| { |
| "paper_id": "P84-1044", |
| "header": { |
| "generated_with": "S2ORC 1.0.0", |
| "date_generated": "2023-01-19T08:20:41.982114Z" |
| }, |
| "title": "REPRESENTING KNOWLEDGE ABOUT KNOWLEDGE AND MUTUAL KNOWLEDGE", |
| "authors": [ |
| { |
| "first": "Sald", |
| "middle": [], |
| "last": "Soulhi", |
| "suffix": "", |
| "affiliation": { |
| "laboratory": "", |
| "institution": "Equipe de Comprehension du Raisonnement Naturel LSI -UPS llg route de Narbonne", |
| "location": { |
| "postCode": "31062", |
| "settlement": "Toulouse", |
| "country": "FRANCE" |
| } |
| }, |
| "email": "" |
| } |
| ], |
| "year": "", |
| "venue": null, |
| "identifiers": {}, |
| "abstract": "In order to represent speech acts, in a multi-agent context, we choose a knowledge representation based on the modal logic of knowledge KT4 which is defined by Sato. Such a formalism allows us to reason about knowledge and represent knowledge about knowledge, the notions of truth value and of definite reference.", |
| "pdf_parse": { |
| "paper_id": "P84-1044", |
| "_pdf_hash": "", |
| "abstract": [ |
| { |
| "text": "In order to represent speech acts, in a multi-agent context, we choose a knowledge representation based on the modal logic of knowledge KT4 which is defined by Sato. Such a formalism allows us to reason about knowledge and represent knowledge about knowledge, the notions of truth value and of definite reference.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Abstract", |
| "sec_num": null |
| } |
| ], |
| "body_text": [ |
| { |
| "text": "Speech act representation and the language planning require that the system can reason about intensional concepts like knowledge and belief. A problem resolver must understand the concept of knowledge and know for example what knowledge it needs to achieve specific goals. Our assumption is that a theory of language is part of a theory of action (Austin [4] ).", |
| "cite_spans": [ |
| { |
| "start": 347, |
| "end": 358, |
| "text": "(Austin [4]", |
| "ref_id": null |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "I INTRODUCTION", |
| "sec_num": null |
| }, |
| { |
| "text": "Reasoning about knowledge encounters the problem of intensionality. One aspect of this problem is the indirect reference introduced by Frege ~] during the last century. Mc Carthy [15] presents this problem by giving the following example : Let the two phrases : Pat knows Mike's telephone number (I) and", |
| "cite_spans": [ |
| { |
| "start": 179, |
| "end": 183, |
| "text": "[15]", |
| "ref_id": null |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "I INTRODUCTION", |
| "sec_num": null |
| }, |
| { |
| "text": "Pat dialled Mike's telephone number (2) The meaning of the proposition \"Mike's telephone number\" in (I) is the concept of the telephone number, whereas its meaning in (2) is the number itself. Then if we have : \"Mary's telephone number = Mike's telephone number\", we can deduce that :", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "I INTRODUCTION", |
| "sec_num": null |
| }, |
| { |
| "text": "\"Pat dialled Mary's telephone number\" but we cannot deduce that :", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "I INTRODUCTION", |
| "sec_num": null |
| }, |
| { |
| "text": "\"Pat knows Mary's telephone number\", because Pat may not have known the equality mentioned above.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "I INTRODUCTION", |
| "sec_num": null |
| }, |
| { |
| "text": "Thus there are verbs like \"to know\", \"to believe\" and \"to want\" that create an \"opaque\" context. For Frege a sentence is a name, refe-rence of a sentence is its truth value, the sense of a sentence is the proposition. In an oblique context, the reference becomes the proposition. For example the referent of the sentence p in the indirect context \"A knows that p\" is a proposition and no longer a truth value.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "I INTRODUCTION", |
| "sec_num": null |
| }, |
| { |
| "text": "Me Carthy [15] and Konolige [I I] have adopted Frege's approach. They consider the concepts like objects of a firstorder language. Thus one term will denote Mike's telephone number and another will denote the concept of Mike's telephone number. The problem of replacing equalities by equalities is then avoided because the concept of Mike's telephone number and the number itself are different entities.", |
| "cite_spans": [ |
