W. Marek. Stable theories in autoepistemic logic. Fundamenta Informaticae, 12:243--254, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....case this result has already been established by Konolige [61] Corollary 3. 15 A set of sentences Sigma L has exactly one stable expansion SE Sigma ( Note that in the propositional case SE Sigma ( fOE 2 L ae j Sigma j= L OEg gives a compact characterization of Marek s operator E [79] which produces the unique stable expansion of an L free set of premises. Theorem 3.16 Let the underlying logic be a first order predicate calculus. Let Sigma be a set of sentences of L. Then SE Sigma ( is the unique stable set S such that Cn cl ( Sigma) S L. Proof. As SE Sigma ( is a ....

....and their relationships. The arrow is interpreted as implies. For example, strong groundedness implies stable set minimality. It seems clear that in order to obtain more tightly grounded expansions the construction of an expansion should start from the formulae without the L operator. Marek [79] presents such a construction (the operator E) which builds a stable set out of its L free part. The construction can be described compactly using the consequence relation j= L as discussed in Chapter 3. However, the method does not seem to be applicable to building expansions from premises ....

W. Marek. Stable theories in autoepistemic logic. Fundamenta Informaticae, 12:243--254, 1989.

....(#) of Example 2.2 the classical provability relation # by a new many valued one. At the moment we do not see a direct relationship between his and our approach. 3 Characterization of extensions In this section we will generalize methods developed in [17] which are related to results of [11] and [9] The main result is, that an autoepistemic extension E of a set T is uniquely determined by the values of E on belief subformulas of T . An immediate consequence of this result will be that an objective theory T has exactly one multi valued autoepistemic extension. The trivial key ....

V. W. Marek. Stable theories in autoepistemic logic. Fundamenta Informaticae, 12:243--254, 1989. 16

....F 2 Eg [ f:LF : F 62 Eg) Here F ranges over arbitrary formulas, and th(X) denotes the set of propositional consequences of X. If all formulas in A are objective, then (i) A has exactly one stable expansion E, and (ii) an objective formula belongs to E i it follows from A in propositional logic ([8], 6] For any logic program (without variables) I( stands for the set of formulas of autoepistemic logic obtained from by inserting L after every negation [5] By At we denote the set of atoms occurring in . Theorem 3. If a logic program has a unique stable model M , then I( has a ....

W. Marek, Stable Theories in Autoepistemic Logic, Unpublished Note, Department of Computer Science, University of Kentucky, 1986.

....by Reiter [Rei80] forms one of the most important formalisms for non monotonic reasoning. Recently, the relationship between default logics and other forms of non monotonic reasoning have been studied intensely (in particular the work of Konolige [Kon88] and the work of Marek and Truszczynski [MT89a, Mar89] show that default and auto epistemic logics are not that different after all) More recently, the relationship between logic program semantics and non monotonic logics has been studied by Marek and Truszczynski [MT89b] and by Marek and Subrahmanian [MS88] In all these cases, correspondences ....

W. Marek. Stable theories in autoepistemic logic. Fundamenta Informaticae, 12:243--254, 1989.

....the completed program. 5 Stable Models Gelfond introduced an approach to negation through stable models [10] and motivated it by appealing to autoepistemic logic, as developed by Moore [26] The theory has been further developed by Gelfond and Lifschitz [11] and also by Marek and Truszczynski [24, 23]. In this section we follow the definition of [11] which defines stability without reference to autoepistemic logic. We show that if a program has a total well founded model, that model is the unique stable model. We also discuss two programs which do not have total well founded models but do ....

W. Marek. Stable theories in autoepistemic logic. Technical report, University of Kentucky, 1986. (manuscript).

....with the stable expansions of ff in autoepistemic logic. It is not particularly difficult to add a similar operator to KL Gamma . The details have been worked out in the case of OBLIQUE . Epistemic states in autoepistemic logic have been characterized syntactically as so called stable sets [19]. It would be interesting to obtain a similar definition for KL Gamma . The main insight would be a precise characterization of a weaker form of first order logical consequence as a closure condition for those stable sets. Finally, a proof theory for KL Gamma still needs to be found. ....

Marek, W., Stable Theories in Autoepistemic Logic, Fundamenta Informaticae 12, 1989, pp. 243--254.

