 is not true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true. Proposition g2_0 is true. Proposition g1_1 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition t7 is true if proposition btrue is true and proposition wide is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s3 is true. Proposition s3 is true if proposition g3_0 is true. Proposition g3_0 is true. Proposition g0_1 is true. Proposition g1_0 is true. Proposition g0_2 is true. Proposition x3 is true if proposition x0 is not true. Proposition g2_1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k4-i0-skept::skept", "rec_id": "hv2-cyclen-odd_k4-i0", "axis": "cyclen", "difficulty": "odd_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition s0 is true if proposition g0_0 is true and proposition g0_1 is true and proposition g0_2 is true. Proposition cq is true if proposition x0 is true. Proposition t6 is true if proposition t7 is true. Proposition g0_0 is true. Proposition btrue is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition q is true if proposition t0 is true. Proposition p0 is true if proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition t4 is true. Proposition t1 is true if proposition t2 is true. Proposition t0 is true if proposition t1 is true. Proposition t4 is true if proposition t5 is true. Proposition t5 is true if proposition t6 is true and proposition cq is true. Proposition x2 is true if proposition x3 is not true. Proposition x0 is true if proposition x1 is not true. Proposition x1 is true if proposition x2 is not true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true. Proposition g2_0 is true. Proposition g1_1 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition t7 is true if proposition btrue is true and proposition wide is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s3 is true. Proposition s3 is true if proposition g3_0 is true. Proposition g3_0 is true. Proposition g0_1 is true. Proposition g1_0 is true. Proposition g0_2 is true. Proposition x3 is true if proposition x0 is not true. Proposition g2_1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k4-i0-wfs::wfs", "rec_id": "hv2-cyclen-odd_k4-i0", "axis": "cyclen", "difficulty": "odd_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition s0 is true if proposition g0_0 is true and proposition g0_1 is true and proposition g0_2 is true. Proposition cq is true if proposition x0 is true. Proposition t6 is true if proposition t7 is true. Proposition g0_0 is true. Proposition btrue is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition q is true if proposition t0 is true. Proposition p0 is true if proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t2 is true if proposition t3 is true. Proposition t3 is true if proposition t4 is true. Proposition t1 is true if proposition t2 is true. Proposition t0 is true if proposition t1 is true. Proposition t4 is true if proposition t5 is true. Proposition t5 is true if proposition t6 is true and proposition cq is true. Proposition x2 is true if proposition x3 is not true. Proposition x0 is true if proposition x1 is not true. Proposition x1 is true if proposition x2 is not true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true. Proposition g2_0 is true. Proposition g1_1 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition t7 is true if proposition btrue is true and proposition wide is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s3 is true. Proposition s3 is true if proposition g3_0 is true. Proposition g3_0 is true. Proposition g0_1 is true. Proposition g1_0 is true. Proposition g0_2 is true. Proposition x3 is true if proposition x0 is not true. Proposition g2_1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k4-i1-none::none", "rec_id": "hv2-cyclen-odd_k4-i1", "axis": "cyclen", "difficulty": "odd_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition t3 is true if proposition t4 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition g0_0 is true. Proposition t1 is true if proposition t2 is true and proposition cq is true. Proposition q is true if proposition t0 is true. Proposition t4 is true if proposition t5 is true. Proposition x3 is true if proposition etrue is true and proposition x0 is not true. Proposition g3_1 is true. Proposition t0 is true if proposition t1 is true. Proposition g0_1 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition cq is true if proposition x0 is true. Proposition g3_0 is true. Proposition g2_0 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g2_1 is true. Proposition t2 is true if proposition t3 is true. Proposition g1_1 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition btrue is true. Proposition g1_0 is true. Proposition t5 is true if proposition t6 is true and proposition wide is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition t6 is true if proposition t7 is true. Proposition t7 is true if proposition btrue is true. Proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k4-i1-closed_world::closed_world", "rec_id": "hv2-cyclen-odd_k4-i1", "axis": "cyclen", "difficulty": "odd_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition t3 is true if proposition t4 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition g0_0 is true. Proposition t1 is true if proposition t2 is true and proposition cq is true. Proposition q is true if proposition t0 is true. Proposition t4 is true if proposition t5 is true. Proposition x3 is true if proposition etrue is true and proposition x0 is not true. Proposition g3_1 is true. Proposition t0 is true if proposition t1 is true. Proposition g0_1 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition cq is true if proposition x0 is true. Proposition g3_0 is true. Proposition g2_0 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g2_1 is true. Proposition t2 is true if proposition t3 is true. Proposition g1_1 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition btrue is true. Proposition g1_0 is true. Proposition t5 is true if proposition t6 is true and proposition wide is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition t6 is true if proposition t7 is true. Proposition t7 is true if proposition btrue is true. Proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k4-i1-cred::cred", "rec_id": "hv2-cyclen-odd_k4-i1", "axis": "cyclen", "difficulty": "odd_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition t3 is true if proposition t4 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition g0_0 is true. Proposition t1 is true if proposition t2 is true and proposition cq is true. Proposition q is true if proposition t0 is true. Proposition t4 is true if proposition t5 is true. Proposition x3 is true if proposition etrue is true and proposition x0 is not true. Proposition g3_1 is true. Proposition t0 is true if proposition t1 is true. Proposition g0_1 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition cq is true if proposition x0 is true. Proposition g3_0 is true. Proposition g2_0 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g2_1 is true. Proposition t2 is true if proposition t3 is true. Proposition g1_1 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition btrue is true. Proposition g1_0 is true. Proposition t5 is true if proposition t6 is true and proposition wide is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition t6 is true if proposition t7 is true. Proposition t7 is true if proposition btrue is true. Proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k4-i1-skept::skept", "rec_id": "hv2-cyclen-odd_k4-i1", "axis": "cyclen", "difficulty": "odd_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition t3 is true if proposition t4 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition g0_0 is true. Proposition t1 is true if proposition t2 is true and proposition cq is true. Proposition q is true if proposition t0 is true. Proposition t4 is true if proposition t5 is true. Proposition x3 is true if proposition etrue is true and proposition x0 is not true. Proposition g3_1 is true. Proposition t0 is true if proposition t1 is true. Proposition g0_1 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition cq is true if proposition x0 is true. Proposition g3_0 is true. Proposition g2_0 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g2_1 is true. Proposition t2 is true if proposition t3 is true. Proposition g1_1 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition btrue is true. Proposition g1_0 is true. Proposition t5 is true if proposition t6 is true and proposition wide is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition t6 is true if proposition t7 is true. Proposition t7 is true if proposition btrue is true. Proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k4-i1-wfs::wfs", "rec_id": "hv2-cyclen-odd_k4-i1", "axis": "cyclen", "difficulty": "odd_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition t3 is true if proposition t4 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition g0_0 is true. Proposition t1 is true if proposition t2 is true and proposition cq is true. Proposition q is true if proposition t0 is true. Proposition t4 is true if proposition t5 is true. Proposition x3 is true if proposition etrue is true and proposition x0 is not true. Proposition g3_1 is true. Proposition t0 is true if proposition t1 is true. Proposition g0_1 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition cq is true if proposition x0 is true. Proposition g3_0 is true. Proposition g2_0 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g2_1 is true. Proposition t2 is true if proposition t3 is true. Proposition g1_1 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition btrue is true. Proposition g1_0 is true. Proposition t5 is true if proposition t6 is true and proposition wide is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition t6 is true if proposition t7 is true. Proposition t7 is true if proposition btrue is true. Proposition etrue is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k4-i2-none::none", "rec_id": "hv2-cyclen-odd_k4-i2", "axis": "cyclen", "difficulty": "odd_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition x1 is true if proposition x2 is not true. Proposition t5 is true if proposition t6 is true and proposition cq is true. Proposition g0_1 is true. Proposition g2_1 is true. Proposition x3 is true if proposition x0 is not true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition btrue is true. Proposition x2 is true if proposition x3 is not true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true and proposition g0_2 is true. Proposition t1 is true if proposition t2 is true. Proposition t4 is true if proposition t5 is true. Proposition t3 is true if proposition t4 is true. Proposition g2_0 is true. Proposition g0_0 is true. Proposition cq is true if proposition x0 is true. Proposition g3_1 is true. Proposition t6 is true if proposition t7 is true. Proposition t0 is true if proposition t1 is true. Proposition x0 is true if proposition x1 is not true. Proposition q is true if proposition t0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t2 is true if proposition t3 is true. Proposition g3_0 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition s1 is true if proposition g1_0 is true. Proposition g1_0 is true. Proposition g0_2 is true. Proposition t7 is true if proposition btrue is true and proposition wide is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k4-i2-closed_world::closed_world", "rec_id": "hv2-cyclen-odd_k4-i2", "axis": "cyclen", "difficulty": "odd_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition x1 is true if proposition x2 is not true. Proposition t5 is true if proposition t6 is true and proposition cq is true. Proposition g0_1 is true. Proposition g2_1 is true. Proposition x3 is true if proposition x0 is not true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition btrue is true. Proposition x2 is true if proposition x3 is not true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true and proposition g0_2 is true. Proposition t1 is true if proposition t2 is true. Proposition t4 is true if proposition t5 is true. Proposition t3 is true if proposition t4 is true. Proposition g2_0 is true. Proposition g0_0 is true. Proposition cq is true if proposition x0 is true. Proposition g3_1 is true. Proposition t6 is true if proposition t7 is true. Proposition t0 is true if proposition t1 is true. Proposition x0 is true if proposition x1 is not true. Proposition q is true if proposition t0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t2 is true if proposition t3 is true. Proposition g3_0 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition s1 is true if proposition g1_0 is true. Proposition g1_0 is true. Proposition g0_2 is true. Proposition t7 is true if proposition btrue is true and proposition wide is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k4-i2-cred::cred", "rec_id": "hv2-cyclen-odd_k4-i2", "axis": "cyclen", "difficulty": "odd_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition x1 is true if proposition x2 is not true. Proposition t5 is true if proposition t6 is true and proposition cq is true. Proposition g0_1 is true. Proposition g2_1 is true. Proposition x3 is true if proposition x0 is not true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition btrue is true. Proposition x2 is true if proposition x3 is not true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true and proposition g0_2 is true. Proposition t1 is true if proposition t2 is true. Proposition t4 is true if proposition t5 is true. Proposition t3 is true if proposition t4 is true. Proposition g2_0 is true. Proposition g0_0 is true. Proposition cq is true if proposition x0 is true. Proposition g3_1 is true. Proposition t6 is true if proposition t7 is true. Proposition t0 is true if proposition t1 is true. Proposition x0 is true if proposition x1 is not true. Proposition q is true if proposition t0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t2 is true if proposition t3 is true. Proposition g3_0 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition s1 is true if proposition g1_0 is true. Proposition g1_0 is true. Proposition g0_2 is true. Proposition t7 is true if proposition btrue is true and proposition wide is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k4-i2-skept::skept", "rec_id": "hv2-cyclen-odd_k4-i2", "axis": "cyclen", "difficulty": "odd_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition x1 is true if proposition x2 is not true. Proposition t5 is true if proposition t6 is true and proposition cq is true. Proposition g0_1 is true. Proposition g2_1 is true. Proposition x3 is true if proposition x0 is not true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition btrue is true. Proposition x2 is true if proposition x3 is not true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true and proposition g0_2 is true. Proposition t1 is true if proposition t2 is true. Proposition t4 is true if proposition t5 is true. Proposition t3 is true if proposition t4 is true. Proposition g2_0 is true. Proposition g0_0 is true. Proposition cq is true if proposition x0 is true. Proposition g3_1 is true. Proposition t6 is true if proposition t7 is true. Proposition t0 is true if proposition t1 is true. Proposition x0 is true if proposition x1 is not true. Proposition q is true if proposition t0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t2 is true if proposition t3 is true. Proposition g3_0 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition s1 is true if proposition g1_0 is true. Proposition g1_0 is true. Proposition g0_2 is true. Proposition t7 is true if proposition btrue is true and proposition wide is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k4-i2-wfs::wfs", "rec_id": "hv2-cyclen-odd_k4-i2", "axis": "cyclen", "difficulty": "odd_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition x1 is true if proposition x2 is not true. Proposition t5 is true if proposition t6 is true and proposition cq is true. Proposition g0_1 is true. Proposition g2_1 is true. Proposition x3 is true if proposition x0 is not true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition btrue is true. Proposition x2 is true if proposition x3 is not true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true and proposition g0_2 is true. Proposition t1 is true if proposition t2 is true. Proposition t4 is true if proposition t5 is true. Proposition t3 is true if proposition t4 is true. Proposition g2_0 is true. Proposition g0_0 is true. Proposition cq is true if proposition x0 is true. Proposition g3_1 is true. Proposition t6 is true if proposition t7 is true. Proposition t0 is true if proposition t1 is true. Proposition x0 is true if proposition x1 is not true. Proposition q is true if proposition t0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t2 is true if proposition t3 is true. Proposition g3_0 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition s1 is true if proposition g1_0 is true. Proposition g1_0 is true. Proposition g0_2 is true. Proposition t7 is true if proposition btrue is true and proposition wide is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k4-i3-none::none", "rec_id": "hv2-cyclen-odd_k4-i3", "axis": "cyclen", "difficulty": "odd_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition g3_0 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true and proposition g1_3 is true. Proposition cq is true if proposition x0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition t1 is true if proposition t2 is true. Proposition btrue is true. Proposition t6 is true if proposition t7 is true and proposition cq is true. Proposition t2 is true if proposition t3 is true. Proposition p0 is true if proposition s1 is true and proposition s2 is true. Proposition g1_3 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition p1 is true if proposition s3 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition t0 is true if proposition t1 is true and proposition wide is true. Proposition t3 is true if proposition t4 is true. Proposition g0_0 is true. Proposition t7 is true if proposition btrue is true. Proposition g1_1 is true. Proposition t4 is true if proposition t5 is true. Proposition g3_1 is true. Proposition g1_0 is true. Proposition s0 is true if proposition g0_0 is true. Proposition s2 is true if proposition g2_0 is true. Proposition g1_2 is true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition etrue is true. Proposition t5 is true if proposition t6 is true. Proposition p3 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true. Proposition g2_0 is true. Proposition q is true if proposition t0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true and proposition p3 is true. Proposition x3 is true if proposition etrue is true and proposition x0 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k4-i3-closed_world::closed_world", "rec_id": "hv2-cyclen-odd_k4-i3", "axis": "cyclen", "difficulty": "odd_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition g3_0 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true and proposition g1_3 is true. Proposition cq is true if proposition x0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition t1 is true if proposition t2 is true. Proposition btrue is true. Proposition t6 is true if proposition t7 is true and proposition cq is true. Proposition t2 is true if proposition t3 is true. Proposition p0 is true if proposition s1 is true and proposition s2 is true. Proposition g1_3 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition p1 is true if proposition s3 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition t0 is true if proposition t1 is true and proposition wide is true. Proposition t3 is true if proposition t4 is true. Proposition g0_0 is true. Proposition t7 is true if proposition btrue is true. Proposition g1_1 is true. Proposition t4 is true if proposition t5 is true. Proposition g3_1 is true. Proposition g1_0 is true. Proposition s0 is true if proposition g0_0 is true. Proposition s2 is true if proposition g2_0 is true. Proposition g1_2 is true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition etrue is true. Proposition t5 is true if proposition t6 is true. Proposition p3 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true. Proposition g2_0 is true. Proposition q is true if proposition t0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true and proposition p3 is true. Proposition x3 is true if proposition etrue is true and proposition x0 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k4-i3-cred::cred", "rec_id": "hv2-cyclen-odd_k4-i3", "axis": "cyclen", "difficulty": "odd_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition g3_0 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true and proposition g1_3 is true. Proposition cq is true if proposition x0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition t1 is true if proposition t2 is true. Proposition btrue is true. Proposition t6 is true if proposition t7 is true and proposition cq is true. Proposition t2 is true if proposition t3 is true. Proposition p0 is true if proposition s1 is true and proposition s2 is true. Proposition g1_3 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition p1 is true if proposition s3 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition t0 is true if proposition t1 is true and proposition wide is true. Proposition t3 is true if proposition t4 is true. Proposition g0_0 is true. Proposition t7 is true if proposition btrue is true. Proposition g1_1 is true. Proposition t4 is true if proposition t5 is true. Proposition g3_1 is true. Proposition g1_0 is true. Proposition s0 is true if proposition g0_0 is true. Proposition s2 is true if proposition g2_0 is true. Proposition g1_2 is true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition etrue is true. Proposition t5 is true if proposition t6 is true. Proposition p3 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true. Proposition g2_0 is true. Proposition q is true if proposition t0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true and proposition p3 is true. Proposition x3 is true if proposition etrue is true and proposition x0 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k4-i3-skept::skept", "rec_id": "hv2-cyclen-odd_k4-i3", "axis": "cyclen", "difficulty": "odd_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition g3_0 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true and proposition g1_3 is true. Proposition cq is true if proposition x0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition t1 is true if proposition t2 is true. Proposition btrue is true. Proposition t6 is true if proposition t7 is true and proposition cq is true. Proposition t2 is true if proposition t3 is true. Proposition p0 is true if proposition s1 is true and proposition s2 is true. Proposition g1_3 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition p1 is true if proposition s3 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition t0 is true if proposition t1 is true and proposition wide is true. Proposition t3 is true if proposition t4 is true. Proposition g0_0 is true. Proposition t7 is true if proposition btrue is true. Proposition g1_1 is true. Proposition t4 is true if proposition t5 is true. Proposition g3_1 is true. Proposition g1_0 is true. Proposition s0 is true if proposition g0_0 is true. Proposition s2 is true if proposition g2_0 is true. Proposition g1_2 is true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition etrue is true. Proposition t5 is true if proposition t6 is true. Proposition p3 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true. Proposition g2_0 is true. Proposition q is true if proposition t0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true and proposition p3 is true. Proposition x3 is true if proposition etrue is true and proposition x0 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k4-i3-wfs::wfs", "rec_id": "hv2-cyclen-odd_k4-i3", "axis": "cyclen", "difficulty": "odd_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition g3_0 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true and proposition g1_3 is true. Proposition cq is true if proposition x0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition t1 is true if proposition t2 is true. Proposition btrue is true. Proposition t6 is true if proposition t7 is true and proposition cq is true. Proposition t2 is true if proposition t3 is true. Proposition p0 is true if proposition s1 is true and proposition s2 is true. Proposition g1_3 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition p1 is true if proposition s3 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition t0 is true if proposition t1 is true and proposition wide is true. Proposition t3 is true if proposition t4 is true. Proposition g0_0 is true. Proposition t7 is true if proposition btrue is true. Proposition g1_1 is true. Proposition t4 is true if proposition t5 is true. Proposition g3_1 is true. Proposition g1_0 is true. Proposition s0 is true if proposition g0_0 is true. Proposition s2 is true if proposition g2_0 is true. Proposition g1_2 is true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition etrue is true. Proposition t5 is true if proposition t6 is true. Proposition p3 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true. Proposition