| { |
| "start": 28, |
| "end": 33, |
| "text": "[I I]", |
| "ref_id": null |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "I INTRODUCTION", |
| "sec_num": null |
| }, |
| { |
| "text": "Mc Carthy's distinction concept/object corresponds to Frege's sense/reference or to modern logicians' intension/extension. Maida and Shapiro [13] adopt the same approach but use propositional semantic networks that are labelled graphs, and that only represent intenslons and not extensions, that is to say individual concepts and propositions and not referents and truth values. We bear in mind that a semantic network is a graph whose nodes represent individuals and whose oriented arcs represent binary relations.", |
| "cite_spans": [ |
| { |
| "start": 123, |
| "end": 145, |
| "text": "Maida and Shapiro [13]", |
| "ref_id": null |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "I INTRODUCTION", |
| "sec_num": null |
| }, |
| { |
| "text": "Cohen E6], being interested in speech act planning, proposes the formalism of partitioned semantic networks as data base to represent an agent's beliefs. A partitioned semantic network is a labelled graph whose nodes and arcs are distributed into spaces. Every node or space is identified by its own label. Hendrix ~9] introduced it to represent the situations requiring the delimitation of information sub-sets. In this way Cohen succeeds in avoiding the problems raised by the data base approach. These problems are clearly identified by Moore FI7, 18] . For example to represent 'A does-not believe P', Cohen asserts Believe (A,P) in a global data base, entirely separated from any agent's knowledge base. But as Appelt ~] notes, this solution raised problems when one needs to combine facts from a particular data base with global facts to prove a single assertion. For example, from the assertion :", |
| "cite_spans": [ |
| { |
| "start": 540, |
| "end": 550, |
| "text": "Moore FI7,", |
| "ref_id": null |
| }, |
| { |
| "start": 551, |
| "end": 554, |
| "text": "18]", |
| "ref_id": null |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "I INTRODUCTION", |
| "sec_num": null |
| }, |
| { |
| "text": "know (John,Q) & know (John,P ~Q)", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "I INTRODUCTION", |
| "sec_num": null |
| }, |
| { |
| "text": "where P~ Q is in John's data base and ~ know (John,Q) is in the global data base, it should be possible to conclude % know (John,P) but a good strategy must be found ! In a nutshell, in this first approach which we will call a syntactical one, an agent's beliefs are identified with formulas in a first-order language, and propositional attitudes are modelled as relations between an agent and a formula in the object language, but Montague showed that modalities cannot consistently be treated as predicates applying to nouns of propositions.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "I INTRODUCTION", |
| "sec_num": null |
| }, |
| { |
| "text": "The other approach no longer considers the intenslon as an object but as a function from possible worlds to entities. For instance the intension of a predicate P is the function which to each possible world W (or more generally a point of reference, see Scott [23] ) associates the extension of P in W.", |
| "cite_spans": [ |
| { |
| "start": 254, |
| "end": 264, |
| "text": "Scott [23]", |
| "ref_id": null |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "I INTRODUCTION", |
| "sec_num": null |
| }, |
| { |
| "text": "This approach is the one that Moore D7,18] adopted. He gave a first-order axiomatization of Kripke The fundamental assumption that makes this translation possible, is that an attribution of any propositional attitude like \"to know\", \"to believe\", \"to remember\", \"to strive\" entails a division of the set of possible worlds into two classes : the possible worlds that go with the propositional attitude that is considered, and those that are incompatible with it. Thus \"A knows that P\" is equivalent to \"P is true in every world compatible with what A knows\".", |
| "cite_spans": [ |
| { |
| "start": 92, |
| "end": 98, |
| "text": "Kripke", |
| "ref_id": null |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "I INTRODUCTION", |
| "sec_num": null |
| }, |
| { |