....the following three conditions: ST1 T = Cn(T ) ST2 If 2 T then L 2 T , ST3 If 62 T then :L 2 T . Stable theories capture the intuition of belief sets of an agent with full introspection capabilities, and are of fundamental importance in nonmonotonic modal formalisms. It is well known ([Moo85, Mar89]) that for every theory S L there is a unique stable theory T such that T L = Cn(S) We denote this unique stable theory by St(S) Stable sets have the following property, which we will refer to as the disjunctive property: if T is stable and formulas and are propositional combinations ....

W. Marek. Stable theories in autoepistemic logic. Fundamenta Informaticae, 12:243--254, 1989.

....mentioned above which we think are important for applications. The idea of interpreting the belief operator L by an operator on a lattice can also be found in [3] 3 Characterization of extensions In this section we will generalize methods developed in [13] which are related to results of [7] and [5] The main result is, that an autoepistemic extension E of a set T is uniquely determined by the values of E on belief subformulas of T . An immediate consequence of this result will be that an objective theory T has exactly one multi valued autoepistemic extension. Definition 4 Let T be ....

W. Marek. Stable theories in autoepistemic logic. Fundamenta Informaticae, 12:243-- 254, 1989.

....[GL88] developed the concept of stable semantics. Gelfond, Przymusinski and Przymusinska [GPP89] have extended the CWA to stratified programs and have developed the concept of an Iterated Closed World Assumption (ICWA) defined by iteration on the level of predicates. Marek and his co workers [MS88, Mar, MT89] have made contributions to auto epistemic reasoning and its relationship to logic programming and other forms of default rea soning. Van Gelder, Ross and Schlipf have developed the concept of the well founded approach [VRS88] that gives semantics to all general Horn logic programs including ....

W. Marek. Stable theories in autoepistemic logic. to appear in Fundamenta Informaticae.

....case this result has already been established by Konolige [61] Corollary 3. 15 A set of sentences Sigma L has exactly one stable expansion SE Sigma ( Note that in the propositional case SE Sigma ( fOE 2 L ae j Sigma j= L OEg gives a compact characterization of Marek s operator E [79] which produces the unique stable expansion of an L free set of premises. Theorem 3.16 Let the underlying logic be a first order predicate calculus. Let Sigma be a set of sentences of L. Then SE Sigma ( is the unique stable set S such that Cn cl ( Sigma) S L. Proof. As SE Sigma ( is a ....

....and their relationships. The arrow is interpreted as implies. For example, strong groundedness implies stable set minimality. It seems clear that in order to obtain more tightly grounded expansions the construction of an expansion should start from the formulae without the L operator. Marek [79] presents such a construction (the operator E) which builds a stable set out of its L free part. The construction can be described compactly using the consequence relation j= L as discussed in Chapter 3. However, the method does not seem to be applicable to building expansions from premises ....

W. Marek. Stable theories in autoepistemic logic. Fundamenta Informaticae, 12:243--254, 1989.

....) Cn L(H) T [ fBF : BF 2 L(H) and L(T Pi ) j= min Fg [ fLF : LF 2 L(H) and L(T Pi ) j= Fg [ f:LF : LF 2 L(H) and L(T Pi ) 6j= Fg) Proof. The proof is similar to the proof of the result establishing the existence of a unique stable autoepistemic expansion of an objective theory [Mar89] and thus it is only sketched in here. Let T 0 = T Pi and suppose that T n was already constructed for some natural n. Let: T n 1 = Cn(T n [ fLF : T n j= Fg [ f:LF : T n 6j= Fg) where the F s range over all formulae which involve at most n levels of nesting of the knowledge operator L and ....

....j= F j T j= LF: Similarly, an objective formula F is minimally entailed by T if and only if the belief atom BF belongs to T : T j= min F j T j= min F j T j= BF: Proof. It is known that any consistent objective theory T has a unique consistent stable autoepistemic expansion T [Mar89]. By the previous Theorem, there is a unique static expansion T of T such that T = T jKL . If T had another consistent static expansion T then its restriction to KL would have to coincide with T and thus T would have to coincide with T . In any static expansion T j= F j ....

W. Marek. Stable Theories in Autoepistemic Logic. Fundamenta Informaticae 12(1989), 243-254.

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