g2_0 is true. Proposition q is true if proposition t0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true and proposition p3 is true. Proposition x3 is true if proposition etrue is true and proposition x0 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k4-i4-none::none", "rec_id": "hv2-cyclen-odd_k4-i4", "axis": "cyclen", "difficulty": "odd_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition t1 is true if proposition t2 is true. Proposition t5 is true if proposition t6 is true. Proposition g1_2 is true. Proposition q is true if proposition t0 is true. Proposition btrue is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true. Proposition g1_1 is true. Proposition t7 is true if proposition btrue is true. Proposition etrue is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition t6 is true if proposition t7 is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true. Proposition x3 is true if proposition etrue is true and proposition x0 is not true. Proposition g2_1 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition t4 is true if proposition t5 is true and proposition wide is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g2_0 is true. Proposition g1_0 is true. Proposition g0_1 is true. Proposition g3_0 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition t0 is true if proposition t1 is true and proposition cq is true. Proposition s3 is true if proposition g3_0 is true. Proposition t3 is true if proposition t4 is true. Proposition g0_0 is true. Proposition cq is true if proposition x0 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t2 is true if proposition t3 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k4-i4-closed_world::closed_world", "rec_id": "hv2-cyclen-odd_k4-i4", "axis": "cyclen", "difficulty": "odd_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition t1 is true if proposition t2 is true. Proposition t5 is true if proposition t6 is true. Proposition g1_2 is true. Proposition q is true if proposition t0 is true. Proposition btrue is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true. Proposition g1_1 is true. Proposition t7 is true if proposition btrue is true. Proposition etrue is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition t6 is true if proposition t7 is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true. Proposition x3 is true if proposition etrue is true and proposition x0 is not true. Proposition g2_1 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition t4 is true if proposition t5 is true and proposition wide is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g2_0 is true. Proposition g1_0 is true. Proposition g0_1 is true. Proposition g3_0 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition t0 is true if proposition t1 is true and proposition cq is true. Proposition s3 is true if proposition g3_0 is true. Proposition t3 is true if proposition t4 is true. Proposition g0_0 is true. Proposition cq is true if proposition x0 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t2 is true if proposition t3 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k4-i4-cred::cred", "rec_id": "hv2-cyclen-odd_k4-i4", "axis": "cyclen", "difficulty": "odd_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition t1 is true if proposition t2 is true. Proposition t5 is true if proposition t6 is true. Proposition g1_2 is true. Proposition q is true if proposition t0 is true. Proposition btrue is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true. Proposition g1_1 is true. Proposition t7 is true if proposition btrue is true. Proposition etrue is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition t6 is true if proposition t7 is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true. Proposition x3 is true if proposition etrue is true and proposition x0 is not true. Proposition g2_1 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition t4 is true if proposition t5 is true and proposition wide is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g2_0 is true. Proposition g1_0 is true. Proposition g0_1 is true. Proposition g3_0 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition t0 is true if proposition t1 is true and proposition cq is true. Proposition s3 is true if proposition g3_0 is true. Proposition t3 is true if proposition t4 is true. Proposition g0_0 is true. Proposition cq is true if proposition x0 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t2 is true if proposition t3 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k4-i4-skept::skept", "rec_id": "hv2-cyclen-odd_k4-i4", "axis": "cyclen", "difficulty": "odd_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition t1 is true if proposition t2 is true. Proposition t5 is true if proposition t6 is true. Proposition g1_2 is true. Proposition q is true if proposition t0 is true. Proposition btrue is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true. Proposition g1_1 is true. Proposition t7 is true if proposition btrue is true. Proposition etrue is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition t6 is true if proposition t7 is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true. Proposition x3 is true if proposition etrue is true and proposition x0 is not true. Proposition g2_1 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition t4 is true if proposition t5 is true and proposition wide is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g2_0 is true. Proposition g1_0 is true. Proposition g0_1 is true. Proposition g3_0 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition t0 is true if proposition t1 is true and proposition cq is true. Proposition s3 is true if proposition g3_0 is true. Proposition t3 is true if proposition t4 is true. Proposition g0_0 is true. Proposition cq is true if proposition x0 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t2 is true if proposition t3 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k4-i4-wfs::wfs", "rec_id": "hv2-cyclen-odd_k4-i4", "axis": "cyclen", "difficulty": "odd_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition t1 is true if proposition t2 is true. Proposition t5 is true if proposition t6 is true. Proposition g1_2 is true. Proposition q is true if proposition t0 is true. Proposition btrue is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true. Proposition g1_1 is true. Proposition t7 is true if proposition btrue is true. Proposition etrue is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition t6 is true if proposition t7 is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true. Proposition x3 is true if proposition etrue is true and proposition x0 is not true. Proposition g2_1 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition t4 is true if proposition t5 is true and proposition wide is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g2_0 is true. Proposition g1_0 is true. Proposition g0_1 is true. Proposition g3_0 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition t0 is true if proposition t1 is true and proposition cq is true. Proposition s3 is true if proposition g3_0 is true. Proposition t3 is true if proposition t4 is true. Proposition g0_0 is true. Proposition cq is true if proposition x0 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t2 is true if proposition t3 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k4-i5-none::none", "rec_id": "hv2-cyclen-odd_k4-i5", "axis": "cyclen", "difficulty": "odd_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition g1_1 is true. Proposition g1_2 is true. Proposition p3 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g3_1 is true. Proposition g3_0 is true. Proposition p2 is true if proposition s1 is true and proposition s3 is true. Proposition x1 is true if proposition x2 is not true. Proposition x2 is true if proposition x3 is not true. Proposition g1_0 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true. Proposition t2 is true if proposition t3 is true and proposition wide is true. Proposition t5 is true if proposition t6 is true. Proposition g2_0 is true. Proposition x3 is true if proposition x0 is not true. Proposition s0 is true if proposition g0_0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition g2_1 is true. Proposition q is true if proposition t0 is true. Proposition g0_0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true and proposition p3 is true. Proposition btrue is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t1 is true if proposition t2 is true and proposition cq is true. Proposition t0 is true if proposition t1 is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t7 is true if proposition btrue is true. Proposition t3 is true if proposition t4 is true. Proposition x0 is true if proposition x1 is not true. Proposition t6 is true if proposition t7 is true. Proposition cq is true if proposition x0 is true. Proposition t4 is true if proposition t5 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k4-i5-closed_world::closed_world", "rec_id": "hv2-cyclen-odd_k4-i5", "axis": "cyclen", "difficulty": "odd_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition g1_1 is true. Proposition g1_2 is true. Proposition p3 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g3_1 is true. Proposition g3_0 is true. Proposition p2 is true if proposition s1 is true and proposition s3 is true. Proposition x1 is true if proposition x2 is not true. Proposition x2 is true if proposition x3 is not true. Proposition g1_0 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true. Proposition t2 is true if proposition t3 is true and proposition wide is true. Proposition t5 is true if proposition t6 is true. Proposition g2_0 is true. Proposition x3 is true if proposition x0 is not true. Proposition s0 is true if proposition g0_0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition g2_1 is true. Proposition q is true if proposition t0 is true. Proposition g0_0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true and proposition p3 is true. Proposition btrue is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t1 is true if proposition t2 is true and proposition cq is true. Proposition t0 is true if proposition t1 is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t7 is true if proposition btrue is true. Proposition t3 is true if proposition t4 is true. Proposition x0 is true if proposition x1 is not true. Proposition t6 is true if proposition t7 is true. Proposition cq is true if proposition x0 is true. Proposition t4 is true if proposition t5 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k4-i5-cred::cred", "rec_id": "hv2-cyclen-odd_k4-i5", "axis": "cyclen", "difficulty": "odd_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition g1_1 is true. Proposition g1_2 is true. Proposition p3 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g3_1 is true. Proposition g3_0 is true. Proposition p2 is true if proposition s1 is true and proposition s3 is true. Proposition x1 is true if proposition x2 is not true. Proposition x2 is true if proposition x3 is not true. Proposition g1_0 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true. Proposition t2 is true if proposition t3 is true and proposition wide is true. Proposition t5 is true if proposition t6 is true. Proposition g2_0 is true. Proposition x3 is true if proposition x0 is not true. Proposition s0 is true if proposition g0_0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition g2_1 is true. Proposition q is true if proposition t0 is true. Proposition g0_0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true and proposition p3 is true. Proposition btrue is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t1 is true if proposition t2 is true and proposition cq is true. Proposition t0 is true if proposition t1 is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t7 is true if proposition btrue is true. Proposition t3 is true if proposition t4 is true. Proposition x0 is true if proposition x1 is not true. Proposition t6 is true if proposition t7 is true. Proposition cq is true if proposition x0 is true. Proposition t4 is true if proposition t5 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k4-i5-skept::skept", "rec_id": "hv2-cyclen-odd_k4-i5", "axis": "cyclen", "difficulty": "odd_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition g1_1 is true. Proposition g1_2 is true. Proposition p3 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g3_1 is true. Proposition g3_0 is true. Proposition p2 is true if proposition s1 is true and proposition s3 is true. Proposition x1 is true if proposition x2 is not true. Proposition x2 is true if proposition x3 is not true. Proposition g1_0 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true. Proposition t2 is true if proposition t3 is true and proposition wide is true. Proposition t5 is true if proposition t6 is true. Proposition g2_0 is true. Proposition x3 is true if proposition x0 is not true. Proposition s0 is true if proposition g0_0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition g2_1 is true. Proposition q is true if proposition t0 is true. Proposition g0_0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true and proposition p3 is true. Proposition btrue is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t1 is true if proposition t2 is true and proposition cq is true. Proposition t0 is true if proposition t1 is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t7 is true if proposition btrue is true. Proposition t3 is true if proposition t4 is true. Proposition x0 is true if proposition x1 is not true. Proposition t6 is true if proposition t7 is true. Proposition cq is true if proposition x0 is true. Proposition t4 is true if proposition t5 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k4-i5-wfs::wfs", "rec_id": "hv2-cyclen-odd_k4-i5", "axis": "cyclen", "difficulty": "odd_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition g1_1 is true. Proposition g1_2 is true. Proposition p3 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g3_1 is true. Proposition g3_0 is true. Proposition p2 is true if proposition s1 is true and proposition s3 is true. Proposition x1 is true if proposition x2 is not true. Proposition x2 is true if proposition x3 is not true. Proposition g1_0 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true. Proposition t2 is true if proposition t3 is true and proposition wide is true. Proposition t5 is true if proposition t6 is true. Proposition g2_0 is true. Proposition x3 is true if proposition x0 is not true. Proposition s0 is true if proposition g0_0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition g2_1 is true. Proposition q is true if proposition t0 is true. Proposition g0_0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true and proposition p3 is true. Proposition btrue is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t1 is true if proposition t2 is true and proposition cq is true. Proposition t0 is true if proposition t1 is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t7 is true if proposition btrue is true. Proposition t3 is true if proposition t4 is true. Proposition x0 is true if proposition x1 is not true. Proposition t6 is true if proposition t7 is true. Proposition cq is true if proposition x0 is true. Proposition t4 is true if proposition t5 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k4-i6-none::none", "rec_id": "hv2-cyclen-odd_k4-i6", "axis": "cyclen", "difficulty": "odd_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition t1 is true if proposition t2 is true and proposition cq is true. Proposition t7 is true if proposition btrue is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition t6 is true if proposition t7 is true and proposition wide is true. Proposition g1_1 is true. Proposition cq is true if proposition x0 is true. Proposition g3_0 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition x0 is true if proposition x1 is not true. Proposition g2_1 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition t2 is true if proposition t3 is true. Proposition g3_1 is true. Proposition t4 is true if proposition t5 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t5 is true if proposition t6 is true. Proposition btrue is true. Proposition x1 is true if proposition x2 is not true. Proposition q is true if proposition t0 is true. Proposition g0_1 is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition x3 is true if proposition x0 is not true. Proposition t0 is true if proposition t1 is true. Proposition g2_0 is true. Proposition g0_0 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition t3 is true if proposition t4 is true. Proposition g1_0 is true. Proposition x2 is true if proposition x3 is not true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k4-i6-closed_world::closed_world", "rec_id": "hv2-cyclen-odd_k4-i6", "axis": "cyclen", "difficulty": "odd_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition t1 is true if proposition t2 is true and proposition cq is true. Proposition t7 is true if proposition btrue is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition t6 is true if proposition t7 is true and proposition wide is true. Proposition g1_1 is true. Proposition cq is true if proposition x0 is true. Proposition g3_0 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition x0 is true if proposition x1 is not true. Proposition g2_1 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition t2 is true if proposition t3 is true. Proposition g3_1 is true. Proposition t4 is true if proposition t5 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t5 is true if proposition t6 is true. Proposition btrue is true. Proposition x1 is true if proposition x2 is not true. Proposition q is true if proposition t0 is true. Proposition g0_1 is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition x3 is true if proposition x0 is not true. Proposition t0 is true if proposition t1 is true. Proposition g2_0 is true. Proposition g0_0 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition t3 is true if proposition t4 is true. Proposition g1_0 is true. Proposition x2 is true if proposition x3 is not true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k4-i6-cred::cred", "rec_id": "hv2-cyclen-odd_k4-i6", "axis": "cyclen", "difficulty": "odd_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition t1 is true if proposition t2 is true and proposition cq is true. Proposition t7 is true if proposition btrue is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition t6 is true if proposition t7 is true and proposition wide is true. Proposition g1_1 is true. Proposition cq is true if proposition x0 is true. Proposition g3_0 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition x0 is true if proposition x1 is not true. Proposition g2_1 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition t2 is true if proposition t3 is true. Proposition g3_1 is true. Proposition t4 is true if proposition t5 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t5 is true if proposition t6 is true. Proposition btrue is true. Proposition x1 is true if proposition x2 is not true. Proposition q is true if proposition t0 is true. Proposition g0_1 is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition x3 is true if proposition x0 is not true. Proposition t0 is true if proposition t1 is true. Proposition g2_0 is true. Proposition g0_0 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition t3 is true if proposition t4 is true. Proposition g1_0 is true. Proposition x2 is true if proposition x3 is not true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k4-i6-skept::skept", "rec_id": "hv2-cyclen-odd_k4-i6", "axis": "cyclen", "difficulty": "odd_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition t1 is true if proposition t2 is true and proposition cq is true. Proposition t7 is true if proposition btrue is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition t6 is true if proposition t7 is true and proposition wide is true. Proposition g1_1 is true. Proposition cq is true if proposition x0 is true. Proposition g3_0 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition x0 is true if proposition x1 is not true. Proposition g2_1 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition t2 is true if proposition t3 is true. Proposition g3_1 is true. Proposition t4 is true if proposition t5 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t5 is true if proposition t6 is true. Proposition btrue is true. Proposition x1 is true if proposition x2 is not true. Proposition q is true if proposition t0 is true. Proposition g0_1 is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition x3 is true if proposition x0 is not true. Proposition t0 is true if proposition t1 is true. Proposition g2_0 is true. Proposition g0_0 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition t3 is true if proposition t4 is true. Proposition g1_0 is true. Proposition x2 is true if proposition x3 is not true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k4-i6-wfs::wfs", "rec_id": "hv2-cyclen-odd_k4-i6", "axis": "cyclen", "difficulty": "odd_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition t1 is true if proposition t2 is true and proposition cq is true. Proposition t7 is true if proposition btrue is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition t6 is true if proposition t7 is true and proposition wide is true. Proposition g1_1 is true. Proposition cq is true if proposition x0 is true. Proposition g3_0 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition x0 is true if proposition x1 is not true. Proposition g2_1 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition t2 is true if proposition t3 is true. Proposition g3_1 is true. Proposition t4 is true if proposition t5 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t5 is true if proposition t6 is true. Proposition btrue is true. Proposition x1 is true if proposition x2 is not true. Proposition q is true if proposition t0 is true. Proposition g0_1 is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition x3 is true if proposition x0 is not true. Proposition t0 is true if proposition t1 is true. Proposition g2_0 is true. Proposition g0_0 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition t3 is true if proposition t4 is true. Proposition g1_0 is true. Proposition x2 is true if proposition x3 is not true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k4-i7-none::none", "rec_id": "hv2-cyclen-odd_k4-i7", "axis": "cyclen", "difficulty": "odd_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition g0_0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true. Proposition s2 is true if proposition g2_0 is true. Proposition g1_1 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition etrue is true. Proposition x3 is true if proposition etrue is true and proposition x0 is not true. Proposition g3_1 is true. Proposition t3 is true if proposition t4 is true and proposition wide is true. Proposition t6 is true if proposition t7 is true. Proposition p1 is true if proposition s2 is true and proposition s3 is true. Proposition t2 is true if proposition t3 is true. Proposition g0_1 is true. Proposition t4 is true if proposition t5 is true. Proposition cq is true if proposition x0 is true. Proposition t7 is true if proposition btrue is true and proposition cq is true. Proposition g3_2 is true. Proposition g3_0 is true. Proposition t1 is true if proposition t2 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition t0 is true if proposition t1 is true. Proposition t5 is true if proposition t6 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true and proposition g3_2 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition q is true if proposition t0 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition g1_0 is true. Proposition g2_0 is true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s3 is true. Proposition btrue is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k4-i7-closed_world::closed_world", "rec_id": "hv2-cyclen-odd_k4-i7", "axis": "cyclen", "difficulty": "odd_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition g0_0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true. Proposition s2 is true if proposition g2_0 is true. Proposition g1_1 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition etrue is true. Proposition x3 is true if proposition etrue is true and proposition x0 is not true. Proposition g3_1 is true. Proposition t3 is true if proposition t4 is true and proposition wide is true. Proposition t6 is true if proposition t7 is true. Proposition p1 is true if proposition s2 is true and proposition s3 is true. Proposition t2 is true if proposition t3 is true. Proposition g0_1 is true. Proposition t4 is true if proposition t5 is true. Proposition cq is true if proposition x0 is true. Proposition t7 is true if proposition btrue is true and proposition cq is true. Proposition g3_2 is true. Proposition g3_0 is true. Proposition t1 is true if proposition t2 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition t0 is true if proposition t1 is true. Proposition t5 is true if proposition t6 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true and proposition g3_2 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition q is true if proposition t0 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition g1_0 is true. Proposition g2_0 is true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s3 is true. Proposition btrue is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k4-i7-cred::cred", "rec_id": "hv2-cyclen-odd_k4-i7", "axis": "cyclen", "difficulty": "odd_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition g0_0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true. Proposition s2 is true if proposition g2_0 is true. Proposition g1_1 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition etrue is true. Proposition x3 is true if proposition etrue is true and proposition x0 is not true. Proposition g3_1 is true. Proposition t3 is true if proposition t4 is true and proposition wide is true. Proposition t6 is true if proposition t7 is true. Proposition p1 is true if proposition s2 is true and proposition s3 is true. Proposition t2 is true if proposition t3 is true. Proposition g0_1 is true. Proposition t4 is true if proposition t5 is true. Proposition cq is true if proposition x0 is true. Proposition t7 is true if proposition btrue is true and proposition cq is true. Proposition g3_2 is true. Proposition g3_0 is true. Proposition t1 is true if proposition t2 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition t0 is true if proposition t1 is true. Proposition t5 is true if proposition t6 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true and proposition g3_2 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition q is true if proposition t0 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition g1_0 is true. Proposition g2_0 is true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s3 is true. Proposition btrue is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k4-i7-skept::skept", "rec_id": "hv2-cyclen-odd_k4-i7", "axis": "cyclen", "difficulty": "odd_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition g0_0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true. Proposition s2 is true if proposition g2_0 is true. Proposition g1_1 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition etrue is true. Proposition x3 is true if proposition etrue is true and proposition x0 is not true. Proposition g3_1 is true. Proposition t3 is true if proposition t4 is true and proposition wide is true. Proposition t6 is true if proposition t7 is true. Proposition p1 is true if proposition s2 is true and proposition s3 is true. Proposition t2 is true if proposition t3 is true. Proposition g0_1 is true. Proposition t4 is true if proposition t5 is true. Proposition cq is true if proposition x0 is true. Proposition t7 is true if proposition btrue is true and proposition cq is true. Proposition g3_2 is true. Proposition g3_0 is true. Proposition t1 is true if proposition t2 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition t0 is true if proposition t1 is true. Proposition t5 is true if proposition t6 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true and proposition g3_2 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition q is true if proposition t0 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition g1_0 is true. Proposition g2_0 is true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s3 is true. Proposition btrue is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k4-i7-wfs::wfs", "rec_id": "hv2-cyclen-odd_k4-i7", "axis": "cyclen", "difficulty": "odd_k4", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition g0_0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true. Proposition s2 is true if proposition g2_0 is true. Proposition g1_1 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition etrue is true. Proposition x3 is true if proposition etrue is true and proposition x0 is not true. Proposition g3_1 is true. Proposition t3 is true if proposition t4 is true and proposition wide is true. Proposition t6 is true if proposition t7 is true. Proposition p1 is true if proposition s2 is true and proposition s3 is true. Proposition t2 is true if proposition t3 is true. Proposition g0_1 is true. Proposition t4 is true if proposition t5 is true. Proposition cq is true if proposition x0 is true. Proposition t7 is true if proposition btrue is true and proposition cq is true. Proposition g3_2 is true. Proposition g3_0 is true. Proposition t1 is true if proposition t2 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition t0 is true if proposition t1 is true. Proposition t5 is true if proposition t6 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true and proposition g3_2 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition q is true if proposition t0 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition g1_0 is true. Proposition g2_0 is true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s3 is true. Proposition btrue is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k8-i0-none::none", "rec_id": "hv2-cyclen-even_one_sided_k8-i0", "axis": "cyclen", "difficulty": "even_one_sided_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition g1_1 is true. Proposition t3 is true if proposition t4 is true. Proposition g2_1 is true. Proposition x7 is true if proposition x0 is not true. Proposition x4 is true if proposition x5 is not true. Proposition s3 is true if proposition g3_0 is true. Proposition t1 is true if proposition t2 is true. Proposition g3_0 is true. Proposition p0 is true if proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition btrue is true. Proposition t6 is true if proposition t7 is true. Proposition g0_0 is true. Proposition x5 is true if proposition x6 is not true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition x3 is true if proposition x4 is not true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s3 is true. Proposition q is true if proposition t0 is true. Proposition t2 is true if proposition t3 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true. Proposition x2 is true if proposition x3 is not true. Proposition x0 is true if proposition x1 is not true. Proposition t5 is true if proposition t6 is true and proposition cq is true. Proposition x1 is true if proposition x2 is not true. Proposition cq is true if proposition x0 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true and proposition g0_2 is true. Proposition g0_1 is true. Proposition g0_2 is true. Proposition g1_0 is true. Proposition t7 is true if proposition btrue is true and proposition wide is true. Proposition t4 is true if proposition t5 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition g2_0 is true. Proposition t0 is true if proposition t1 is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition x6 is true if proposition x7 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k8-i0-closed_world::closed_world", "rec_id": "hv2-cyclen-even_one_sided_k8-i0", "axis": "cyclen", "difficulty": "even_one_sided_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition g1_1 is true. Proposition t3 is true if proposition t4 is true. Proposition g2_1 is true. Proposition x7 is true if proposition x0 is not true. Proposition x4 is true if proposition x5 is not true. Proposition s3 is true if proposition g3_0 is true. Proposition t1 is true if proposition t2 is true. Proposition g3_0 is true. Proposition p0 is true if proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition btrue is true. Proposition t6 is true if proposition t7 is true. Proposition g0_0 is true. Proposition x5 is true if proposition x6 is not true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition x3 is true if proposition x4 is not true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s3 is true. Proposition q is true if proposition t0 is true. Proposition t2 is true if proposition t3 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true. Proposition x2 is true if proposition x3 is not true. Proposition x0 is true if proposition x1 is not true. Proposition t5 is true if proposition t6 is true and proposition cq is true. Proposition x1 is true if proposition x2 is not true. Proposition cq is true if proposition x0 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true and proposition g0_2 is true. Proposition g0_1 is true. Proposition g0_2 is true. Proposition g1_0 is true. Proposition t7 is true if proposition btrue is true and proposition wide is true. Proposition t4 is true if proposition t5 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition g2_0 is true. Proposition t0 is true if proposition t1 is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition x6 is true if proposition x7 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k8-i0-cred::cred", "rec_id": "hv2-cyclen-even_one_sided_k8-i0", "axis": "cyclen", "difficulty": "even_one_sided_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition g1_1 is true. Proposition t3 is true if proposition t4 is true. Proposition g2_1 is true. Proposition x7 is true if proposition x0 is not true. Proposition x4 is true if proposition x5 is not true. Proposition s3 is true if proposition g3_0 is true. Proposition t1 is true if proposition t2 is true. Proposition g3_0 is true. Proposition p0 is true if proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition btrue is true. Proposition t6 is true if proposition t7 is true. Proposition g0_0 is true. Proposition x5 is true if proposition x6 is not true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition x3 is true if proposition x4 is not true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s3 is true. Proposition q is true if proposition t0 is true. Proposition t2 is true if proposition t3 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true. Proposition x2 is true if proposition x3 is not true. Proposition x0 is true if proposition x1 is not true. Proposition t5 is true if proposition t6 is true and proposition cq is true. Proposition x1 is true if proposition x2 is not true. Proposition cq is true if proposition x0 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true and proposition g0_2 is true. Proposition g0_1 is true. Proposition g0_2 is true. Proposition g1_0 is true. Proposition t7 is true if proposition btrue is true and proposition wide is true. Proposition t4 is true if proposition t5 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition g2_0 is true. Proposition t0 is true if proposition t1 is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition x6 is true if proposition x7 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k8-i0-skept::skept", "rec_id": "hv2-cyclen-even_one_sided_k8-i0", "axis": "cyclen", "difficulty": "even_one_sided_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition g1_1 is true. Proposition t3 is true if proposition t4 is true. Proposition g2_1 is true. Proposition x7 is true if proposition x0 is not true. Proposition x4 is true if proposition x5 is not true. Proposition s3 is true if proposition g3_0 is true. Proposition t1 is true if proposition t2 is true. Proposition g3_0 is true. Proposition p0 is true if proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition btrue is true. Proposition t6 is true if proposition t7 is true. Proposition g0_0 is true. Proposition x5 is true if proposition x6 is not true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition x3 is true if proposition x4 is not true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s3 is true. Proposition q is true if proposition t0 is true. Proposition t2 is true if proposition t3 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true. Proposition x2 is true if proposition x3 is not true. Proposition x0 is true if proposition x1 is not true. Proposition t5 is true if proposition t6 is true and proposition cq is true. Proposition x1 is true if proposition x2 is not true. Proposition cq is true if proposition x0 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true and proposition g0_2 is true. Proposition g0_1 is true. Proposition g0_2 is true. Proposition g1_0 is true. Proposition t7 is true if proposition btrue is true and proposition wide is true. Proposition t4 is true if proposition t5 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition g2_0 is true. Proposition t0 is true if proposition t1 is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition x6 is true if proposition x7 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k8-i0-wfs::wfs", "rec_id": "hv2-cyclen-even_one_sided_k8-i0", "axis": "cyclen", "difficulty": "even_one_sided_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition g1_1 is true. Proposition t3 is true if proposition t4 is true. Proposition g2_1 is true. Proposition x7 is true if proposition x0 is not true. Proposition x4 is true if proposition x5 is not true. Proposition s3 is true if proposition g3_0 is true. Proposition t1 is true if proposition t2 is true. Proposition g3_0 is true. Proposition p0 is true if proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition btrue is true. Proposition t6 is true if proposition t7 is true. Proposition g0_0 is true. Proposition x5 is true if proposition x6 is not true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition x3 is true if proposition x4 is not true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s3 is true. Proposition q is true if proposition t0 is true. Proposition t2 is true if proposition t3 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true. Proposition x2 is true if proposition x3 is not true. Proposition x0 is true if proposition x1 is not true. Proposition t5 is true if proposition t6 is true and proposition cq is true. Proposition x1 is true if proposition x2 is not true. Proposition cq is true if proposition x0 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true and proposition g0_2 is true. Proposition g0_1 is true. Proposition g0_2 is true. Proposition g1_0 is true. Proposition t7 is true if proposition btrue is true and proposition wide is true. Proposition t4 is true if proposition t5 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition g2_0 is true. Proposition t0 is true if proposition t1 is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition x6 is true if proposition x7 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k8-i1-none::none", "rec_id": "hv2-cyclen-even_one_sided_k8-i1", "axis": "cyclen", "difficulty": "even_one_sided_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition btrue is true. Proposition x7 is true if proposition etrue is true and proposition x0 is not true. Proposition g2_1 is true. Proposition cq is true if proposition x0 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition x4 is true if proposition etrue is true and proposition x5 is not true. Proposition g3_0 is true. Proposition t5 is true if proposition t6 is true and proposition wide is true. Proposition q is true if proposition t0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition x3 is true if proposition etrue is true and proposition x4 is not true. Proposition g2_0 is true. Proposition t2 is true if proposition t3 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t0 is true if proposition t1 is true. Proposition g1_1 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition g0_1 is true. Proposition x6 is true if proposition etrue is true and proposition x7 is not true. Proposition g1_0 is true. Proposition t4 is true if proposition t5 is true. Proposition g3_1 is true. Proposition g0_0 is true. Proposition x5 is true if proposition etrue is true and proposition x6 is not true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t1 is true if proposition t2 is true and proposition cq is true. Proposition t3 is true if proposition t4 is true. Proposition etrue is true. Proposition t6 is true if proposition t7 is true. Proposition t7 is true if proposition btrue is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k8-i1-closed_world::closed_world", "rec_id": "hv2-cyclen-even_one_sided_k8-i1", "axis": "cyclen", "difficulty": "even_one_sided_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition btrue is true. Proposition x7 is true if proposition etrue is true and proposition x0 is not true. Proposition g2_1 is true. Proposition cq is true if proposition x0 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition x4 is true if proposition etrue is true and proposition x5 is not true. Proposition g3_0 is true. Proposition t5 is true if proposition t6 is true and proposition wide is true. Proposition q is true if proposition t0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition x3 is true if proposition etrue is true and proposition x4 is not true. Proposition g2_0 is true. Proposition t2 is true if proposition t3 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t0 is true if proposition t1 is true. Proposition g1_1 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition g0_1 is true. Proposition x6 is true if proposition etrue is true and proposition x7 is not true. Proposition g1_0 is true. Proposition t4 is true if proposition t5 is true. Proposition g3_1 is true. Proposition g0_0 is true. Proposition x5 is true if proposition etrue is true and proposition x6 is not true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t1 is true if proposition t2 is true and proposition cq is true. Proposition t3 is true if proposition t4 is true. Proposition etrue is true. Proposition t6 is true if proposition t7 is true. Proposition t7 is true if proposition btrue is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k8-i1-cred::cred", "rec_id": "hv2-cyclen-even_one_sided_k8-i1", "axis": "cyclen", "difficulty": "even_one_sided_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition btrue is true. Proposition x7 is true if proposition etrue is true and proposition x0 is not true. Proposition g2_1 is true. Proposition cq is true if proposition x0 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition x4 is true if proposition etrue is true and proposition x5 is not true. Proposition g3_0 is true. Proposition t5 is true if proposition t6 is true and proposition wide is true. Proposition q is true if proposition t0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition x3 is true if proposition etrue is true and proposition x4 is not true. Proposition g2_0 is true. Proposition t2 is true if proposition t3 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t0 is true if proposition t1 is true. Proposition g1_1 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition g0_1 is true. Proposition x6 is true if proposition etrue is true and proposition x7 is not true. Proposition g1_0 is true. Proposition t4 is true if proposition t5 is true. Proposition g3_1 is true. Proposition g0_0 is true. Proposition x5 is true if proposition etrue is true and proposition x6 is not true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t1 is true if proposition t2 is true and proposition cq is true. Proposition t3 is true if proposition t4 is true. Proposition etrue is true. Proposition t6 is true if proposition t7 is true. Proposition t7 is true if proposition btrue is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k8-i1-skept::skept", "rec_id": "hv2-cyclen-even_one_sided_k8-i1", "axis": "cyclen", "difficulty": "even_one_sided_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition btrue is true. Proposition x7 is true if proposition etrue is true and proposition x0 is not true. Proposition g2_1 is true. Proposition cq is true if proposition x0 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition x4 is true if proposition etrue is true and proposition x5 is not true. Proposition g3_0 is true. Proposition t5 is true if proposition t6 is true and proposition wide is true. Proposition q is true if proposition t0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition x3 is true if proposition etrue is true and proposition x4 is not true. Proposition g2_0 is true. Proposition t2 is true if proposition t3 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t0 is true if proposition t1 is true. Proposition g1_1 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition g0_1 is true. Proposition x6 is true if proposition etrue is true and proposition x7 is not true. Proposition g1_0 is true. Proposition t4 is true if proposition t5 is true. Proposition g3_1 is true. Proposition g0_0 is true. Proposition x5 is true if proposition etrue is true and proposition x6 is not true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t1 is true if proposition t2 is true and proposition cq is true. Proposition t3 is true if proposition t4 is true. Proposition etrue is true. Proposition t6 is true if proposition t7 is true. Proposition t7 is true if proposition btrue is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k8-i1-wfs::wfs", "rec_id": "hv2-cyclen-even_one_sided_k8-i1", "axis": "cyclen", "difficulty": "even_one_sided_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition btrue is true. Proposition x7 is true if proposition etrue is true and proposition x0 is not true. Proposition g2_1 is true. Proposition cq is true if proposition x0 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition x4 is true if proposition etrue is true and proposition x5 is not true. Proposition g3_0 is true. Proposition t5 is true if proposition t6 is true and proposition wide is true. Proposition q is true if proposition t0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition x3 is true if proposition etrue is true and proposition x4 is not true. Proposition g2_0 is true. Proposition t2 is true if proposition t3 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t0 is true if proposition t1 is true. Proposition g1_1 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition g0_1 is true. Proposition x6 is true if proposition etrue is true and proposition x7 is not true. Proposition g1_0 is true. Proposition t4 is true if proposition t5 is true. Proposition g3_1 is true. Proposition g0_0 is true. Proposition x5 is true if proposition etrue is true and proposition x6 is not true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t1 is true if proposition t2 is true and proposition cq is true. Proposition t3 is true if proposition t4 is true. Proposition etrue is true. Proposition t6 is true if proposition t7 is true. Proposition t7 is true if proposition btrue is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k8-i2-none::none", "rec_id": "hv2-cyclen-even_one_sided_k8-i2", "axis": "cyclen", "difficulty": "even_one_sided_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition x3 is true if proposition x4 is not true. Proposition g3_1 is true. Proposition x1 is true if proposition x2 is not true. Proposition g0_1 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true and proposition g0_2 is true. Proposition g2_0 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g2_1 is true. Proposition g0_0 is true. Proposition x6 is true if proposition x7 is not true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition x2 is true if proposition x3 is not true. Proposition g0_2 is true. Proposition t7 is true if proposition btrue is true and proposition wide is true. Proposition cq is true if proposition x0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t0 is true if proposition t1 is true. Proposition t3 is true if proposition t4 is true. Proposition x4 is true if proposition x5 is not true. Proposition g3_0 is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition q is true if proposition t0 is true. Proposition x0 is true if proposition x1 is not true. Proposition t6 is true if proposition t7 is true. Proposition x7 is true if proposition x0 is not true. Proposition t5 is true if proposition t6 is true and proposition cq is true. Proposition s1 is true if proposition g1_0 is true. Proposition x5 is true if proposition x6 is not true. Proposition g1_0 is true. Proposition t4 is true if proposition t5 is true. Proposition t2 is true if proposition t3 is true. Proposition t1 is true if proposition t2 is true. Proposition btrue is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k8-i2-closed_world::closed_world", "rec_id": "hv2-cyclen-even_one_sided_k8-i2", "axis": "cyclen", "difficulty": "even_one_sided_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition x3 is true if proposition x4 is not true. Proposition g3_1 is true. Proposition x1 is true if proposition x2 is not true. Proposition g0_1 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true and proposition g0_2 is true. Proposition g2_0 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g2_1 is true. Proposition g0_0 is true. Proposition x6 is true if proposition x7 is not true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition x2 is true if proposition x3 is not true. Proposition g0_2 is true. Proposition t7 is true if proposition btrue is true and proposition wide is true. Proposition cq is true if proposition x0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t0 is true if proposition t1 is true. Proposition t3 is true if proposition t4 is true. Proposition x4 is true if proposition x5 is not true. Proposition g3_0 is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition q is true if proposition t0 is true. Proposition x0 is true if proposition x1 is not true. Proposition t6 is true if proposition t7 is true. Proposition x7 is true if proposition x0 is not true. Proposition t5 is true if proposition t6 is true and proposition cq is true. Proposition s1 is true if proposition g1_0 is true. Proposition