| "text": "We think that possible worlds language is complicated and unintuitive, since, rather than reasoning directly about facts that someone knows, we reason about the possible worlds compatible with what he knows. This translation also presents some problems for the planning. For instance to establish that A knows that P, we must make P true in every world which is compatible with A's knowledge. This set of worlds is a potentially infinite set.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "I INTRODUCTION", |
| "sec_num": null |
| }, |
| { |
| "text": "The most important advantage of Moore's approach [17, 183 is that it gives a smart axiomatization of the interaction between knowledge and action.", |
| "cite_spans": [ |
| { |
| "start": 32, |
| "end": 53, |
| "text": "Moore's approach [17,", |
| "ref_id": null |
| }, |
| { |
| "start": 54, |
| "end": 57, |
| "text": "183", |
| "ref_id": null |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "I INTRODUCTION", |
| "sec_num": null |
| }, |
| { |
| "text": "Our approach is comprised in the general framework of the second approach, but instead of encoding Hintikka's modal logic of knowledge in a first-order language, we consider the logic of knowledge proposed by Mc Carthy, the decidability of which was proved by Sato [21] and we propose a prover of this logic, based on natural deduction.", |
| "cite_spans": [ |
| { |
| "start": 260, |
| "end": 269, |
| "text": "Sato [21]", |
| "ref_id": null |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "II PRESENTATION OF OUR APPROACH", |
| "sec_num": null |
| }, |
| { |
| "text": "We bear in mind that the idea of using the modal logic of knowledge in A.I. was proposed for the first time by Mc Carthy and Hayes [14] .", |
| "cite_spans": [ |
| { |
| "start": 131, |
| "end": 135, |
| "text": "[14]", |
| "ref_id": null |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "II PRESENTATION OF OUR APPROACH", |
| "sec_num": null |
| }, |
| { |
| "text": "A language L is a triple (Pr,Sp,T) where :", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "A. Languages", |
| "sec_num": null |
| }, |
| { |
| "text": ".Pr is the set of propositional variables, .Sp is the set of persons, .T is the set of positive integers. The language of classical propositional calculus is L = (Pr,6,~). SoCSp will also be denoted by 0 and will be called \"FOOL\".", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "A. Languages", |
| "sec_num": null |
| }, |
| { |
| "text": "The set of well formed formulas is defined to be the least set Wff such as :", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "B. Well Formed Formulas", |
| "sec_num": null |
| }, |
| { |
| "text": "(W|) PrC Wff (W 2) a,b-~ Wff implies aD b eWff (W 3) S6_Sp,t 6.T,aeWff implles(St)a~_Wff The symbol D denotes \"implication\". axiom. Now, we give the meaning of axioms :", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "B. Well Formed Formulas", |
| "sec_num": null |
| }, |
| { |
| "text": "(A2) says that what is known is true, that is to say that it is impossible to have false knowledge. If P is false, we cannot say : \"John knows that P\" but we can say \"John believes that P\". This axiom is the main difference between knowledge and belief. This distinction is important for planning because when an agent achieves his goals, the beliefs on which he bases his actions must generally be true.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "B. Well Formed Formulas", |
| "sec_num": null |
| }, |
| { |
| "text": "(A3) says that what FOOL knows at time t, FOOL knows at time t that anyone knows it at time t. FOOL's knowledge represents universal knowledge, that is to say all agents knowledge. (A4) says that what is known will remain true and that every agent can apply modus ponens, that is, he knows all the logical consequences of his knowledge. (A5) says that if someone knows something then he knows that he knows it. This axiom is often required to reason about plans composed of several steps. It will be referred to as the positive introspective axiom. (A6) is the rule of inference.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "B. Well Formed Formulas", |
| "sec_num": null |
| }, |
| { |
| "text": "lue.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "D. Representation of the notion of truth va-", |
| "sec_num": null |
| }, |
| { |
| "text": "We give a great importance to the representation of the notion of truth value of a proposition, for example the utterance :", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "D. Representation of the notion of truth va-", |
