Datasets:
AlienLogic: the public demo world
The proof format, the standard rules, and the reference manual are those of the evaluation world (explorationbench/sandboxes/logic/engine.py). The demo world adds three side conditions; the evaluation world adds 24 of its own, and no demo condition is one of them.
Side conditions
| Id | Name | Condition |
|---|---|---|
| X01 | AND_E_LEFT_ONLY |
AND_E2 is disabled: from AND(φ, ψ) only the left conjunct φ can be extracted, with AND_E1. |
| X02 | IMPL_E_MAJOR_FIRST |
IMPL_E must cite the implication line first and its antecedent second: [IMPL_E i, j] where line i is IMPL(φ, ψ) and line j is φ. |
| X03 | BOT_E_TOP_LEVEL_ONLY |
BOT_E is refused inside an assumption box; it is only legal at indent 0. |
Fixed worked examples
Three accepted proofs and two refused ones, with the verdict the checker returns. A refusal carries only an opaque reason id.
S01. Modus Ponens, implication cited first. Verdict: ACCEPTED.
PROOF
premises: IMPL(p, q), p
goal: q
1. IMPL(p, q) [premise]
2. p [premise]
3. q [IMPL_E 1, 2]
END
S02. Left projection of a conjunction. Verdict: ACCEPTED.
PROOF
premises: AND(p, q)
goal: p
1. AND(p, q) [premise]
2. p [AND_E1 1]
END
S03. Explosion at the top level. Verdict: ACCEPTED.
PROOF
premises: p, NOT(p)
goal: q
1. p [premise]
2. NOT(p) [premise]
3. BOT [NOT_E 1, 2]
4. q [BOT_E 3]
END
R01. 用 AND_E2 取出右侧合取项。 Verdict: ALIEN_AXIOM_DISABLED / X01.
PROOF
premises: AND(p, q)
goal: q
1. AND(p, q) [premise]
2. q [AND_E2 1]
END
R02. IMPL_E 先引用前件，再引用蕴含式。 Verdict: ALIEN_SIDE_CONDITION / X02.
PROOF
premises: p, IMPL(p, q)
goal: q
1. p [premise]
2. IMPL(p, q) [premise]
3. q [IMPL_E 1, 2]
END
Held-out theorems
P01 · sanity
Premises: a, b. Goal: AND(a, b). Built around: none.
Reference proof:
PROOF
premises: a, b
goal: AND(a, b)
1. a [premise]
2. b [premise]
3. AND(a, b) [AND_I 1, 2]
END
P02 · single_rule
Premises: r, IMPL(r, t). Goal: t. Built around: IMPL_E_MAJOR_FIRST.
Reference proof:
PROOF
premises: r, IMPL(r, t)
goal: t
1. r [premise]
2. IMPL(r, t) [premise]
3. t [IMPL_E 2, 1]
END
The standard-calculus proof below is refused here (X02):
PROOF
premises: r, IMPL(r, t)
goal: t
1. r [premise]
2. IMPL(r, t) [premise]
3. t [IMPL_E 1, 2]
END
P03 · single_rule
Premises: NOT(p). Goal: IMPL(p, q). Built around: BOT_E_TOP_LEVEL_ONLY.
Reference proof:
PROOF
premises: NOT(p)
goal: IMPL(p, q)
1. NOT(p) [premise]
2. | p [assume]
3. | | NOT(q) [assume]
4. | | BOT [NOT_E 2, 1]
5. | NOT(NOT(q)) [NOT_I 3-4]
6. | q [DNE 5]
7. IMPL(p, q) [IMPL_I 2-6]
END
The standard-calculus proof below is refused here (X03):
PROOF
premises: NOT(p)
goal: IMPL(p, q)
1. NOT(p) [premise]
2. | p [assume]
3. | BOT [NOT_E 2, 1]
4. | q [BOT_E 3]
5. IMPL(p, q) [IMPL_I 2-4]
END
P04 · two_rule
Premises: AND(s, t), IMPL(s, t). Goal: t. Built around: AND_E_LEFT_ONLY, IMPL_E_MAJOR_FIRST.
Reference proof:
PROOF
premises: AND(s, t), IMPL(s, t)
goal: t
1. AND(s, t) [premise]
2. IMPL(s, t) [premise]
3. s [AND_E1 1]
4. t [IMPL_E 2, 3]
END
The standard-calculus proof below is refused here (X01):
PROOF
premises: AND(s, t), IMPL(s, t)
goal: t
1. AND(s, t) [premise]
2. IMPL(s, t) [premise]
3. t [AND_E2 1]
END
U01 · unprovable
Premises: AND(p, q). Goal: q. Built around: AND_E_LEFT_ONLY.
Unprovable. The correct answer is ANSWER: UNPROVABLE. Certificate: with AND_E2 gone every remaining rule is sound when AND(φ, ψ) is read as φ, and under that reading the assignment {'p': True, 'q': False} makes the premises true and the goal false.
A naive attempt the checker refuses:
PROOF
premises: AND(p, q)
goal: q
1. AND(p, q) [premise]
2. q [AND_E2 1]
END