x5 is true if proposition x6 is not true. Proposition g1_0 is true. Proposition t4 is true if proposition t5 is true. Proposition t2 is true if proposition t3 is true. Proposition t1 is true if proposition t2 is true. Proposition btrue is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k8-i2-cred::cred", "rec_id": "hv2-cyclen-even_one_sided_k8-i2", "axis": "cyclen", "difficulty": "even_one_sided_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition x3 is true if proposition x4 is not true. Proposition g3_1 is true. Proposition x1 is true if proposition x2 is not true. Proposition g0_1 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true and proposition g0_2 is true. Proposition g2_0 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g2_1 is true. Proposition g0_0 is true. Proposition x6 is true if proposition x7 is not true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition x2 is true if proposition x3 is not true. Proposition g0_2 is true. Proposition t7 is true if proposition btrue is true and proposition wide is true. Proposition cq is true if proposition x0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t0 is true if proposition t1 is true. Proposition t3 is true if proposition t4 is true. Proposition x4 is true if proposition x5 is not true. Proposition g3_0 is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition q is true if proposition t0 is true. Proposition x0 is true if proposition x1 is not true. Proposition t6 is true if proposition t7 is true. Proposition x7 is true if proposition x0 is not true. Proposition t5 is true if proposition t6 is true and proposition cq is true. Proposition s1 is true if proposition g1_0 is true. Proposition x5 is true if proposition x6 is not true. Proposition g1_0 is true. Proposition t4 is true if proposition t5 is true. Proposition t2 is true if proposition t3 is true. Proposition t1 is true if proposition t2 is true. Proposition btrue is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k8-i2-skept::skept", "rec_id": "hv2-cyclen-even_one_sided_k8-i2", "axis": "cyclen", "difficulty": "even_one_sided_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition x3 is true if proposition x4 is not true. Proposition g3_1 is true. Proposition x1 is true if proposition x2 is not true. Proposition g0_1 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true and proposition g0_2 is true. Proposition g2_0 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g2_1 is true. Proposition g0_0 is true. Proposition x6 is true if proposition x7 is not true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition x2 is true if proposition x3 is not true. Proposition g0_2 is true. Proposition t7 is true if proposition btrue is true and proposition wide is true. Proposition cq is true if proposition x0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t0 is true if proposition t1 is true. Proposition t3 is true if proposition t4 is true. Proposition x4 is true if proposition x5 is not true. Proposition g3_0 is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition q is true if proposition t0 is true. Proposition x0 is true if proposition x1 is not true. Proposition t6 is true if proposition t7 is true. Proposition x7 is true if proposition x0 is not true. Proposition t5 is true if proposition t6 is true and proposition cq is true. Proposition s1 is true if proposition g1_0 is true. Proposition x5 is true if proposition x6 is not true. Proposition g1_0 is true. Proposition t4 is true if proposition t5 is true. Proposition t2 is true if proposition t3 is true. Proposition t1 is true if proposition t2 is true. Proposition btrue is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k8-i2-wfs::wfs", "rec_id": "hv2-cyclen-even_one_sided_k8-i2", "axis": "cyclen", "difficulty": "even_one_sided_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition x3 is true if proposition x4 is not true. Proposition g3_1 is true. Proposition x1 is true if proposition x2 is not true. Proposition g0_1 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true and proposition g0_2 is true. Proposition g2_0 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g2_1 is true. Proposition g0_0 is true. Proposition x6 is true if proposition x7 is not true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition x2 is true if proposition x3 is not true. Proposition g0_2 is true. Proposition t7 is true if proposition btrue is true and proposition wide is true. Proposition cq is true if proposition x0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t0 is true if proposition t1 is true. Proposition t3 is true if proposition t4 is true. Proposition x4 is true if proposition x5 is not true. Proposition g3_0 is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition q is true if proposition t0 is true. Proposition x0 is true if proposition x1 is not true. Proposition t6 is true if proposition t7 is true. Proposition x7 is true if proposition x0 is not true. Proposition t5 is true if proposition t6 is true and proposition cq is true. Proposition s1 is true if proposition g1_0 is true. Proposition x5 is true if proposition x6 is not true. Proposition g1_0 is true. Proposition t4 is true if proposition t5 is true. Proposition t2 is true if proposition t3 is true. Proposition t1 is true if proposition t2 is true. Proposition btrue is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k8-i3-none::none", "rec_id": "hv2-cyclen-even_one_sided_k8-i3", "axis": "cyclen", "difficulty": "even_one_sided_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition t2 is true if proposition t3 is true. Proposition etrue is true. Proposition x4 is true if proposition etrue is true and proposition x5 is not true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true and proposition g1_3 is true. Proposition x6 is true if proposition etrue is true and proposition x7 is not true. Proposition t5 is true if proposition t6 is true. Proposition cq is true if proposition x0 is true. Proposition x7 is true if proposition etrue is true and proposition x0 is not true. Proposition t0 is true if proposition t1 is true and proposition wide is true. Proposition g3_0 is true. Proposition q is true if proposition t0 is true. Proposition x5 is true if proposition etrue is true and proposition x6 is not true. Proposition t6 is true if proposition t7 is true and proposition cq is true. Proposition s0 is true if proposition g0_0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition t1 is true if proposition t2 is true. Proposition g1_3 is true. Proposition t3 is true if proposition t4 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition p1 is true if proposition s3 is true. Proposition t7 is true if proposition btrue is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true and proposition p3 is true. Proposition g1_1 is true. Proposition btrue is true. Proposition p3 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g3_1 is true. Proposition g2_0 is true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s2 is true if proposition g2_0 is true. Proposition t4 is true if proposition t5 is true. Proposition g0_0 is true. Proposition g1_0 is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true. Proposition g1_2 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition p0 is true if proposition s1 is true and proposition s2 is true. Proposition x3 is true if proposition etrue is true and proposition x4 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k8-i3-closed_world::closed_world", "rec_id": "hv2-cyclen-even_one_sided_k8-i3", "axis": "cyclen", "difficulty": "even_one_sided_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition t2 is true if proposition t3 is true. Proposition etrue is true. Proposition x4 is true if proposition etrue is true and proposition x5 is not true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true and proposition g1_3 is true. Proposition x6 is true if proposition etrue is true and proposition x7 is not true. Proposition t5 is true if proposition t6 is true. Proposition cq is true if proposition x0 is true. Proposition x7 is true if proposition etrue is true and proposition x0 is not true. Proposition t0 is true if proposition t1 is true and proposition wide is true. Proposition g3_0 is true. Proposition q is true if proposition t0 is true. Proposition x5 is true if proposition etrue is true and proposition x6 is not true. Proposition t6 is true if proposition t7 is true and proposition cq is true. Proposition s0 is true if proposition g0_0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition t1 is true if proposition t2 is true. Proposition g1_3 is true. Proposition t3 is true if proposition t4 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition p1 is true if proposition s3 is true. Proposition t7 is true if proposition btrue is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true and proposition p3 is true. Proposition g1_1 is true. Proposition btrue is true. Proposition p3 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g3_1 is true. Proposition g2_0 is true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s2 is true if proposition g2_0 is true. Proposition t4 is true if proposition t5 is true. Proposition g0_0 is true. Proposition g1_0 is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true. Proposition g1_2 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition p0 is true if proposition s1 is true and proposition s2 is true. Proposition x3 is true if proposition etrue is true and proposition x4 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k8-i3-cred::cred", "rec_id": "hv2-cyclen-even_one_sided_k8-i3", "axis": "cyclen", "difficulty": "even_one_sided_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition t2 is true if proposition t3 is true. Proposition etrue is true. Proposition x4 is true if proposition etrue is true and proposition x5 is not true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true and proposition g1_3 is true. Proposition x6 is true if proposition etrue is true and proposition x7 is not true. Proposition t5 is true if proposition t6 is true. Proposition cq is true if proposition x0 is true. Proposition x7 is true if proposition etrue is true and proposition x0 is not true. Proposition t0 is true if proposition t1 is true and proposition wide is true. Proposition g3_0 is true. Proposition q is true if proposition t0 is true. Proposition x5 is true if proposition etrue is true and proposition x6 is not true. Proposition t6 is true if proposition t7 is true and proposition cq is true. Proposition s0 is true if proposition g0_0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition t1 is true if proposition t2 is true. Proposition g1_3 is true. Proposition t3 is true if proposition t4 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition p1 is true if proposition s3 is true. Proposition t7 is true if proposition btrue is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true and proposition p3 is true. Proposition g1_1 is true. Proposition btrue is true. Proposition p3 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g3_1 is true. Proposition g2_0 is true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s2 is true if proposition g2_0 is true. Proposition t4 is true if proposition t5 is true. Proposition g0_0 is true. Proposition g1_0 is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true. Proposition g1_2 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition p0 is true if proposition s1 is true and proposition s2 is true. Proposition x3 is true if proposition etrue is true and proposition x4 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k8-i3-skept::skept", "rec_id": "hv2-cyclen-even_one_sided_k8-i3", "axis": "cyclen", "difficulty": "even_one_sided_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition t2 is true if proposition t3 is true. Proposition etrue is true. Proposition x4 is true if proposition etrue is true and proposition x5 is not true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true and proposition g1_3 is true. Proposition x6 is true if proposition etrue is true and proposition x7 is not true. Proposition t5 is true if proposition t6 is true. Proposition cq is true if proposition x0 is true. Proposition x7 is true if proposition etrue is true and proposition x0 is not true. Proposition t0 is true if proposition t1 is true and proposition wide is true. Proposition g3_0 is true. Proposition q is true if proposition t0 is true. Proposition x5 is true if proposition etrue is true and proposition x6 is not true. Proposition t6 is true if proposition t7 is true and proposition cq is true. Proposition s0 is true if proposition g0_0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition t1 is true if proposition t2 is true. Proposition g1_3 is true. Proposition t3 is true if proposition t4 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition p1 is true if proposition s3 is true. Proposition t7 is true if proposition btrue is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true and proposition p3 is true. Proposition g1_1 is true. Proposition btrue is true. Proposition p3 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g3_1 is true. Proposition g2_0 is true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s2 is true if proposition g2_0 is true. Proposition t4 is true if proposition t5 is true. Proposition g0_0 is true. Proposition g1_0 is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true. Proposition g1_2 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition p0 is true if proposition s1 is true and proposition s2 is true. Proposition x3 is true if proposition etrue is true and proposition x4 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k8-i3-wfs::wfs", "rec_id": "hv2-cyclen-even_one_sided_k8-i3", "axis": "cyclen", "difficulty": "even_one_sided_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition t2 is true if proposition t3 is true. Proposition etrue is true. Proposition x4 is true if proposition etrue is true and proposition x5 is not true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true and proposition g1_3 is true. Proposition x6 is true if proposition etrue is true and proposition x7 is not true. Proposition t5 is true if proposition t6 is true. Proposition cq is true if proposition x0 is true. Proposition x7 is true if proposition etrue is true and proposition x0 is not true. Proposition t0 is true if proposition t1 is true and proposition wide is true. Proposition g3_0 is true. Proposition q is true if proposition t0 is true. Proposition x5 is true if proposition etrue is true and proposition x6 is not true. Proposition t6 is true if proposition t7 is true and proposition cq is true. Proposition s0 is true if proposition g0_0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition t1 is true if proposition t2 is true. Proposition g1_3 is true. Proposition t3 is true if proposition t4 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition p1 is true if proposition s3 is true. Proposition t7 is true if proposition btrue is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true and proposition p3 is true. Proposition g1_1 is true. Proposition btrue is true. Proposition p3 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g3_1 is true. Proposition g2_0 is true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s2 is true if proposition g2_0 is true. Proposition t4 is true if proposition t5 is true. Proposition g0_0 is true. Proposition g1_0 is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true. Proposition g1_2 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition p0 is true if proposition s1 is true and proposition s2 is true. Proposition x3 is true if proposition etrue is true and proposition x4 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k8-i4-none::none", "rec_id": "hv2-cyclen-even_one_sided_k8-i4", "axis": "cyclen", "difficulty": "even_one_sided_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition t1 is true if proposition t2 is true. Proposition g1_2 is true. Proposition g0_1 is true. Proposition btrue is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition x5 is true if proposition etrue is true and proposition x6 is not true. Proposition t7 is true if proposition btrue is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition t3 is true if proposition t4 is true. Proposition q is true if proposition t0 is true. Proposition etrue is true. Proposition x3 is true if proposition etrue is true and proposition x4 is not true. Proposition t2 is true if proposition t3 is true. Proposition g0_0 is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true. Proposition x7 is true if proposition etrue is true and proposition x0 is not true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition s3 is true if proposition g3_0 is true. Proposition x4 is true if proposition etrue is true and proposition x5 is not true. Proposition x6 is true if proposition etrue is true and proposition x7 is not true. Proposition g1_0 is true. Proposition t6 is true if proposition t7 is true. Proposition g2_1 is true. Proposition t5 is true if proposition t6 is true. Proposition t4 is true if proposition t5 is true and proposition wide is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g3_0 is true. Proposition t0 is true if proposition t1 is true and proposition cq is true. Proposition g2_0 is true. Proposition cq is true if proposition x0 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition g1_1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k8-i4-closed_world::closed_world", "rec_id": "hv2-cyclen-even_one_sided_k8-i4", "axis": "cyclen", "difficulty": "even_one_sided_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition t1 is true if proposition t2 is true. Proposition g1_2 is true. Proposition g0_1 is true. Proposition btrue is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition x5 is true if proposition etrue is true and proposition x6 is not true. Proposition t7 is true if proposition btrue is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition t3 is true if proposition t4 is true. Proposition q is true if proposition t0 is true. Proposition etrue is true. Proposition x3 is true if proposition etrue is true and proposition x4 is not true. Proposition t2 is true if proposition t3 is true. Proposition g0_0 is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true. Proposition x7 is true if proposition etrue is true and proposition x0 is not true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition s3 is true if proposition g3_0 is true. Proposition x4 is true if proposition etrue is true and proposition x5 is not true. Proposition x6 is true if proposition etrue is true and proposition x7 is not true. Proposition g1_0 is true. Proposition t6 is true if proposition t7 is true. Proposition g2_1 is true. Proposition t5 is true if proposition t6 is true. Proposition t4 is true if proposition t5 is true and proposition wide is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g3_0 is true. Proposition t0 is true if proposition t1 is true and proposition cq is true. Proposition g2_0 is true. Proposition cq is true if proposition x0 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition g1_1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k8-i4-cred::cred", "rec_id": "hv2-cyclen-even_one_sided_k8-i4", "axis": "cyclen", "difficulty": "even_one_sided_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition t1 is true if proposition t2 is true. Proposition g1_2 is true. Proposition g0_1 is true. Proposition btrue is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition x5 is true if proposition etrue is true and proposition x6 is not true. Proposition t7 is true if proposition btrue is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition t3 is true if proposition t4 is true. Proposition q is true if proposition t0 is true. Proposition etrue is true. Proposition x3 is true if proposition etrue is true and proposition x4 is not true. Proposition t2 is true if proposition t3 is true. Proposition g0_0 is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true. Proposition x7 is true if proposition etrue is true and proposition x0 is not true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition s3 is true if proposition g3_0 is true. Proposition x4 is true if proposition etrue is true and proposition x5 is not true. Proposition x6 is true if proposition etrue is true and proposition x7 is not true. Proposition g1_0 is true. Proposition t6 is true if proposition t7 is true. Proposition g2_1 is true. Proposition t5 is true if proposition t6 is true. Proposition t4 is true if proposition t5 is true and proposition wide is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g3_0 is true. Proposition t0 is true if proposition t1 is true and proposition cq is true. Proposition g2_0 is true. Proposition cq is true if proposition x0 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition g1_1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k8-i4-skept::skept", "rec_id": "hv2-cyclen-even_one_sided_k8-i4", "axis": "cyclen", "difficulty": "even_one_sided_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition t1 is true if proposition t2 is true. Proposition g1_2 is true. Proposition g0_1 is true. Proposition btrue is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition x5 is true if proposition etrue is true and proposition x6 is not true. Proposition t7 is true if proposition btrue is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition t3 is true if proposition t4 is true. Proposition q is true if proposition t0 is true. Proposition etrue is true. Proposition x3 is true if proposition etrue is true and proposition x4 is not true. Proposition t2 is true if proposition t3 is true. Proposition g0_0 is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true. Proposition x7 is true if proposition etrue is true and proposition x0 is not true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition s3 is true if proposition g3_0 is true. Proposition x4 is true if proposition etrue is true and proposition x5 is not true. Proposition x6 is true if proposition etrue is true and proposition x7 is not true. Proposition g1_0 is true. Proposition t6 is true if proposition t7 is true. Proposition g2_1 is true. Proposition t5 is true if proposition t6 is true. Proposition t4 is true if proposition t5 is true and proposition wide is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g3_0 is true. Proposition t0 is true if proposition t1 is true and proposition cq is true. Proposition g2_0 is true. Proposition cq is true if proposition x0 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition g1_1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k8-i4-wfs::wfs", "rec_id": "hv2-cyclen-even_one_sided_k8-i4", "axis": "cyclen", "difficulty": "even_one_sided_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition t1 is true if proposition t2 is true. Proposition g1_2 is true. Proposition g0_1 is true. Proposition btrue is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition x5 is true if proposition etrue is true and proposition x6 is not true. Proposition t7 is true if proposition btrue is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition t3 is true if proposition t4 is true. Proposition q is true if proposition t0 is true. Proposition etrue is true. Proposition x3 is true if proposition etrue is true and proposition x4 is not true. Proposition t2 is true if proposition t3 is true. Proposition g0_0 is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true. Proposition x7 is true if proposition etrue is true and proposition x0 is not true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition s3 is true if proposition g3_0 is true. Proposition x4 is true if proposition etrue is true and proposition x5 is not true. Proposition x6 is true if proposition etrue is true and proposition x7 is not true. Proposition g1_0 is true. Proposition t6 is true if proposition t7 is true. Proposition g2_1 is true. Proposition t5 is true if proposition t6 is true. Proposition t4 is true if proposition t5 is true and proposition wide is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g3_0 is true. Proposition t0 is true if proposition t1 is true and proposition cq is true. Proposition g2_0 is true. Proposition cq is true if proposition x0 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition g1_1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k8-i5-none::none", "rec_id": "hv2-cyclen-even_one_sided_k8-i5", "axis": "cyclen", "difficulty": "even_one_sided_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition x3 is true if proposition x4 is not true. Proposition t5 is true if proposition t6 is true. Proposition g1_0 is true. Proposition g0_0 is true. Proposition x7 is true if proposition x0 is not true. Proposition s0 is true if proposition g0_0 is true. Proposition t1 is true if proposition t2 is true and proposition cq is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true and proposition p3 is true. Proposition x1 is true if proposition x2 is not true. Proposition x4 is true if proposition x5 is not true. Proposition p2 is true if proposition s1 is true and proposition s3 is true. Proposition x2 is true if proposition x3 is not true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition g1_2 is true. Proposition t3 is true if proposition t4 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition p3 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true. Proposition btrue is true. Proposition g3_0 is true. Proposition g2_0 is true. Proposition g2_1 is true. Proposition g1_1 is true. Proposition t7 is true if proposition btrue is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition q is true if proposition t0 is true. Proposition x5 is true if proposition x6 is not true. Proposition t2 is true if proposition t3 is true and proposition wide is true. Proposition g3_1 is true. Proposition x6 is true if proposition x7 is not true. Proposition t4 is true if proposition t5 is true. Proposition x0 is true if proposition x1 is not true. Proposition t6 is true if proposition t7 is true. Proposition cq is true if proposition x0 is true. Proposition t0 is true if proposition t1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k8-i5-closed_world::closed_world", "rec_id": "hv2-cyclen-even_one_sided_k8-i5", "axis": "cyclen", "difficulty": "even_one_sided_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition x3 is true if proposition x4 is not true. Proposition t5 is true if proposition t6 is true. Proposition g1_0 is true. Proposition g0_0 is true. Proposition x7 is true if proposition x0 is not true. Proposition s0 is true if proposition g0_0 is true. Proposition t1 is true if proposition t2 is true and proposition cq is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true and proposition p3 is true. Proposition x1 is true if proposition x2 is not true. Proposition x4 is true if proposition x5 is not true. Proposition p2 is true if proposition s1 is true and proposition s3 is true. Proposition x2 is true if proposition x3 is not true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition g1_2 is true. Proposition t3 is true if proposition t4 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition p3 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true. Proposition btrue is true. Proposition g3_0 is true. Proposition g2_0 is true. Proposition g2_1 is true. Proposition g1_1 is true. Proposition t7 is true if proposition btrue is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition q is true if proposition t0 is true. Proposition x5 is true if proposition x6 is not true. Proposition t2 is true if proposition t3 is true and proposition wide is true. Proposition g3_1 is true. Proposition x6 is true if proposition x7 is not true. Proposition t4 is true if proposition t5 is true. Proposition x0 is true if proposition x1 is not true. Proposition t6 is true if proposition t7 is true. Proposition cq is true if proposition x0 is true. Proposition t0 is true if proposition t1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k8-i5-cred::cred", "rec_id": "hv2-cyclen-even_one_sided_k8-i5", "axis": "cyclen", "difficulty": "even_one_sided_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition x3 is true if proposition x4 is not true. Proposition t5 is true if proposition t6 is true. Proposition g1_0 is true. Proposition g0_0 is true. Proposition x7 is true if proposition x0 is not true. Proposition s0 is true if proposition g0_0 is true. Proposition t1 is true if proposition t2 is true and proposition cq is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true and proposition p3 is true. Proposition x1 is true if proposition x2 is not true. Proposition x4 is true if proposition x5 is not true. Proposition p2 is true if proposition s1 is true and proposition s3 is true. Proposition x2 is true if proposition x3 is not true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition g1_2 is true. Proposition t3 is true if proposition t4 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition p3 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true. Proposition btrue is true. Proposition g3_0 is true. Proposition g2_0 is true. Proposition g2_1 is true. Proposition g1_1 is true. Proposition t7 is true if proposition btrue is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition q is true if proposition t0 is true. Proposition x5 is true if proposition x6 is not true. Proposition t2 is true if proposition t3 is true and proposition wide is true. Proposition g3_1 is true. Proposition x6 is true if proposition x7 is not true. Proposition t4 is true if proposition t5 is true. Proposition x0 is true if proposition x1 is not true. Proposition t6 is true if proposition t7 is true. Proposition cq is true if proposition x0 is true. Proposition t0 is true if proposition t1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k8-i5-skept::skept", "rec_id": "hv2-cyclen-even_one_sided_k8-i5", "axis": "cyclen", "difficulty": "even_one_sided_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition x3 is true if proposition x4 is not true. Proposition t5 is true if proposition t6 is true. Proposition g1_0 is true. Proposition g0_0 is true. Proposition x7 is true if proposition x0 is not true. Proposition s0 is true if proposition g0_0 is true. Proposition t1 is true if proposition t2 is true and proposition cq is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true and proposition p3 is true. Proposition x1 is true if proposition x2 is not true. Proposition x4 is true if proposition x5 is not true. Proposition p2 is true if proposition s1 is true and proposition s3 is true. Proposition x2 is true if proposition x3 is not true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition g1_2 is true. Proposition t3 is true if proposition t4 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition p3 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true. Proposition btrue is true. Proposition g3_0 is true. Proposition g2_0 is true. Proposition g2_1 is true. Proposition g1_1 is true. Proposition t7 is true if proposition btrue is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition q is true if proposition t0 is true. Proposition x5 is true if proposition x6 is not true. Proposition t2 is true if proposition t3 is true and proposition wide is true. Proposition g3_1 is true. Proposition x6 is true if proposition x7 is not true. Proposition t4 is true if proposition t5 is true. Proposition x0 is true if proposition x1 is not true. Proposition t6 is true if proposition t7 is true. Proposition cq is true if proposition x0 is true. Proposition t0 is true if proposition t1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k8-i5-wfs::wfs", "rec_id": "hv2-cyclen-even_one_sided_k8-i5", "axis": "cyclen", "difficulty": "even_one_sided_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition x3 is true if proposition x4 is not true. Proposition t5 is true if proposition t6 is true. Proposition g1_0 is true. Proposition g0_0 is true. Proposition x7 is true if proposition x0 is not true. Proposition s0 is true if proposition g0_0 is true. Proposition t1 is true if proposition t2 is true and proposition cq is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true and proposition p3 is true. Proposition x1 is true if proposition x2 is not true. Proposition x4 is true if proposition x5 is not true. Proposition p2 is true if proposition s1 is true and proposition s3 is true. Proposition x2 is true if proposition x3 is not true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition g1_2 is true. Proposition t3 is true if proposition t4 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition p3 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true. Proposition btrue is true. Proposition g3_0 is true. Proposition g2_0 is true. Proposition g2_1 is true. Proposition g1_1 is true. Proposition t7 is true if proposition btrue is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition q is true if proposition t0 is true. Proposition x5 is true if proposition x6 is not true. Proposition t2 is true if proposition t3 is true and proposition wide is true. Proposition g3_1 is true. Proposition x6 is true if proposition x7 is not true. Proposition t4 is true if proposition t5 is true. Proposition x0 is true if proposition x1 is not true. Proposition t6 is true if proposition t7 is true. Proposition cq is true if proposition x0 is true. Proposition t0 is true if proposition t1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k8-i6-none::none", "rec_id": "hv2-cyclen-even_one_sided_k8-i6", "axis": "cyclen", "difficulty": "even_one_sided_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition x7 is true if proposition x0 is not true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition x2 is true if proposition x3 is not true. Proposition g0_0 is true. Proposition x4 is true if proposition x5 is not true. Proposition g2_0 is true. Proposition t6 is true if proposition t7 is true and proposition wide is true. Proposition x0 is true if proposition x1 is not true. Proposition g1_1 is true. Proposition t7 is true if proposition btrue is true. Proposition t1 is true if proposition t2 is true and proposition cq is true. Proposition g2_1 is true. Proposition t3 is true if proposition t4 is true. Proposition t0 is true if proposition t1 is true. Proposition g1_0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t5 is true if proposition t6 is true. Proposition x1 is true if proposition x2 is not true. Proposition btrue is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition t2 is true if proposition t3 is true. Proposition x6 is true if proposition x7 is not true. Proposition x3 is true if proposition x4 is not true. Proposition g3_1 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition cq is true if proposition x0 is true. Proposition t4 is true if proposition t5 is true. Proposition g0_1 is true. Proposition q is true if proposition t0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition x5 is true if proposition x6 is not true. Proposition g3_0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k8-i6-closed_world::closed_world", "rec_id": "hv2-cyclen-even_one_sided_k8-i6", "axis": "cyclen", "difficulty": "even_one_sided_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition x7 is true if proposition x0 is not true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition x2 is true if proposition x3 is not true. Proposition g0_0 is true. Proposition x4 is true if proposition x5 is not true. Proposition g2_0 is true. Proposition t6 is true if proposition t7 is true and proposition wide is true. Proposition x0 is true if proposition x1 is not true. Proposition g1_1 is true. Proposition t7 is true if proposition btrue is true. Proposition t1 is true if proposition t2 is true and proposition cq is true. Proposition g2_1 is true. Proposition t3 is true if proposition t4 is true. Proposition t0 is true if proposition t1 is true. Proposition g1_0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t5 is true if proposition t6 is true. Proposition x1 is true if proposition x2 is not true. Proposition btrue is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition t2 is true if proposition t3 is true. Proposition x6 is true if proposition x7 is not true. Proposition x3 is true if proposition x4 is not true. Proposition g3_1 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition cq is true if proposition x0 is true. Proposition t4 is true if proposition t5 is true. Proposition g0_1 is true. Proposition q is true if proposition t0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition x5 is true if proposition x6 is not true. Proposition g3_0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k8-i6-cred::cred", "rec_id": "hv2-cyclen-even_one_sided_k8-i6", "axis": "cyclen", "difficulty": "even_one_sided_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition x7 is true if proposition x0 is not true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition x2 is true if proposition x3 is not true. Proposition g0_0 is true. Proposition x4 is true if proposition x5 is not true. Proposition g2_0 is true. Proposition t6 is true if proposition t7 is true and proposition wide is true. Proposition x0 is true if proposition x1 is not true. Proposition g1_1 is true. Proposition t7 is true if proposition btrue is true. Proposition t1 is true if proposition t2 is true and proposition cq is true. Proposition g2_1 is true. Proposition t3 is true if proposition t4 is true. Proposition t0 is true if proposition t1 is true. Proposition g1_0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t5 is true if proposition t6 is true. Proposition x1 is true if proposition x2 is not true. Proposition btrue is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition t2 is true if proposition t3 is true. Proposition x6 is true if proposition x7 is not true. Proposition x3 is true if proposition x4 is not true. Proposition g3_1 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition cq is true if proposition x0 is true. Proposition t4 is true if proposition t5 is true. Proposition g0_1 is true. Proposition q is true if proposition t0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition x5 is true if proposition x6 is not true. Proposition g3_0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k8-i6-skept::skept", "rec_id": "hv2-cyclen-even_one_sided_k8-i6", "axis": "cyclen", "difficulty": "even_one_sided_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition x7 is true if proposition x0 is not true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition x2 is true if proposition x3 is not true. Proposition g0_0 is true. Proposition x4 is true if proposition x5 is not true. Proposition g2_0 is true. Proposition t6 is true if proposition t7 is true and proposition wide is true. Proposition x0 is true if proposition x1 is not true. Proposition g1_1 is true. Proposition t7 is true if proposition btrue is true. Proposition t1 is true if proposition t2 is true and proposition cq is true. Proposition g2_1 is true. Proposition t3 is true if proposition t4 is true. Proposition t0 is true if proposition t1 is true. Proposition g1_0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t5 is true if proposition t6 is true. Proposition x1 is true if proposition x2 is not true. Proposition btrue is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition t2 is true if proposition t3 is true. Proposition x6 is true if proposition x7 is not true. Proposition x3 is true if proposition x4 is not true. Proposition g3_1 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition cq is true if proposition x0 is true. Proposition t4 is true if proposition t5 is true. Proposition g0_1 is true. Proposition q is true if proposition t0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition x5 is true if proposition x6 is not true. Proposition g3_0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k8-i6-wfs::wfs", "rec_id": "hv2-cyclen-even_one_sided_k8-i6", "axis": "cyclen", "difficulty": "even_one_sided_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition x7 is true if proposition x0 is not true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition x2 is true if proposition x3 is not true. Proposition g0_0 is true. Proposition x4 is true if proposition x5 is not true. Proposition g2_0 is true. Proposition t6 is true if proposition t7 is true and proposition wide is true. Proposition x0 is true if proposition x1 is not true. Proposition g1_1 is true. Proposition t7 is true if proposition btrue is true. Proposition t1 is true if proposition t2 is true and proposition cq is true. Proposition g2_1 is true. Proposition t3 is true if proposition t4 is true. Proposition t0 is true if proposition t1 is true. Proposition g1_0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t5 is true if proposition t6 is true. Proposition x1 is true if proposition x2 is not true. Proposition btrue is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition t2 is true if proposition t3 is true. Proposition x6 is true if proposition x7 is not true. Proposition x3 is true if proposition x4 is not true. Proposition g3_1 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition cq is true if proposition x0 is true. Proposition t4 is true if proposition t5 is true. Proposition g0_1 is true. Proposition q is true if proposition t0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition x5 is true if proposition x6 is not true. Proposition g3_0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k8-i7-none::none", "rec_id": "hv2-cyclen-even_one_sided_k8-i7", "axis": "cyclen", "difficulty": "even_one_sided_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition g0_0 is true. Proposition x5 is true if proposition etrue is true and proposition x6 is not true. Proposition x4 is true if proposition etrue is true and proposition x5 is not true. Proposition g2_0 is true. Proposition cq is true if proposition x0 is true. Proposition etrue is true. Proposition x3 is true if proposition etrue is true and proposition x4 is not true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true. Proposition g3_0 is true. Proposition g1_1 is true. Proposition g3_2 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition g0_1 is true. Proposition t6 is true if proposition t7 is true. Proposition p1 is true if proposition s2 is true and proposition s3 is true. Proposition x6 is true if proposition etrue is true and proposition x7 is not true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t7 is true if proposition btrue is true and proposition cq is true. Proposition q is true if proposition t0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true and proposition g3_2 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition t3 is true if proposition t4 is true and proposition wide is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition t4 is true if proposition t5 is true. Proposition x7 is true if proposition etrue is true and proposition x0 is not true. Proposition s2 is true if proposition g2_0 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition g3_1 is true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s3 is true. Proposition t1 is true if proposition t2 is true. Proposition t0 is true if proposition t1 is true. Proposition t2 is true if proposition t3 is true. Proposition btrue is true. Proposition t5 is true if proposition t6 is true. Proposition g1_0 is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k8-i7-closed_world::closed_world", "rec_id": "hv2-cyclen-even_one_sided_k8-i7", "axis": "cyclen", "difficulty": "even_one_sided_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition g0_0 is true. Proposition x5 is true if proposition etrue is true and proposition x6 is not true. Proposition x4 is true if proposition etrue is true and proposition x5 is not true. Proposition g2_0 is true. Proposition cq is true if proposition x0 is true. Proposition etrue is true. Proposition x3 is true if proposition etrue is true and proposition x4 is not true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true. Proposition g3_0 is true. Proposition g1_1 is true. Proposition g3_2 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition g0_1 is true. Proposition t6 is true if proposition t7 is true. Proposition p1 is true if proposition s2 is true and proposition s3 is true. Proposition x6 is true if proposition etrue is true and proposition x7 is not true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t7 is true if proposition btrue is true and proposition cq is true. Proposition q is true if proposition t0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true and proposition g3_2 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition t3 is true if proposition t4 is true and proposition wide is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition t4 is true if proposition t5 is true. Proposition x7 is true if proposition etrue is true and proposition x0 is not true. Proposition s2 is true if proposition g2_0 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition g3_1 is true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s3 is true. Proposition t1 is true if proposition t2 is true. Proposition t0 is true if proposition t1 is true. Proposition t2 is true if proposition t3 is true. Proposition btrue is true. Proposition t5 is true if proposition t6 is true. Proposition g1_0 is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k8-i7-cred::cred", "rec_id": "hv2-cyclen-even_one_sided_k8-i7", "axis": "cyclen", "difficulty": "even_one_sided_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition g0_0 is true. Proposition x5 is true if proposition etrue is true and proposition x6 is not true. Proposition x4 is true if proposition etrue is true and proposition x5 is not true. Proposition g2_0 is true. Proposition cq is true if proposition x0 is true. Proposition etrue is true. Proposition x3 is true if proposition etrue is true and proposition x4 is not true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true. Proposition g3_0 is true. Proposition g1_1 is true. Proposition g3_2 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition g0_1 is true. Proposition t6 is true if proposition t7 is true. Proposition p1 is true if proposition s2 is true and proposition s3 is true. Proposition x6 is true if proposition etrue is true and proposition x7 is not true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t7 is true if proposition btrue is true and proposition cq is true. Proposition q is true if proposition t0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true and proposition g3_2 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition t3 is true if proposition t4 is true and proposition wide is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition t4 is true if proposition t5 is true. Proposition x7 is true if proposition etrue is true and proposition x0 is not true. Proposition s2 is true if proposition g2_0 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition g3_1 is true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s3 is true. Proposition t1 is true if proposition t2 is true. Proposition t0 is true if proposition t1 is true. Proposition t2 is true if proposition t3 is true. Proposition btrue is true. Proposition t5 is true if proposition t6 is true. Proposition g1_0 is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k8-i7-skept::skept", "rec_id": "hv2-cyclen-even_one_sided_k8-i7", "axis": "cyclen", "difficulty": "even_one_sided_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition g0_0 is true. Proposition x5 is true if proposition etrue is true and proposition x6 is not true. Proposition x4 is true if proposition etrue is true and proposition x5 is not true. Proposition g2_0 is true. Proposition cq is true if proposition x0 is true. Proposition etrue is true. Proposition x3 is true if proposition etrue is true and proposition x4 is not true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true. Proposition g3_0 is true. Proposition g1_1 is true. Proposition g3_2 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition g0_1 is true. Proposition t6 is true if proposition t7 is true. Proposition p1 is true if proposition s2 is true and proposition s3 is true. Proposition x6 is true if proposition etrue is true and proposition x7 is not true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t7 is true if proposition btrue is true and proposition cq is true. Proposition q is true if proposition t0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true and proposition g3_2 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition t3 is true if proposition t4 is true and proposition wide is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition t4 is true if proposition t5 is true. Proposition x7 is true if proposition etrue is true and proposition x0 is not true. Proposition s2 is true if proposition g2_0 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition g3_1 is true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s3 is true. Proposition t1 is true if proposition t2 is true. Proposition t0 is true if proposition t1 is true. Proposition t2 is true if proposition t3 is true. Proposition btrue is true. Proposition t5 is true if proposition t6 is true. Proposition g1_0 is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-even_one_sided_k8-i7-wfs::wfs", "rec_id": "hv2-cyclen-even_one_sided_k8-i7", "axis": "cyclen", "difficulty": "even_one_sided_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition g0_0 is true. Proposition x5 is true if proposition etrue is true and proposition x6 is not true. Proposition x4 is true if proposition etrue is true and proposition x5 is not true. Proposition g2_0 is true. Proposition cq is true if proposition x0 is true. Proposition etrue is true. Proposition x3 is true if proposition etrue is true and proposition x4 is not true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true. Proposition g3_0 is true. Proposition g1_1 is true. Proposition g3_2 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition g0_1 is true. Proposition t6 is true if proposition t7 is true. Proposition p1 is true if proposition s2 is true and proposition s3 is true. Proposition x6 is true if proposition etrue is true and proposition x7 is not true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t7 is true if proposition btrue is true and proposition cq is true. Proposition q is true if proposition t0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true and proposition g3_2 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition t3 is true if proposition t4 is true and proposition wide is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition t4 is true if proposition t5 is true. Proposition x7 is true if proposition etrue is true and proposition x0 is not true. Proposition s2 is true if proposition g2_0 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition g3_1 is true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s3 is true. Proposition t1 is true if proposition t2 is true. Proposition t0 is true if proposition t1 is true. Proposition t2 is true if proposition t3 is true. Proposition btrue is true. Proposition t5 is true if proposition t6 is true. Proposition g1_0 is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k8-i0-none::none", "rec_id": "hv2-cyclen-odd_k8-i0", "axis": "cyclen", "difficulty": "odd_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition g1_1 is true. Proposition t3 is true if proposition t4 is true. Proposition g2_1 is true. Proposition x7 is true if proposition x0 is not true. Proposition x4 is true if proposition x5 is not true. Proposition s3 is true if proposition g3_0 is true. Proposition t1 is true if proposition t2 is true. Proposition g3_0 is true. Proposition p0 is true if proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition btrue is true. Proposition t6 is true if proposition t7 is true. Proposition g0_0 is true. Proposition x5 is true if proposition x6 is not true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition x3 is true if proposition x4 is not true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s3 is true. Proposition q is true if proposition t0 is true. Proposition t2 is true if proposition t3 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true. Proposition x2 is true if proposition x3 is not true. Proposition x0 is true if proposition x1 is not true. Proposition t5 is true if proposition t6 is true and proposition cq is true. Proposition x1 is true if proposition x2 is not true. Proposition cq is true if proposition x0 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true and proposition g0_2 is true. Proposition g0_1 is true. Proposition g0_2 is true. Proposition g1_0 is true. Proposition t7 is true if proposition btrue is true and proposition wide is true. Proposition t4 is true if proposition t5 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition g2_0 is true. Proposition t0 is true if proposition t1 is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition x6 is true if proposition x7 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k8-i0-closed_world::closed_world", "rec_id": "hv2-cyclen-odd_k8-i0", "axis": "cyclen", "difficulty": "odd_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition g1_1 is true. Proposition t3 is true if proposition t4 is true. Proposition g2_1 is true. Proposition x7 is true if proposition x0 is not true. Proposition x4 is true if proposition x5 is not true. Proposition s3 is true if proposition g3_0 is true. Proposition t1 is true if proposition t2 is true. Proposition g3_0 is true. Proposition p0 is true if proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition btrue is true. Proposition t6 is true if proposition t7 is true. Proposition g0_0 is true. Proposition x5 is true if proposition x6 is not true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition x3 is true if proposition x4 is not true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s3 is true. Proposition q is true if proposition t0 is true. Proposition t2 is true if proposition t3 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true. Proposition x2 is true if proposition x3 is not true. Proposition x0 is true if proposition x1 is not true. Proposition t5 is true if proposition t6 is true and proposition cq is true. Proposition x1 is true if proposition x2 is not true. Proposition cq is true if proposition x0 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true and proposition g0_2 is true. Proposition g0_1 is true. Proposition g0_2 is true. Proposition g1_0 is true. Proposition t7 is true if proposition btrue is true and proposition wide is true. Proposition t4 is true if proposition t5 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition g2_0 is true. Proposition t0 is true if proposition t1 is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition x6 is true if proposition x7 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k8-i0-cred::cred", "rec_id": "hv2-cyclen-odd_k8-i0", "axis": "cyclen", "difficulty": "odd_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition g1_1 is true. Proposition t3 is true if proposition t4 is true. Proposition g2_1 is true. Proposition x7 is true if proposition x0 is not true. Proposition x4 is true if proposition x5 is not true. Proposition s3 is true if proposition g3_0 is true. Proposition t1 is true if proposition t2 is true. Proposition g3_0 is true. Proposition p0 is true if proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition btrue is true. Proposition t6 is true if proposition t7 is true. Proposition g0_0 is true. Proposition x5 is true if proposition x6 is not true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition x3 is true if proposition x4 is not true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s3 is true. Proposition q is true if proposition t0 is true. Proposition t2 is true if proposition t3 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true. Proposition x2 is true if proposition x3 is not true. Proposition x0 is true if proposition x1 is not true. Proposition t5 is true if proposition t6 is true and proposition cq is true. Proposition x1 is true if proposition x2 is not true. Proposition cq is true if proposition x0 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true and proposition g0_2 is true. Proposition g0_1 is true. Proposition g0_2 is true. Proposition g1_0 is true. Proposition t7 is true if proposition btrue is true and proposition wide is true. Proposition t4 is true if proposition t5 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition g2_0 is true. Proposition t0 is true if proposition t1 is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition x6 is true if proposition x7 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k8-i0-skept::skept", "rec_id": "hv2-cyclen-odd_k8-i0", "axis": "cyclen", "difficulty": "odd_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition g1_1 is true. Proposition t3 is true if proposition t4 is true. Proposition g2_1 is true. Proposition x7 is true if proposition x0 is not true. Proposition x4 is true if proposition x5 is not true. Proposition s3 is true if proposition g3_0 is true. Proposition t1 is true if proposition t2 is true. Proposition g3_0 is true. Proposition p0 is true if proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition btrue is true. Proposition t6 is true if proposition t7 is true. Proposition g0_0 is true. Proposition x5 is true if proposition x6 is not true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition x3 is true if proposition x4 is not true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s3 is true. Proposition q is true if proposition t0 is true. Proposition t2 is true if proposition t3 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true. Proposition x2 is true if proposition x3 is not true. Proposition x0 is true if proposition x1 is not true. Proposition t5 is true if proposition t6 is true and proposition cq is true. Proposition x1 is true if proposition x2 is not true. Proposition cq is true if proposition x0 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true and proposition g0_2 is true. Proposition g0_1 is true. Proposition g0_2 is true. Proposition g1_0 is true. Proposition t7 is true if proposition btrue is true and proposition wide is true. Proposition t4 is true if proposition t5 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition g2_0 is true. Proposition t0 is true if proposition t1 is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition x6 is true if proposition x7 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k8-i0-wfs::wfs", "rec_id": "hv2-cyclen-odd_k8-i0", "axis": "cyclen", "difficulty": "odd_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition g1_1 is true. Proposition t3 is true if proposition t4 is true. Proposition g2_1 is true. Proposition x7 is true if proposition x0 is not true. Proposition x4 is true if proposition x5 is not true. Proposition s3 is true if proposition g3_0 is true. Proposition t1 is true if proposition t2 is true. Proposition g3_0 is true. Proposition p0 is true if proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition btrue is true. Proposition t6 is true if proposition t7 is true. Proposition g0_0 is true. Proposition x5 is true if proposition x6 is not true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition x3 is true if proposition x4 is not true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s3 is true. Proposition q is true if proposition t0 is true. Proposition t2 is true if proposition t3 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true. Proposition x2 is true if proposition x3 is not true. Proposition x0 is true if proposition x1 is not true. Proposition t5 is true if proposition t6 is true and proposition cq is true. Proposition x1 is true if proposition x2 is not true. Proposition cq is true if proposition x0 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true and proposition g0_2 is true. Proposition g0_1 is true. Proposition g0_2 is true. Proposition g1_0 is true. Proposition t7 is true if proposition btrue is true and proposition wide is true. Proposition t4 is true if proposition t5 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition g2_0 is true. Proposition t0 is true if proposition t1 is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition x6 is true if proposition x7 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k8-i1-none::none", "rec_id": "hv2-cyclen-odd_k8-i1", "axis": "cyclen", "difficulty": "odd_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition btrue is true. Proposition x7 is true if proposition etrue is true and proposition x0 is not true. Proposition g2_1 is true. Proposition cq is true if proposition x0 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition x4 is true if proposition etrue is true and proposition x5 is not true. Proposition g3_0 is true. Proposition t5 is true if proposition t6 is true and proposition wide is true. Proposition q is true if proposition t0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition x3 is true if proposition etrue is true and proposition x4 is not true. Proposition g2_0 is true. Proposition t2 is true if proposition t3 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t0 is true if proposition t1 is true. Proposition g1_1 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition g0_1 is true. Proposition x6 is true if proposition etrue is true and proposition x7 is not true. Proposition g1_0 is true. Proposition t4 is true if proposition t5 is true. Proposition g3_1 is true. Proposition g0_0 is true. Proposition x5 is true if proposition etrue is true and proposition x6 is not true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t1 is true if proposition t2 is true and proposition cq is true. Proposition t3 is true if proposition t4 is true. Proposition etrue is true. Proposition t6 is true if proposition t7 is true. Proposition t7 is true if proposition btrue is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k8-i1-closed_world::closed_world", "rec_id": "hv2-cyclen-odd_k8-i1", "axis": "cyclen", "difficulty": "odd_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition btrue is true. Proposition x7 is true if proposition etrue is true and proposition x0 is not true. Proposition g2_1 is true. Proposition cq is true if proposition x0 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition x4 is true if proposition etrue is true and proposition x5 is not true. Proposition g3_0 is true. Proposition t5 is true if proposition t6 is true and proposition wide is true. Proposition q is true if proposition t0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition x3 is true if proposition etrue is true and proposition x4 is not true. Proposition g2_0 is true. Proposition t2 is true if proposition t3 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t0 is true if proposition t1 is true. Proposition g1_1 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition g0_1 is true. Proposition x6 is true if proposition etrue is true and proposition x7 is not true. Proposition g1_0 is true. Proposition t4 is true if proposition t5 is true. Proposition g3_1 is true. Proposition g0_0 is true. Proposition x5 is true if proposition etrue is true and proposition x6 is not true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t1 is true if proposition t2 is true and proposition cq is true. Proposition t3 is true if proposition t4 is true. Proposition etrue is true. Proposition t6 is true if proposition t7 is true. Proposition t7 is true if proposition btrue is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k8-i1-cred::cred", "rec_id": "hv2-cyclen-odd_k8-i1", "axis": "cyclen", "difficulty": "odd_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition btrue is true. Proposition x7 is true if proposition etrue is true and proposition x0 is not true. Proposition g2_1 is true. Proposition cq is true if proposition x0 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition x4 is true if proposition etrue is true and proposition x5 is not true. Proposition g3_0 is true. Proposition t5 is true if proposition t6 is true and proposition wide is true. Proposition q is true if proposition t0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition x3 is true if proposition etrue is true and proposition x4 is not true. Proposition g2_0 is true. Proposition t2 is true if proposition t3 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t0 is true if proposition t1 is true. Proposition g1_1 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition g0_1 is true. Proposition x6 is true if proposition etrue is true and proposition x7 is not true. Proposition g1_0 is true. Proposition t4 is true if proposition t5 is true. Proposition g3_1 is true. Proposition g0_0 is true. Proposition x5 is true if proposition etrue is true and proposition x6 is not true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t1 is true if proposition t2 is true and proposition cq is true. Proposition t3 is true if proposition t4 is true. Proposition etrue is true. Proposition t6 is true if proposition t7 is true. Proposition t7 is true if proposition btrue is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k8-i1-skept::skept", "rec_id": "hv2-cyclen-odd_k8-i1", "axis": "cyclen", "difficulty": "odd_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition btrue is true. Proposition x7 is true if proposition etrue is true and proposition x0 is not true. Proposition g2_1 is true. Proposition cq is true if proposition x0 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition x4 is true if proposition etrue is true and proposition x5 is not true. Proposition g3_0 is true. Proposition t5 is true if proposition t6 is true and proposition wide is true. Proposition q is true if proposition t0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition x3 is true if proposition etrue is true and proposition x4 is not true. Proposition g2_0 is true. Proposition t2 is true if proposition t3 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t0 is true if proposition t1 is true. Proposition g1_1 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition g0_1 is true. Proposition x6 is true if proposition etrue is true and proposition x7 is not true. Proposition g1_0 is true. Proposition t4 is true if proposition t5 is true. Proposition g3_1 is true. Proposition g0_0 is true. Proposition x5 is true if proposition etrue is true and proposition x6 is not true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t1 is true if proposition t2 is true and proposition cq is true. Proposition t3 is true if proposition t4 is true. Proposition etrue is true. Proposition t6 is true if proposition t7 is true. Proposition t7 is true if proposition btrue is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k8-i1-wfs::wfs", "rec_id": "hv2-cyclen-odd_k8-i1", "axis": "cyclen", "difficulty": "odd_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition btrue is true. Proposition x7 is true if proposition etrue is true and proposition x0 is not true. Proposition g2_1 is true. Proposition cq is true if proposition x0 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition x4 is true if proposition etrue is true and proposition x5 is not true. Proposition g3_0 is true. Proposition t5 is true if proposition t6 is true and proposition wide is true. Proposition q is true if proposition t0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition x3 is true if proposition etrue is true and proposition x4 is not true. Proposition g2_0 is true. Proposition t2 is true if proposition t3 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t0 is true if proposition t1 is true. Proposition g1_1 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition g0_1 is true. Proposition x6 is true if proposition etrue is true and proposition x7 is not true. Proposition g1_0 is true. Proposition t4 is true if proposition t5 is true. Proposition g3_1 is true. Proposition g0_0 is true. Proposition x5 is true if proposition etrue is true and proposition x6 is not true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t1 is true if proposition t2 is true and proposition cq is true. Proposition t3 is true if proposition t4 is true. Proposition etrue is true. Proposition t6 is true if proposition t7 is true. Proposition t7 is true if proposition btrue is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k8-i2-none::none", "rec_id": "hv2-cyclen-odd_k8-i2", "axis": "cyclen", "difficulty": "odd_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition x3 is true if proposition x4 is not true. Proposition g3_1 is true. Proposition x1 is true if proposition x2 is not true. Proposition g0_1 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true and proposition g0_2 is true. Proposition g2_0 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g2_1 is true. Proposition g0_0 is true. Proposition x6 is true if proposition x7 is not true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition x2 is true if proposition x3 is not true. Proposition g0_2 is true. Proposition t7 is true if proposition btrue is true and proposition wide is true. Proposition cq is true if proposition x0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t0 is true if proposition t1 is true. Proposition t3 is true if proposition t4 is true. Proposition x4 is true if proposition x5 is not true. Proposition g3_0 is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition q is true if proposition t0 is true. Proposition x0 is true if proposition x1 is not true. Proposition t6 is true if proposition t7 is true. Proposition x7 is true if proposition x0 is not true. Proposition t5 is true if proposition t6 is true and proposition cq is true. Proposition s1 is true if proposition g1_0 is true. Proposition x5 is true if proposition x6 is not true. Proposition g1_0 is true. Proposition t4 is true if proposition t5 is true. Proposition t2 is true if proposition t3 is true. Proposition t1 is true if proposition t2 is true. Proposition btrue is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k8-i2-closed_world::closed_world", "rec_id": "hv2-cyclen-odd_k8-i2", "axis": "cyclen", "difficulty": "odd_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition x3 is true if proposition x4 is not true. Proposition g3_1 is true. Proposition x1 is true if proposition x2 is not true. Proposition g0_1 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true and proposition g0_2 is true. Proposition g2_0 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g2_1 is true. Proposition g0_0 is true. Proposition x6 is true if proposition x7 is not true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition x2 is true if proposition x3 is not true. Proposition g0_2 is true. Proposition t7 is true if proposition btrue is true and proposition wide is true. Proposition cq is true if proposition x0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t0 is true if proposition t1 is true. Proposition t3 is true if proposition t4 is true. Proposition x4 is true if proposition x5 is not true. Proposition g3_0 is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition q is true if proposition t0 is true. Proposition x0 is true if proposition x1 is not true. Proposition t6 is true if proposition t7 is true. Proposition x7 is true if proposition x0 is not true. Proposition t5 is true if proposition t6 is true and proposition cq is true. Proposition s1 is true if proposition g1_0 is true. Proposition x5 is true if proposition x6 is not true. Proposition g1_0 is true. Proposition t4 is true if proposition t5 is true. Proposition t2 is true if proposition t3 is true. Proposition t1 is true if proposition t2 is true. Proposition btrue is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k8-i2-cred::cred", "rec_id": "hv2-cyclen-odd_k8-i2", "axis": "cyclen", "difficulty": "odd_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition x3 is true if proposition x4 is not true. Proposition g3_1 is true. Proposition x1 is true if proposition x2 is not true. Proposition g0_1 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true and proposition g0_2 is true. Proposition g2_0 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g2_1 is true. Proposition g0_0 is true. Proposition x6 is true if proposition x7 is not true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition x2 is true if proposition x3 is not true. Proposition g0_2 is true. Proposition t7 is true if proposition btrue is true and proposition wide is true. Proposition cq is true if proposition x0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t0 is true if proposition t1 is true. Proposition t3 is true if proposition t4 is true. Proposition x4 is true if proposition x5 is not true. Proposition g3_0 is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition q is true if proposition t0 is true. Proposition x0 is true if proposition x1 is not true. Proposition t6 is true if proposition t7 is true. Proposition x7 is true if proposition x0 is not true. Proposition t5 is true if proposition t6 is true and proposition cq is true. Proposition s1 is true if proposition g1_0 is true. Proposition x5 is true if proposition x6 is not true. Proposition g1_0 is true. Proposition t4 is true if proposition t5 is true. Proposition t2 is true if proposition t3 is true. Proposition t1 is true if proposition t2 is true. Proposition btrue is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k8-i2-skept::skept", "rec_id": "hv2-cyclen-odd_k8-i2", "axis": "cyclen", "difficulty": "odd_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition x3 is true if proposition x4 is not true. Proposition g3_1 is true. Proposition x1 is true if proposition x2 is not true. Proposition g0_1 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true and proposition g0_2 is true. Proposition g2_0 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g2_1 is true. Proposition g0_0 is true. Proposition x6 is true if proposition x7 is not true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition x2 is true if proposition x3 is not true. Proposition g0_2 is true. Proposition t7 is true if proposition btrue is true and proposition wide is true. Proposition cq is true if proposition x0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t0 is true if proposition t1 is true. Proposition t3 is true if proposition t4 is true. Proposition x4 is true if proposition x5 is not true. Proposition g3_0 is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition q is true if proposition t0 is true. Proposition x0 is true if proposition x1 is not true. Proposition t6 is true if proposition t7 is true. Proposition x7 is true if proposition x0 is not true. Proposition t5 is true if proposition t6 is true and proposition cq is true. Proposition s1 is true if proposition g1_0 is true. Proposition x5 is true if proposition x6 is not true. Proposition g1_0 is true. Proposition t4 is true if proposition t5 is true. Proposition t2 is true if proposition t3 is true. Proposition t1 is true if proposition t2 is true. Proposition btrue is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k8-i2-wfs::wfs", "rec_id": "hv2-cyclen-odd_k8-i2", "axis": "cyclen", "difficulty": "odd_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition x3 is true if proposition x4 is not true. Proposition g3_1 is true. Proposition x1 is true if proposition x2 is not true. Proposition g0_1 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true and proposition g0_2 is true. Proposition g2_0 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g2_1 is true. Proposition g0_0 is true. Proposition x6 is true if proposition x7 is not true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition x2 is true if proposition x3 is not true. Proposition g0_2 is true. Proposition t7 is true if proposition btrue is true and