| "sec_num": null |
| }, |
| { |
| "text": "John knows whether he is taller than Bill (I) can be considered as an assertion that mentions the truth value of the proposition P = John is taller than Bill, without taking a position as to whether the latter is true or false.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "D. Representation of the notion of truth va-", |
| "sec_num": null |
| }, |
| { |
| "text": "In our formalism (I) is represented by :", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "D. Representation of the notion of truth va-", |
| "sec_num": null |
| }, |
| { |
| "text": "{John} P This disjunctive solution is also adopted by Allen and Perrault D]\" Maida and Shapiro [13] represent this notion by a node because the truth value is a concept (an object of thought).", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "D. Representation of the notion of truth va-", |
| "sec_num": null |
| }, |
| { |
| "text": "The representation of the notion of truth value is useful to plan questions : A speaker can ask a hearer whether a certain proposition is true, if the latter knows whether this proposition is true.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "D. Representation of the notion of truth va-", |
| "sec_num": null |
| }, |
| { |
| "text": "Let us consider a dialogue between two participants : A speaker S and a hearer H. The language is then reduced to : Sp = (O,H,S} and T = {l} Let P stand for the proposition : \"The description D in the context C is uniquely satisfied by E\". Clark and Marshall [5] give examples that show that for S to refer to H to some entity E using some description D in a context C, it is sufficient that P is a mutual knowledge; this condition is tantamount to (O)P is provable. Perrault and Cohen [20] show that this condition is too strong. They claim that an infinite number of conjuncts are necessary for successful reference :", |
| "cite_spans": [ |
| { |
| "start": 240, |
| "end": 262, |
| "text": "Clark and Marshall [5]", |
| "ref_id": null |
| }, |
| { |
| "start": 467, |
| "end": 490, |
| "text": "Perrault and Cohen [20]", |
| "ref_id": null |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "E. Representing definite descriptions in conversational systems :", |
| "sec_num": null |
| }, |
| { |
| "text": "(S) P& (S)(H) e& (S)(H)(S) e & ... with only a finite number of false conjuncts. Finally, Nadathur and Joshi ~9] give the following expression as sufficient condition for using D to refer to E : (S) BD (S)(H) P & ~ ((S) BO(S)~(O)P) where B is the conjunction of the set of sentences that form the core knowledge of S and ~ is the inference symbole.", |
| "cite_spans": [ |
| { |
| "start": 90, |
| "end": 112, |
| "text": "Nadathur and Joshi ~9]", |
| "ref_id": null |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "E. Representing definite descriptions in conversational systems :", |
| "sec_num": null |
| }, |
| { |
| "text": "Gentzen's goal was to build a formalism reflecting most of the logical reasonings that are really used in mathemati-cal proofs\u2022 He is the inventor of natural deduction (for classical and intultionistic logics). Sato ~|] defines Gentzen -type sysmen GT4 which is equivalent to KT4. We consider here, schStte-type system KT4' [22] which is a generalization of S4 and equivalent to GT4 (and thus to KT4), in order to avoid the thinning rule of the system GT4 (which introduces a cumbersome combinatory). Firstly, we are going to give some difinitions to introduce KT4'.", |
| "cite_spans": [ |
| { |
| "start": 324, |
| "end": 328, |
| "text": "[22]", |
| "ref_id": null |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "III SCHOTTE -TYPE SYSTEM KT4'", |
| "sec_num": null |
| }, |
| { |
| "text": "A. Inductive definition of positive and negative parts of a formula F Logical symbols are ~ and V. a. F is a positive part of F. b. If % A is a positive part of F, then A is a negative part of F. c. If ~ A is a negative part of F, then A is a positive part of F. d. If A V B is a positive part of F, then A and B are positive parts of F. Positive parts or negative parts which do not contain any other positive parts or negative parts are called minimal parts.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "III SCHOTTE -TYPE SYSTEM KT4'", |
| "sec_num": null |
| }, |
| { |
