W. Marek and M. Truszczy nski. Autoepistemic logic. Journal of the Association for Computing Machinery, 38:588--619, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....M4 = that has an empty minimal model. Theorem 1. Given a ground RRL program P the question whether there exists a stable model M of P is NP complete. Proof. The NP hardness follows directly from the fact that the problem of existence of a stable model for a normal logic program is NP complete [4] and all normal logic programs are RRL programs. If we guess the model M we can construct the reduct P M in a linear time with respect to the number of literals in rule bodies and the minimal model MM(P M ) can be computed in linear time [2] This implies that the problem is in NP and thus it ....

W. Marek and M. Truszczynski. Autoepistemic logic. Journal of the Association for Computing Machinery, 38:588-619, 1991.

....both configurations have the same weight, the next maximize statement is considered. 3.4 COMPUTATIONAL COMPLEXITY Theorem 3.4.1 The problem of finding a stable model for a ground RL program is NP complete. Proof. As the problem of finding a stable model of a normal logic program is NP complete [21] and all normal programs are also RL programs, the corresponding problem for RL is automatically NP hard. To prove the completeness we show how a RL program P can be reduced into a normal logic program in polynomial time. First, we replace all choice rules of the form fhg l 1 ; l m : ....

W. Marek and M. Truszczy nski. Autoepistemic logic. Journal of the Association for Computing Machinery, 38:588--619, 1991.

....stable model of P , because its reduct is p r and its unique minimal model is fg. However, P does have another stable model, namely fs; qg. Hence, a program may possess multiple stable models, one, or none at all. The problem of deciding whether a ground program has stable models is NP complete [9]. Indeed, to build a stable model it is enough to guess which atoms will appear non negated, and then verify uniqueness in polynomial time using the deductive closure of the reduct of the program with respect to this set. Smodels. smodels is a C implementation of logic programming with stable ....

V. Marek and M. Truszczynski. Autoepistemic logic. Journal of the Association for Computing Machinery, 38(3):588-619, 1991.

....sg is not a stable model of P , because its reduct is p r and its unique minimal model is fg. However P does have another stable model fs; qg. Hence, a program may posses multiple stable models, one or none at all. The problem of deciding whether a ground program has stable models is NPcomplete [10]. Indeed, to build a stable model it is enough to guess which atoms will appear non negated, and then verify uniqueness in polynomial time using the deductive closure of the reduct of the program with respect to this set. Smodels. smodels [12] is a C implementation of logic programming for ....

V. Marek and M. Truszczynski. Autoepistemic logic. Journal of the Association for Computing Machinery, 38(3):588-619, 1991.

....stable model of P , because its reduct is p r and its unique minimal model is fg. However, P does have another stable model, namely fs; qg. Hence, a program may possess multiple stable models, one, or none at all. The problem of deciding whether a ground program has stable models is NP complete [8]. Indeed, to build a stable model it is enough to guess which atoms will appear non negated, and then verify uniqueness in polynomial time using the deductive closure of the reduct of the program with respect to this set. 3 Smodels. smodels is a C implementation of logic programming with ....

V. Marek and M. Truszczynski. Autoepistemic logic. Journal of the Association for Computing Machinery, 38(3):588-619, 1991.

.... stable model is P 2 complete and the complexity remains essentially the same even if negative body literals are not allowed [EG95] However, if disjunctions are not allowed, the complexity drops one level: deciding whether a normal (non disjunctive) program has a stable model is NP complete [MT91] For many practical purposes this subclass of default theories is sucient. Our aim is to employ the techniques devised for default logic and show how they can be further developed to exploit the more limited structure of logic programs. This is the approach used in a system called smodels ....

W. Marek and M. Truszczynski. Autoepistemic logic. Journal of the Association for Computing Machinery, 38:588-619, 1991.

....forms. Using the terminology of default logic 1 , the deductive approach is to take logic program rules as defaults. That is a logic program is taken as a default theory. Then we use concepts like extensions or weak extensions associated with the theory to provide the program with a meaning (see [66]) Note that there are various classifications of different approaches to semantic theories for logic programs with nonmonotonic negation. In [105] semantic theories are classified into the program completion approach and the canonical model approach. In [33] they are divided into ....

....the O semantics [81] The relationship between these semantic theories for logic programming is also studied in [33] and [34] 5. 7 Stable Semantics The stable model semantics was introduced in Gelfond [46] further developed by Gelfond and Lifschitz in [47] and also by Marek and Truszczynski in [66]. The original stable model semantics is based on a two valued framework. Subsequently, various different but equivalent three valued versions of stable model semantics were proposed, including 3 stable models [85] partial stable models [95] and preferred extensions [37] In this section, we only ....

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V. W. Marek and M. Truszczynski. Autoepistemic logic. Journal of the Association for Computing Machinery, 38(3):588--619, 1991.

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W. Marek and M. Truszczy nski. Autoepistemic logic. Journal of the Association for Computing Machinery, 38:588--619, 1991.

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W. Marek and M. Truszczynski. Autoepistemic logic. Journal of the Association for Computing Machinery, 38:588-619, 1991.

....of P . Before addressing the complexity of the process, we comment on the relation between stable model semantics and non monotonic reasoning. Stable model semantics is closely related to a number of knowledge representation formalisms: Stable model semantics relates with autoepistemic logic [15]. A stable model of a program can be interpreted as a possible set of beliefs that a rational agent might hold when the agent s behaviour is de ned by program s rules. Stable model semantics relates to Reiter s default logics [25] Within default logic, stable models are extensions to ....

....and check whether S matches with the closure of the reduct. However, once a candidate set of atoms is provided, it is possible to check if it consists in a stable model in quadratic time. From this, satis ability is easily seen to be decidable in NP. Moreover a matching lower bound was proved in [15]. 19 Theorem 3.13. The problem to decide whether a nite propositional logic program has a stable model is NP complete. But we can ask for the computation of models satisfying certain conditions. In this case we provide a subset of the solution set and the computation searches stable models in ....

V.W. Marek and M. Truszczynski. Autoepistemic logic. Journal of the Association for Computing Machinery, 38(3):588-619, 1991.

....is either xed to some constant d or it is unlimited; and Function symbols are either allowed or not. The main complexity results are presented in Table 1. The model complexities of function free normal logic programs with the stable model semantics have been presented in earlier literature [10, 4]. The corresponding complexity classes of function free restricted programs are the same so we see that at least in these categories restricted programs are as expressive as normal logic programs. Since the model problem of the unrestricted case is 2 NEXP complete, we know that restricted ....

W. Marek and M. Truszczynski. Autoepistemic logic. Journal of the Association for Computing Machinery, 38:588-619, 1991.

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