proposition wide is true. Proposition cq is true if proposition x0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t0 is true if proposition t1 is true. Proposition t3 is true if proposition t4 is true. Proposition x4 is true if proposition x5 is not true. Proposition g3_0 is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition q is true if proposition t0 is true. Proposition x0 is true if proposition x1 is not true. Proposition t6 is true if proposition t7 is true. Proposition x7 is true if proposition x0 is not true. Proposition t5 is true if proposition t6 is true and proposition cq is true. Proposition s1 is true if proposition g1_0 is true. Proposition x5 is true if proposition x6 is not true. Proposition g1_0 is true. Proposition t4 is true if proposition t5 is true. Proposition t2 is true if proposition t3 is true. Proposition t1 is true if proposition t2 is true. Proposition btrue is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k8-i3-none::none", "rec_id": "hv2-cyclen-odd_k8-i3", "axis": "cyclen", "difficulty": "odd_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition t2 is true if proposition t3 is true. Proposition etrue is true. Proposition x4 is true if proposition etrue is true and proposition x5 is not true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true and proposition g1_3 is true. Proposition x6 is true if proposition etrue is true and proposition x7 is not true. Proposition t5 is true if proposition t6 is true. Proposition cq is true if proposition x0 is true. Proposition x7 is true if proposition etrue is true and proposition x0 is not true. Proposition t0 is true if proposition t1 is true and proposition wide is true. Proposition g3_0 is true. Proposition q is true if proposition t0 is true. Proposition x5 is true if proposition etrue is true and proposition x6 is not true. Proposition t6 is true if proposition t7 is true and proposition cq is true. Proposition s0 is true if proposition g0_0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition t1 is true if proposition t2 is true. Proposition g1_3 is true. Proposition t3 is true if proposition t4 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition p1 is true if proposition s3 is true. Proposition t7 is true if proposition btrue is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true and proposition p3 is true. Proposition g1_1 is true. Proposition btrue is true. Proposition p3 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g3_1 is true. Proposition g2_0 is true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s2 is true if proposition g2_0 is true. Proposition t4 is true if proposition t5 is true. Proposition g0_0 is true. Proposition g1_0 is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true. Proposition g1_2 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition p0 is true if proposition s1 is true and proposition s2 is true. Proposition x3 is true if proposition etrue is true and proposition x4 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k8-i3-closed_world::closed_world", "rec_id": "hv2-cyclen-odd_k8-i3", "axis": "cyclen", "difficulty": "odd_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition t2 is true if proposition t3 is true. Proposition etrue is true. Proposition x4 is true if proposition etrue is true and proposition x5 is not true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true and proposition g1_3 is true. Proposition x6 is true if proposition etrue is true and proposition x7 is not true. Proposition t5 is true if proposition t6 is true. Proposition cq is true if proposition x0 is true. Proposition x7 is true if proposition etrue is true and proposition x0 is not true. Proposition t0 is true if proposition t1 is true and proposition wide is true. Proposition g3_0 is true. Proposition q is true if proposition t0 is true. Proposition x5 is true if proposition etrue is true and proposition x6 is not true. Proposition t6 is true if proposition t7 is true and proposition cq is true. Proposition s0 is true if proposition g0_0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition t1 is true if proposition t2 is true. Proposition g1_3 is true. Proposition t3 is true if proposition t4 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition p1 is true if proposition s3 is true. Proposition t7 is true if proposition btrue is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true and proposition p3 is true. Proposition g1_1 is true. Proposition btrue is true. Proposition p3 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g3_1 is true. Proposition g2_0 is true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s2 is true if proposition g2_0 is true. Proposition t4 is true if proposition t5 is true. Proposition g0_0 is true. Proposition g1_0 is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true. Proposition g1_2 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition p0 is true if proposition s1 is true and proposition s2 is true. Proposition x3 is true if proposition etrue is true and proposition x4 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k8-i3-cred::cred", "rec_id": "hv2-cyclen-odd_k8-i3", "axis": "cyclen", "difficulty": "odd_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition t2 is true if proposition t3 is true. Proposition etrue is true. Proposition x4 is true if proposition etrue is true and proposition x5 is not true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true and proposition g1_3 is true. Proposition x6 is true if proposition etrue is true and proposition x7 is not true. Proposition t5 is true if proposition t6 is true. Proposition cq is true if proposition x0 is true. Proposition x7 is true if proposition etrue is true and proposition x0 is not true. Proposition t0 is true if proposition t1 is true and proposition wide is true. Proposition g3_0 is true. Proposition q is true if proposition t0 is true. Proposition x5 is true if proposition etrue is true and proposition x6 is not true. Proposition t6 is true if proposition t7 is true and proposition cq is true. Proposition s0 is true if proposition g0_0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition t1 is true if proposition t2 is true. Proposition g1_3 is true. Proposition t3 is true if proposition t4 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition p1 is true if proposition s3 is true. Proposition t7 is true if proposition btrue is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true and proposition p3 is true. Proposition g1_1 is true. Proposition btrue is true. Proposition p3 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g3_1 is true. Proposition g2_0 is true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s2 is true if proposition g2_0 is true. Proposition t4 is true if proposition t5 is true. Proposition g0_0 is true. Proposition g1_0 is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true. Proposition g1_2 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition p0 is true if proposition s1 is true and proposition s2 is true. Proposition x3 is true if proposition etrue is true and proposition x4 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k8-i3-skept::skept", "rec_id": "hv2-cyclen-odd_k8-i3", "axis": "cyclen", "difficulty": "odd_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition t2 is true if proposition t3 is true. Proposition etrue is true. Proposition x4 is true if proposition etrue is true and proposition x5 is not true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true and proposition g1_3 is true. Proposition x6 is true if proposition etrue is true and proposition x7 is not true. Proposition t5 is true if proposition t6 is true. Proposition cq is true if proposition x0 is true. Proposition x7 is true if proposition etrue is true and proposition x0 is not true. Proposition t0 is true if proposition t1 is true and proposition wide is true. Proposition g3_0 is true. Proposition q is true if proposition t0 is true. Proposition x5 is true if proposition etrue is true and proposition x6 is not true. Proposition t6 is true if proposition t7 is true and proposition cq is true. Proposition s0 is true if proposition g0_0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition t1 is true if proposition t2 is true. Proposition g1_3 is true. Proposition t3 is true if proposition t4 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition p1 is true if proposition s3 is true. Proposition t7 is true if proposition btrue is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true and proposition p3 is true. Proposition g1_1 is true. Proposition btrue is true. Proposition p3 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g3_1 is true. Proposition g2_0 is true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s2 is true if proposition g2_0 is true. Proposition t4 is true if proposition t5 is true. Proposition g0_0 is true. Proposition g1_0 is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true. Proposition g1_2 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition p0 is true if proposition s1 is true and proposition s2 is true. Proposition x3 is true if proposition etrue is true and proposition x4 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k8-i3-wfs::wfs", "rec_id": "hv2-cyclen-odd_k8-i3", "axis": "cyclen", "difficulty": "odd_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition t2 is true if proposition t3 is true. Proposition etrue is true. Proposition x4 is true if proposition etrue is true and proposition x5 is not true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true and proposition g1_3 is true. Proposition x6 is true if proposition etrue is true and proposition x7 is not true. Proposition t5 is true if proposition t6 is true. Proposition cq is true if proposition x0 is true. Proposition x7 is true if proposition etrue is true and proposition x0 is not true. Proposition t0 is true if proposition t1 is true and proposition wide is true. Proposition g3_0 is true. Proposition q is true if proposition t0 is true. Proposition x5 is true if proposition etrue is true and proposition x6 is not true. Proposition t6 is true if proposition t7 is true and proposition cq is true. Proposition s0 is true if proposition g0_0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition t1 is true if proposition t2 is true. Proposition g1_3 is true. Proposition t3 is true if proposition t4 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition p1 is true if proposition s3 is true. Proposition t7 is true if proposition btrue is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true and proposition p3 is true. Proposition g1_1 is true. Proposition btrue is true. Proposition p3 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g3_1 is true. Proposition g2_0 is true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s2 is true if proposition g2_0 is true. Proposition t4 is true if proposition t5 is true. Proposition g0_0 is true. Proposition g1_0 is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true. Proposition g1_2 is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition p0 is true if proposition s1 is true and proposition s2 is true. Proposition x3 is true if proposition etrue is true and proposition x4 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k8-i4-none::none", "rec_id": "hv2-cyclen-odd_k8-i4", "axis": "cyclen", "difficulty": "odd_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition t1 is true if proposition t2 is true. Proposition g1_2 is true. Proposition g0_1 is true. Proposition btrue is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition x5 is true if proposition etrue is true and proposition x6 is not true. Proposition t7 is true if proposition btrue is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition t3 is true if proposition t4 is true. Proposition q is true if proposition t0 is true. Proposition etrue is true. Proposition x3 is true if proposition etrue is true and proposition x4 is not true. Proposition t2 is true if proposition t3 is true. Proposition g0_0 is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true. Proposition x7 is true if proposition etrue is true and proposition x0 is not true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition s3 is true if proposition g3_0 is true. Proposition x4 is true if proposition etrue is true and proposition x5 is not true. Proposition x6 is true if proposition etrue is true and proposition x7 is not true. Proposition g1_0 is true. Proposition t6 is true if proposition t7 is true. Proposition g2_1 is true. Proposition t5 is true if proposition t6 is true. Proposition t4 is true if proposition t5 is true and proposition wide is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g3_0 is true. Proposition t0 is true if proposition t1 is true and proposition cq is true. Proposition g2_0 is true. Proposition cq is true if proposition x0 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition g1_1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k8-i4-closed_world::closed_world", "rec_id": "hv2-cyclen-odd_k8-i4", "axis": "cyclen", "difficulty": "odd_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition t1 is true if proposition t2 is true. Proposition g1_2 is true. Proposition g0_1 is true. Proposition btrue is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition x5 is true if proposition etrue is true and proposition x6 is not true. Proposition t7 is true if proposition btrue is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition t3 is true if proposition t4 is true. Proposition q is true if proposition t0 is true. Proposition etrue is true. Proposition x3 is true if proposition etrue is true and proposition x4 is not true. Proposition t2 is true if proposition t3 is true. Proposition g0_0 is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true. Proposition x7 is true if proposition etrue is true and proposition x0 is not true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition s3 is true if proposition g3_0 is true. Proposition x4 is true if proposition etrue is true and proposition x5 is not true. Proposition x6 is true if proposition etrue is true and proposition x7 is not true. Proposition g1_0 is true. Proposition t6 is true if proposition t7 is true. Proposition g2_1 is true. Proposition t5 is true if proposition t6 is true. Proposition t4 is true if proposition t5 is true and proposition wide is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g3_0 is true. Proposition t0 is true if proposition t1 is true and proposition cq is true. Proposition g2_0 is true. Proposition cq is true if proposition x0 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition g1_1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k8-i4-cred::cred", "rec_id": "hv2-cyclen-odd_k8-i4", "axis": "cyclen", "difficulty": "odd_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition t1 is true if proposition t2 is true. Proposition g1_2 is true. Proposition g0_1 is true. Proposition btrue is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition x5 is true if proposition etrue is true and proposition x6 is not true. Proposition t7 is true if proposition btrue is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition t3 is true if proposition t4 is true. Proposition q is true if proposition t0 is true. Proposition etrue is true. Proposition x3 is true if proposition etrue is true and proposition x4 is not true. Proposition t2 is true if proposition t3 is true. Proposition g0_0 is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true. Proposition x7 is true if proposition etrue is true and proposition x0 is not true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition s3 is true if proposition g3_0 is true. Proposition x4 is true if proposition etrue is true and proposition x5 is not true. Proposition x6 is true if proposition etrue is true and proposition x7 is not true. Proposition g1_0 is true. Proposition t6 is true if proposition t7 is true. Proposition g2_1 is true. Proposition t5 is true if proposition t6 is true. Proposition t4 is true if proposition t5 is true and proposition wide is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g3_0 is true. Proposition t0 is true if proposition t1 is true and proposition cq is true. Proposition g2_0 is true. Proposition cq is true if proposition x0 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition g1_1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k8-i4-skept::skept", "rec_id": "hv2-cyclen-odd_k8-i4", "axis": "cyclen", "difficulty": "odd_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition t1 is true if proposition t2 is true. Proposition g1_2 is true. Proposition g0_1 is true. Proposition btrue is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition x5 is true if proposition etrue is true and proposition x6 is not true. Proposition t7 is true if proposition btrue is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition t3 is true if proposition t4 is true. Proposition q is true if proposition t0 is true. Proposition etrue is true. Proposition x3 is true if proposition etrue is true and proposition x4 is not true. Proposition t2 is true if proposition t3 is true. Proposition g0_0 is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true. Proposition x7 is true if proposition etrue is true and proposition x0 is not true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition s3 is true if proposition g3_0 is true. Proposition x4 is true if proposition etrue is true and proposition x5 is not true. Proposition x6 is true if proposition etrue is true and proposition x7 is not true. Proposition g1_0 is true. Proposition t6 is true if proposition t7 is true. Proposition g2_1 is true. Proposition t5 is true if proposition t6 is true. Proposition t4 is true if proposition t5 is true and proposition wide is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g3_0 is true. Proposition t0 is true if proposition t1 is true and proposition cq is true. Proposition g2_0 is true. Proposition cq is true if proposition x0 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition g1_1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k8-i4-wfs::wfs", "rec_id": "hv2-cyclen-odd_k8-i4", "axis": "cyclen", "difficulty": "odd_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition t1 is true if proposition t2 is true. Proposition g1_2 is true. Proposition g0_1 is true. Proposition btrue is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition x5 is true if proposition etrue is true and proposition x6 is not true. Proposition t7 is true if proposition btrue is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition t3 is true if proposition t4 is true. Proposition q is true if proposition t0 is true. Proposition etrue is true. Proposition x3 is true if proposition etrue is true and proposition x4 is not true. Proposition t2 is true if proposition t3 is true. Proposition g0_0 is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true. Proposition x7 is true if proposition etrue is true and proposition x0 is not true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition s3 is true if proposition g3_0 is true. Proposition x4 is true if proposition etrue is true and proposition x5 is not true. Proposition x6 is true if proposition etrue is true and proposition x7 is not true. Proposition g1_0 is true. Proposition t6 is true if proposition t7 is true. Proposition g2_1 is true. Proposition t5 is true if proposition t6 is true. Proposition t4 is true if proposition t5 is true and proposition wide is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition g3_0 is true. Proposition t0 is true if proposition t1 is true and proposition cq is true. Proposition g2_0 is true. Proposition cq is true if proposition x0 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition g1_1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k8-i5-none::none", "rec_id": "hv2-cyclen-odd_k8-i5", "axis": "cyclen", "difficulty": "odd_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition x3 is true if proposition x4 is not true. Proposition t5 is true if proposition t6 is true. Proposition g1_0 is true. Proposition g0_0 is true. Proposition x7 is true if proposition x0 is not true. Proposition s0 is true if proposition g0_0 is true. Proposition t1 is true if proposition t2 is true and proposition cq is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true and proposition p3 is true. Proposition x1 is true if proposition x2 is not true. Proposition x4 is true if proposition x5 is not true. Proposition p2 is true if proposition s1 is true and proposition s3 is true. Proposition x2 is true if proposition x3 is not true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition g1_2 is true. Proposition t3 is true if proposition t4 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition p3 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true. Proposition btrue is true. Proposition g3_0 is true. Proposition g2_0 is true. Proposition g2_1 is true. Proposition g1_1 is true. Proposition t7 is true if proposition btrue is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition q is true if proposition t0 is true. Proposition x5 is true if proposition x6 is not true. Proposition t2 is true if proposition t3 is true and proposition wide is true. Proposition g3_1 is true. Proposition x6 is true if proposition x7 is not true. Proposition t4 is true if proposition t5 is true. Proposition x0 is true if proposition x1 is not true. Proposition t6 is true if proposition t7 is true. Proposition cq is true if proposition x0 is true. Proposition t0 is true if proposition t1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k8-i5-closed_world::closed_world", "rec_id": "hv2-cyclen-odd_k8-i5", "axis": "cyclen", "difficulty": "odd_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition x3 is true if proposition x4 is not true. Proposition t5 is true if proposition t6 is true. Proposition g1_0 is true. Proposition g0_0 is true. Proposition x7 is true if proposition x0 is not true. Proposition s0 is true if proposition g0_0 is true. Proposition t1 is true if proposition t2 is true and proposition cq is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true and proposition p3 is true. Proposition x1 is true if proposition x2 is not true. Proposition x4 is true if proposition x5 is not true. Proposition p2 is true if proposition s1 is true and proposition s3 is true. Proposition x2 is true if proposition x3 is not true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition g1_2 is true. Proposition t3 is true if proposition t4 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition p3 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true. Proposition btrue is true. Proposition g3_0 is true. Proposition g2_0 is true. Proposition g2_1 is true. Proposition g1_1 is true. Proposition t7 is true if proposition btrue is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition q is true if proposition t0 is true. Proposition x5 is true if proposition x6 is not true. Proposition t2 is true if proposition t3 is true and proposition wide is true. Proposition g3_1 is true. Proposition x6 is true if proposition x7 is not true. Proposition t4 is true if proposition t5 is true. Proposition x0 is true if proposition x1 is not true. Proposition t6 is true if proposition t7 is true. Proposition cq is true if proposition x0 is true. Proposition t0 is true if proposition t1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k8-i5-cred::cred", "rec_id": "hv2-cyclen-odd_k8-i5", "axis": "cyclen", "difficulty": "odd_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition x3 is true if proposition x4 is not true. Proposition t5 is true if proposition t6 is true. Proposition g1_0 is true. Proposition g0_0 is true. Proposition x7 is true if proposition x0 is not true. Proposition s0 is true if proposition g0_0 is true. Proposition t1 is true if proposition t2 is true and proposition cq is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true and proposition p3 is true. Proposition x1 is true if proposition x2 is not true. Proposition x4 is true if proposition x5 is not true. Proposition p2 is true if proposition s1 is true and proposition s3 is true. Proposition x2 is true if proposition x3 is not true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition g1_2 is true. Proposition t3 is true if proposition t4 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition p3 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true. Proposition btrue is true. Proposition g3_0 is true. Proposition g2_0 is true. Proposition g2_1 is true. Proposition g1_1 is true. Proposition t7 is true if proposition btrue is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition q is true if proposition t0 is true. Proposition x5 is true if proposition x6 is not true. Proposition t2 is true if proposition t3 is true and proposition wide is true. Proposition g3_1 is true. Proposition x6 is true if proposition x7 is not true. Proposition t4 is true if proposition t5 is true. Proposition x0 is true if proposition x1 is not true. Proposition t6 is true if proposition t7 is true. Proposition cq is true if proposition x0 is true. Proposition t0 is true if proposition t1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k8-i5-skept::skept", "rec_id": "hv2-cyclen-odd_k8-i5", "axis": "cyclen", "difficulty": "odd_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition x3 is true if proposition x4 is not true. Proposition t5 is true if proposition t6 is true. Proposition g1_0 is true. Proposition g0_0 is true. Proposition x7 is true if proposition x0 is not true. Proposition s0 is true if proposition g0_0 is true. Proposition t1 is true if proposition t2 is true and proposition cq is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true and proposition p3 is true. Proposition x1 is true if proposition x2 is not true. Proposition x4 is true if proposition x5 is not true. Proposition p2 is true if proposition s1 is true and proposition s3 is true. Proposition x2 is true if proposition x3 is not true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition g1_2 is true. Proposition t3 is true if proposition t4 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition p3 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true. Proposition btrue is true. Proposition g3_0 is true. Proposition g2_0 is true. Proposition g2_1 is true. Proposition g1_1 is true. Proposition t7 is true if proposition btrue is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition q is true if proposition t0 is true. Proposition x5 is true if proposition x6 is not true. Proposition t2 is true if proposition t3 is true and proposition wide is true. Proposition g3_1 is true. Proposition x6 is true if proposition x7 is not true. Proposition t4 is true if proposition t5 is true. Proposition x0 is true if proposition x1 is not true. Proposition t6 is true if proposition t7 is true. Proposition cq is true if proposition x0 is true. Proposition t0 is true if proposition t1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k8-i5-wfs::wfs", "rec_id": "hv2-cyclen-odd_k8-i5", "axis": "cyclen", "difficulty": "odd_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition x3 is true if proposition x4 is not true. Proposition t5 is true if proposition t6 is true. Proposition g1_0 is true. Proposition g0_0 is true. Proposition x7 is true if proposition x0 is not true. Proposition s0 is true if proposition g0_0 is true. Proposition t1 is true if proposition t2 is true and proposition cq is true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true and proposition p3 is true. Proposition x1 is true if proposition x2 is not true. Proposition x4 is true if proposition x5 is not true. Proposition p2 is true if proposition s1 is true and proposition s3 is true. Proposition x2 is true if proposition x3 is not true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition g1_2 is true. Proposition t3 is true if proposition t4 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition p3 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true and proposition g1_2 is true. Proposition btrue is true. Proposition g3_0 is true. Proposition g2_0 is true. Proposition g2_1 is true. Proposition g1_1 is true. Proposition t7 is true if proposition btrue is true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition q is true if proposition t0 is true. Proposition x5 is true if proposition x6 is not true. Proposition t2 is true if proposition t3 is true and proposition wide is true. Proposition g3_1 is true. Proposition x6 is true if proposition x7 is not true. Proposition t4 is true if proposition t5 is true. Proposition x0 is true if proposition x1 is not true. Proposition t6 is true if proposition t7 is true. Proposition cq is true if proposition x0 is true. Proposition t0 is true if proposition t1 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k8-i6-none::none", "rec_id": "hv2-cyclen-odd_k8-i6", "axis": "cyclen", "difficulty": "odd_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition x7 is true if proposition x0 is not true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition x2 is true if proposition x3 is not true. Proposition g0_0 is true. Proposition x4 is true if proposition x5 is not true. Proposition g2_0 is true. Proposition t6 is true if proposition t7 is true and proposition wide is true. Proposition x0 is true if proposition x1 is not true. Proposition g1_1 is true. Proposition t7 is true if proposition btrue is true. Proposition t1 is true if proposition t2 is true and proposition cq is true. Proposition g2_1 is true. Proposition t3 is true if proposition t4 is true. Proposition t0 is true if proposition t1 is true. Proposition g1_0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t5 is true if proposition t6 is true. Proposition x1 is true if proposition x2 is not true. Proposition btrue is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition t2 is true if proposition t3 is true. Proposition x6 is true if proposition x7 is not true. Proposition x3 is true if proposition x4 is not true. Proposition g3_1 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition cq is true if proposition x0 is true. Proposition t4 is true if proposition t5 is true. Proposition g0_1 is true. Proposition q is true if proposition t0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition x5 is true if proposition x6 is not true. Proposition g3_0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k8-i6-closed_world::closed_world", "rec_id": "hv2-cyclen-odd_k8-i6", "axis": "cyclen", "difficulty": "odd_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition x7 is true if proposition x0 is not true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition x2 is true if proposition x3 is not true. Proposition g0_0 is true. Proposition x4 is true if proposition x5 is not true. Proposition g2_0 is true. Proposition t6 is true if proposition t7 is true and proposition wide is true. Proposition x0 is true if proposition x1 is not true. Proposition g1_1 is true. Proposition t7 is true if proposition btrue is true. Proposition t1 is true if proposition t2 is true and proposition cq is true. Proposition g2_1 is true. Proposition t3 is true if proposition t4 is true. Proposition t0 is true if proposition t1 is true. Proposition g1_0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t5 is true if proposition t6 is true. Proposition x1 is true if proposition x2 is not true. Proposition btrue is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition t2 is true if proposition t3 is true. Proposition x6 is true if proposition x7 is not true. Proposition x3 is true if proposition x4 is not true. Proposition g3_1 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition cq is true if proposition x0 is true. Proposition t4 is true if proposition t5 is true. Proposition g0_1 is true. Proposition q is true if proposition t0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition x5 is true if proposition x6 is not true. Proposition g3_0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k8-i6-cred::cred", "rec_id": "hv2-cyclen-odd_k8-i6", "axis": "cyclen", "difficulty": "odd_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition x7 is true if proposition x0 is not true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition x2 is true if proposition x3 is not true. Proposition g0_0 is true. Proposition x4 is true if proposition x5 is not true. Proposition g2_0 is true. Proposition t6 is true if proposition t7 is true and proposition wide is true. Proposition x0 is true if proposition x1 is not true. Proposition g1_1 is true. Proposition t7 is true if proposition btrue is true. Proposition t1 is true if proposition t2 is true and proposition cq is true. Proposition g2_1 is true. Proposition t3 is true if proposition t4 is true. Proposition t0 is true if proposition t1 is true. Proposition g1_0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t5 is true if proposition t6 is true. Proposition x1 is true if proposition x2 is not true. Proposition btrue is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition t2 is true if proposition t3 is true. Proposition x6 is true if proposition x7 is not true. Proposition x3 is true if proposition x4 is not true. Proposition g3_1 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition cq is true if proposition x0 is true. Proposition t4 is true if proposition t5 is true. Proposition g0_1 is true. Proposition q is true if proposition t0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition x5 is true if proposition x6 is not true. Proposition g3_0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k8-i6-skept::skept", "rec_id": "hv2-cyclen-odd_k8-i6", "axis": "cyclen", "difficulty": "odd_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition x7 is true if proposition x0 is not true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition x2 is true if proposition x3 is not true. Proposition g0_0 is true. Proposition x4 is true if proposition x5 is not true. Proposition g2_0 is true. Proposition t6 is true if proposition t7 is true and proposition wide is true. Proposition x0 is true if proposition x1 is not true. Proposition g1_1 is true. Proposition t7 is true if proposition btrue is true. Proposition t1 is true if proposition t2 is true and proposition cq is true. Proposition g2_1 is true. Proposition t3 is true if proposition t4 is true. Proposition t0 is true if proposition t1 is true. Proposition g1_0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t5 is true if proposition t6 is true. Proposition x1 is true if proposition x2 is not true. Proposition btrue is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition t2 is true if proposition t3 is true. Proposition x6 is true if proposition x7 is not true. Proposition x3 is true if proposition x4 is not true. Proposition g3_1 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition cq is true if proposition x0 is true. Proposition t4 is true if proposition t5 is true. Proposition g0_1 is true. Proposition q is true if proposition t0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition x5 is true if proposition x6 is not true. Proposition g3_0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k8-i6-wfs::wfs", "rec_id": "hv2-cyclen-odd_k8-i6", "axis": "cyclen", "difficulty": "odd_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition x7 is true if proposition x0 is not true. Proposition p1 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition x2 is true if proposition x3 is not true. Proposition g0_0 is true. Proposition x4 is true if proposition x5 is not true. Proposition g2_0 is true. Proposition t6 is true if proposition t7 is true and proposition wide is true. Proposition x0 is true if proposition x1 is not true. Proposition g1_1 is true. Proposition t7 is true if proposition btrue is true. Proposition t1 is true if proposition t2 is true and proposition cq is true. Proposition g2_1 is true. Proposition t3 is true if proposition t4 is true. Proposition t0 is true if proposition t1 is true. Proposition g1_0 is true. Proposition wide is true if proposition p0 is true and proposition p1 is true. Proposition t5 is true if proposition t6 is true. Proposition x1 is true if proposition x2 is not true. Proposition btrue is true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition s2 is true if proposition g2_0 is true and proposition g2_1 is true. Proposition t2 is true if proposition t3 is true. Proposition x6 is true if proposition x7 is not true. Proposition x3 is true if proposition x4 is not true. Proposition g3_1 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition cq is true if proposition x0 is true. Proposition t4 is true if proposition t5 is true. Proposition g0_1 is true. Proposition q is true if proposition t0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true. Proposition x5 is true if proposition x6 is not true. Proposition g3_0 is true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k8-i7-none::none", "rec_id": "hv2-cyclen-odd_k8-i7", "axis": "cyclen", "difficulty": "odd_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "none", "gold": null, "prompt": "Answer using ordinary commonsense reasoning about the rules below.\n\nRules:\nProposition g0_0 is true. Proposition x5 is true if proposition etrue is true and proposition x6 is not true. Proposition x4 is true if proposition etrue is true and proposition x5 is not true. Proposition g2_0 is true. Proposition cq is true if proposition x0 is true. Proposition etrue is true. Proposition x3 is true if proposition etrue is true and proposition x4 is not true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true. Proposition g3_0 is true. Proposition g1_1 is true. Proposition g3_2 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition g0_1 is true. Proposition t6 is true if proposition t7 is true. Proposition p1 is true if proposition s2 is true and proposition s3 is true. Proposition x6 is true if proposition etrue is true and proposition x7 is not true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t7 is true if proposition btrue is true and proposition cq is true. Proposition q is true if proposition t0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true and proposition g3_2 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition t3 is true if proposition t4 is true and proposition wide is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition t4 is true if proposition t5 is true. Proposition x7 is true if proposition etrue is true and proposition x0 is not true. Proposition s2 is true if proposition g2_0 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition g3_1 is true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s3 is true. Proposition t1 is true if proposition t2 is true. Proposition t0 is true if proposition t1 is true. Proposition t2 is true if proposition t3 is true. Proposition btrue is true. Proposition t5 is true if proposition t6 is true. Proposition g1_0 is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k8-i7-closed_world::closed_world", "rec_id": "hv2-cyclen-odd_k8-i7", "axis": "cyclen", "difficulty": "odd_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "closed_world", "gold": "C", "prompt": "Use the CLOSED-WORLD ASSUMPTION with NEGATION-AS-FAILURE, interpreted operationally as in Prolog-style reasoning. A positive goal is 'true' if it can be derived by a terminating proof from the rules. A positive goal is 'false' if every attempted proof of it finitely fails. The default-negated goal not G is 'true' if the goal G finitely fails, and 'false' if G succeeds. If evaluating the goal does not terminate, gets stuck because of negation (flounders), or otherwise cannot produce a definite success or finite failure, answer 'Cannot be determined.' Answer 'Definitely yes' for true, 'Definitely no' for 'false', and 'Cannot be determined' when no definite operational result is obtained.\n\nRules:\nProposition g0_0 is true. Proposition x5 is true if proposition etrue is true and proposition x6 is not true. Proposition x4 is true if proposition etrue is true and proposition x5 is not true. Proposition g2_0 is true. Proposition cq is true if proposition x0 is true. Proposition etrue is true. Proposition x3 is true if proposition etrue is true and proposition x4 is not true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true. Proposition g3_0 is true. Proposition g1_1 is true. Proposition g3_2 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition g0_1 is true. Proposition t6 is true if proposition t7 is true. Proposition p1 is true if proposition s2 is true and proposition s3 is true. Proposition x6 is true if proposition etrue is true and proposition x7 is not true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t7 is true if proposition btrue is true and proposition cq is true. Proposition q is true if proposition t0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true and proposition g3_2 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition t3 is true if proposition t4 is true and proposition wide is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition t4 is true if proposition t5 is true. Proposition x7 is true if proposition etrue is true and proposition x0 is not true. Proposition s2 is true if proposition g2_0 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition g3_1 is true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s3 is true. Proposition t1 is true if proposition t2 is true. Proposition t0 is true if proposition t1 is true. Proposition t2 is true if proposition t3 is true. Proposition btrue is true. Proposition t5 is true if proposition t6 is true. Proposition g1_0 is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k8-i7-cred::cred", "rec_id": "hv2-cyclen-odd_k8-i7", "axis": "cyclen", "difficulty": "odd_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "cred", "gold": "A", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with CREDULOUS, also called BRAVE, reasoning. An answer set is a self-consistent set of atoms that is closed under the program rules and contains exactly the atoms justified by those rules under the assumptions made in that same set. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' if the statement holds in AT LEAST ONE answer set. Answer 'Definitely no' if the statement holds in no answer sets. If the program has no answer sets, answer 'Definitely no.'\n\nRules:\nProposition g0_0 is true. Proposition x5 is true if proposition etrue is true and proposition x6 is not true. Proposition x4 is true if proposition etrue is true and proposition x5 is not true. Proposition g2_0 is true. Proposition cq is true if proposition x0 is true. Proposition etrue is true. Proposition x3 is true if proposition etrue is true and proposition x4 is not true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true. Proposition g3_0 is true. Proposition g1_1 is true. Proposition g3_2 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition g0_1 is true. Proposition t6 is true if proposition t7 is true. Proposition p1 is true if proposition s2 is true and proposition s3 is true. Proposition x6 is true if proposition etrue is true and proposition x7 is not true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t7 is true if proposition btrue is true and proposition cq is true. Proposition q is true if proposition t0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true and proposition g3_2 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition t3 is true if proposition t4 is true and proposition wide is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition t4 is true if proposition t5 is true. Proposition x7 is true if proposition etrue is true and proposition x0 is not true. Proposition s2 is true if proposition g2_0 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition g3_1 is true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s3 is true. Proposition t1 is true if proposition t2 is true. Proposition t0 is true if proposition t1 is true. Proposition t2 is true if proposition t3 is true. Proposition btrue is true. Proposition t5 is true if proposition t6 is true. Proposition g1_0 is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k8-i7-skept::skept", "rec_id": "hv2-cyclen-odd_k8-i7", "axis": "cyclen", "difficulty": "odd_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "skept", "gold": "B", "prompt": "Use STABLE-MODEL, also called ANSWER-SET, semantics with SKEPTICAL, also called CAUTIOUS, reasoning. Consider all answer sets of the program. Each answer set is a self-consistent scenario that is closed under the rules and contains exactly the atoms justified by those rules under its own assumptions about default negation. Equivalently, after evaluating each default-negated condition not A relative to a candidate set, the candidate must be exactly the minimal set of atoms derivable from the remaining rules. For the queried statement: Answer 'Definitely yes' only if the statement holds in EVERY answer set. Answer 'Definitely no' if there is at least one answer set in which the statement does not hold. If the program has no answer sets, then the statement vacuously holds in every answer set; answer 'Definitely yes.'\n\nRules:\nProposition g0_0 is true. Proposition x5 is true if proposition etrue is true and proposition x6 is not true. Proposition x4 is true if proposition etrue is true and proposition x5 is not true. Proposition g2_0 is true. Proposition cq is true if proposition x0 is true. Proposition etrue is true. Proposition x3 is true if proposition etrue is true and proposition x4 is not true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true. Proposition g3_0 is true. Proposition g1_1 is true. Proposition g3_2 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition g0_1 is true. Proposition t6 is true if proposition t7 is true. Proposition p1 is true if proposition s2 is true and proposition s3 is true. Proposition x6 is true if proposition etrue is true and proposition x7 is not true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t7 is true if proposition btrue is true and proposition cq is true. Proposition q is true if proposition t0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true and proposition g3_2 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition t3 is true if proposition t4 is true and proposition wide is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition t4 is true if proposition t5 is true. Proposition x7 is true if proposition etrue is true and proposition x0 is not true. Proposition s2 is true if proposition g2_0 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition g3_1 is true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s3 is true. Proposition t1 is true if proposition t2 is true. Proposition t0 is true if proposition t1 is true. Proposition t2 is true if proposition t3 is true. Proposition btrue is true. Proposition t5 is true if proposition t6 is true. Proposition g1_0 is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."} {"id": "hv2-cyclen-odd_k8-i7-wfs::wfs", "rec_id": "hv2-cyclen-odd_k8-i7", "axis": "cyclen", "difficulty": "odd_k8", "n_stable_models": 2, "divergence_bin": "even_one_sided", "cond": "wfs", "gold": "C", "prompt": "Use WELL-FOUNDED semantics, with three truth values: 'true', 'false', and 'undefined'. A statement is 'true' if it has founded support: that is, it follows from rules whose positive conditions are 'true' and whose default-negated conditions are 'false', with the justification ultimately grounded rather than relying only on unsupported circular reasoning. A statement is 'false' if all possible rules that could derive it are defeated, inapplicable, or depend only on unfounded circular support. A statement is 'undefined' if it is neither founded true nor founded false, typically because its truth depends on an unresolved cycle through default negation or on other undefined statements. Answer 'Definitely yes' if the statement is 'true'. Answer 'Definitely no' if the statement is 'false'. Answer 'Cannot be determined' if the statement is 'undefined'. For a default-negated query not G, evaluate it using the well-founded truth value of G: not G is 'true' when G is 'false', 'false' when G is 'true', and 'undefined' when G is 'undefined'.\n\nRules:\nProposition g0_0 is true. Proposition x5 is true if proposition etrue is true and proposition x6 is not true. Proposition x4 is true if proposition etrue is true and proposition x5 is not true. Proposition g2_0 is true. Proposition cq is true if proposition x0 is true. Proposition etrue is true. Proposition x3 is true if proposition etrue is true and proposition x4 is not true. Proposition wide is true if proposition p0 is true and proposition p1 is true and proposition p2 is true. Proposition g3_0 is true. Proposition g1_1 is true. Proposition g3_2 is true. Proposition s0 is true if proposition g0_0 is true and proposition g0_1 is true. Proposition g0_1 is true. Proposition t6 is true if proposition t7 is true. Proposition p1 is true if proposition s2 is true and proposition s3 is true. Proposition x6 is true if proposition etrue is true and proposition x7 is not true. Proposition p0 is true if proposition s0 is true and proposition s1 is true and proposition s2 is true and proposition s3 is true. Proposition t7 is true if proposition btrue is true and proposition cq is true. Proposition q is true if proposition t0 is true. Proposition s3 is true if proposition g3_0 is true and proposition g3_1 is true and proposition g3_2 is true. Proposition s1 is true if proposition g1_0 is true and proposition g1_1 is true. Proposition t3 is true if proposition t4 is true and proposition wide is true. Proposition x1 is true if proposition etrue is true and proposition x2 is not true. Proposition t4 is true if proposition t5 is true. Proposition x7 is true if proposition etrue is true and proposition x0 is not true. Proposition s2 is true if proposition g2_0 is true. Proposition x2 is true if proposition etrue is true and proposition x3 is not true. Proposition g3_1 is true. Proposition p2 is true if proposition s0 is true and proposition s1 is true and proposition s3 is true. Proposition t1 is true if proposition t2 is true. Proposition t0 is true if proposition t1 is true. Proposition t2 is true if proposition t3 is true. Proposition btrue is true. Proposition t5 is true if proposition t6 is true. Proposition g1_0 is true. Proposition x0 is true if proposition etrue is true and proposition x1 is not true.\n\nQuestion: Is proposition q true?\n\nChoose exactly one:\n A. Definitely yes\n B. Definitely no\n C. Cannot be determined\n\nThink step by step, then end with a line 'ANSWER: X' where X is A, B, or C."}