| "text": "The truth of a positive part implies the truth of the formula which contains this positive part.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "B. Semantic property", |
| "sec_num": null |
| }, |
| { |
| "text": "The falsehood of a negative part implies the truth of the formula which contains this negative part. ", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "B. Semantic property", |
| "sec_num": null |
| }, |
| { |
| "text": "An axiom is any formula of the form F[P+,P-] where P is a propositional variable.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "E. Axiom", |
| "sec_num": null |
| }, |
| { |
| "text": "F. Inference rules Any KT4' proof-figure can be transformed into a KT4' proof-figure with the same conclusion and without any cut as a rule of inference (hence, the rule (R4) is superfluous. The proof of this theorem is an extension of Sch~tte's one for $4'. This theorem allows derivations \"without detour\"\u2022", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "E. Axiom", |
| "sec_num": null |
| }, |
| { |
| "text": "(R!) F[(A V B)j V ~ A, FI(A V B) ] v ~ B ~ FL(A V B) J -- (R2) F[(St)A 3 V~A ~ FT(st)A~ (PO) ~(Su)A 1V ... V ~(Su)Am V ~(Ou)B. V ... V ~(Ou)Bn V", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "E. Axiom", |
| "sec_num": null |
| }, |
| { |
| "text": "A logical axiom is a formula of the form F[P+,P-]. A proof is an single-rooted tree of formulas all of whose leaves are logical axioms. It is grown upwards from the root, the rules (RI), (R2) and (R3) must be applied in a reverse sense. These reversal rules will be used as \"production rules\"\u2022 The meaning of each production expressed in terms of the progranting language PROLOG is an implication\u2022 It can be shown [24J that the following strategy is a complete proof procedure :", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "IV DECISION PROCEDURE", |
| "sec_num": null |
| }, |
| { |
| "text": "\u2022 The formula to prove is at the star-ring node;", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "IV DECISION PROCEDURE", |
| "sec_num": null |
| }, |
| { |
| "text": "\u2022 Queue the minimal parts in the given formula; \u2022 Grow the tree by using the rule (R|) in priority , followed by the rule (R2), then by the rule (R3).", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "IV DECISION PROCEDURE", |
| "sec_num": null |
| }, |
| { |
| "text": "The choice of the rule to apply can be done intelligently. In general, the choice of (RI) then (R2) increases the likelihood to find a proof because these (reversal) rules give more complex formulas. In the case where (R3) does not lead to a loss of formulas, it is more efficient to choose it at first\u2022 The following example is given to illustrate this strategy : We would like to express our sincerest thanks to Professor AndrOs Raggio who has guided and adviced us to achieve this work. We would like to express our hearty thanks to Professors Mario Borillo, Jacques Virbel and Luis Fari~as Del Cerro for their encouragments.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "IV DECISION PROCEDURE", |
| "sec_num": null |
| } |
| ], |
| "back_matter": [], |
| "bib_entries": { |
| "BIBREF0": { |
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| "text": "St)a means \"S knows a at time t\" <St>a (= % (St) ~ a) means \"a is possible for S at time t\". {St}a (= (St)a V (St) % a) means \"S knows whether a at time t\". C. Hilbert-type System KT4 The axiom schemata for KT4 are : At. Axioms of ordinary propositional logic A2. (St)a \u2022 a A3. (Ot)a ~ (Or) (St)a A4. (St) (a D b) ~ ((Su)a D(Su)b), where t 6 u A5. (St)a ~ (St) (St)a A6. If a is an axiom, then (St)a is an", |
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| "text": "C where (Su)A I ..... (Su)Am, (Ou)B I , ..., (Ou) B6 must appear as neg6tire parts in the conclusion,", |
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| "text": "ExampleTake (A4) as an example and let Fo denotes its equivalent version in our language (Fo is at the start node) :Fo = ~(St)(~a V b) V ~(Su)a V(Su)b where t < u P~ denotes positive parts and P? denotes I negative parts l P+ = {~(St)(~ a V b), %(Su)a,(Su)b}; 2 P = {(St)(~ a V b),(Su)a}; O By (R3) we have (no losses of formulas) : F l = ~(St)(% a V b) V %(Su)a V b \u00f7 PI = {%(St)(~ a V b), ~(Su)a,b} F-= {(St)(% a V b),(Su)a} By (~2) we have : F~ = F~ V ~,(~a V b", |
| "uris": null, |
| "type_str": "figure" |
| } |
| } |
| } |
| } |
