Michael Gelfond, Halina Przymusinska, and Teodor Przymusinski. On the relationship between circumscription and negation as failure. Artificial Intelligence, 38(1):75--94, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the Careful Closed World Assumption (CCWA) by Gelfond Mailing address: TU Wien, Paniglgasse 16, A 1040 Wien, Austria. Internet e mail: feiter,gottlobg vexpert.dbai.tuwien.ac. at and Przymusinska [13] and the Extended Closed World Assumption (ECWA) by Gelfond, Przymusinska, and Przymusinski [12]. Circumscription was introduced by McCarthy in [22] It is known that circumscription as de ned in [36] coincides with the ECWA in the case of propositional logic [12] While much work has been devoted to the study of the logical properties of such forms of closed world reasoning and of their ....

.... and Przymusinska [13] and the Extended Closed World Assumption (ECWA) by Gelfond, Przymusinska, and Przymusinski [12] Circumscription was introduced by McCarthy in [22] It is known that circumscription as de ned in [36] coincides with the ECWA in the case of propositional logic [12]. While much work has been devoted to the study of the logical properties of such forms of closed world reasoning and of their applicability in di erent contexts, the interest in a complexity analysis of these methods has emerged only more recently [2, 20, 21, 29] Papalaskari and Weinstein [37] ....

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M. Gelfond, H. Przymusinksa, and T. Przymusinski. On the Relationship Between Circumscription and Negation as Failure. Arti cial Intelligence, 38:75-94, 1989.

....are properly included in the atomic sentences from P in I. This notion of minimality is a propositional version of the minimality in parallel circumscription [71] where the atomic sentences P are minimized and the atomic sentences not in F or P are allowed to vary. Furthermore, Gelfond and et al. [36] show that parallel circumscription corresponds to a particular form of the negation as failure rule (the extended closed world assumption) We use the following notations. The complement OE of a formula OE of the form :OE is OE and for other formulae it is :OE. For a set of formulae P we ....

M. Gelfond, H. Przymusinska, and T. Przymusinski. On the relationship between circumscription and negation as failure. Artificial Intelligence, 38:75--94, 1989.

....about the inputs that later may be adjusted by other (higher level) layers. These assumptions typically take the form of the Closed World Assumption (CWA) by minimizing a predicate in the layer s input language (Extended CWA, a generalization of CWA, was shown to be equivalent to circumscription [8]) More formally, for a set of axioms, A, let L(A) be the set of nonlogical symbols (predicates, functions, and constants) that appear in A. Also, let (A) be the FOL language built using the symbols in L(A) a language here is the set of all FOL sentences that can be built from those symbols) ....

M. Gelfond, H. Przymusinska, and T. C. Przymusinski. On the relationship between circumscription and negation as failure. Artificial Intelligence, 38 ( 1):75-94, Feb. 1989.

....Section 6. The problem of dealing with preferences in the context of nonmonotonic rule systems has attracted extensive interest in the past decades. In fact, for almost every nonmonotonic approach there exist prioritised versions designed to handle preference information of some form (see, e.g. [Gelfond et al. 1989; Konolige, 1988; Rintanen, 1994; Nebel, 1998; Eiter and Gottlob, 1995; Brewka, 1989; Brewka, 1996] Prioritised versions of default logic and logic programming 33 under the answer set semantics includes [Baader and Hollunder, 1993; Brewka, 1994; Rintanen, 1998b; Sakama and Inoue, 1996; ....

M. Gelfond, H. Przymusinska, and T. Przymusinski. On the relationship between circumscription and negation as failure. Artificial Intelligence, 38(1):75--94, February 1989.

....by McCarthy [McC80] is perhaps the most well known and extensively studied such formalism. It enjoys high expressive power and thus is suitable for modeling a wide variety of problems requiring nonmonotonic reasoning. Moreover, propositional circumscription has been shown by Gelfond et al. [GPP89] to coincide with reasoning under the extended closed world assumption (ECWA) which is one of the main formalisms for reasoning with incomplete information. Research partially supported by the Research Committee of the University of Patras and by the Computer Technology Institute. y ....

M. Gelfond, H. Przymusinska, and T. Przymusinksi. On the relationship between circumscription and negation as failure. Artificial Intelligence, 38:273--287, 1989.

....where preferences among defaults can be represented via an explicit preference relation and where the logical machinery is extended accordingly. Indeed, for all major nonmonotonic formalisms, such prioritized versions have been proposed in the past. Among them are prioritized circumscription [12], hierarchic autoepistemic logic [15] prioritized theory revision [2, 19] prioritized logic programming [25, 28] or prioritized abduction [10] Also several prioritized versions of Reiter s default logic, the logic we are dealing with in this paper, have been described in the literature [16, ....

M. Gelfond, H. Przymusinska, and T. Przymusinski. On the Relationship Between Circumscription and Negation as Failure. AIJ, 38:75--94, 1989.

....model of an extended disjunctive logic program. The Partial Disjunctive Stable Model Semantics (PDSM) by Przymusinski [49] which extends the Well Founded Semantics of van Gelder, Ross, and Schlipf [65] The Iterated Closed World Assumption (ICWA) by Gelfond, Przymusinska, and Przymusinski [28]. The Perfect Models Semantics (PERF) by Przymusinski [48] Various forms of the Closed World Assumption (CWA) including the Generalized CWA (GCWA) by Minker [43] the Extended GCWA (EGCWA) by Yahya and Henschen [67] the Careful CWA (CCWA) by Gelfond and Przymusinska [26] and the Extended ....

.... [48] Various forms of the Closed World Assumption (CWA) including the Generalized CWA (GCWA) by Minker [43] the Extended GCWA (EGCWA) by Yahya and Henschen [67] the Careful CWA (CCWA) by Gelfond and Przymusinska [26] and the Extended CWA (ECWA) by Gelfond, Przymusinska, and Przymusinski [28], which coincides in the nite propositional case with McCarthy s circumscription (CIRC) 41, 42] and further variants such as the Disjunctive Database Rule (DDR) of Ross and Topor [55] which is equivalent to the Weak GCWA (WGCWA) of Rajasekar, Lobo, and Minker [51] and the Possible Models ....

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M. Gelfond, H. Przymusinska, and T. Przymusinski. On the Relationship Between Circumscription and Negation as Failure. Articial Intelligence, 38:75-94, 1989.

....where preferences among defaults can be represented via an explicit preference relation and where the logical machinery is extended accordingly. Indeed, for all major nonmonotonic formalisms,such prioritized versions have been proposed in the past. Among them are prioritized circumscription (Gelfond et al. 1989), hierarchic autoepistemic logic (Konolige, 1988) prioritized theory revision (Benferhat et al. 1993; Nebel, 1998) prioritized logic programming 1 GERHARD BREWKA AND THOMAS EITER (Sakama and Inoue, 1996; Zhang and Foo, 1997) or prioritized abduction (Eiter and Gottlob, 1995) Also several ....

Gelfond, M., Przymusinska, H., and Przymusinski, T. (1989). On the Relationship Between Circumscription and Negation as Failure. Artificial Intelligence, 38:75--94.

....consistent) normal programs T into inconsistent theories CWA(T , L) In the context of stratified programs, the situation can be significantly improved by replacing the general closed world assumption CWA by the stratified closed world assumption SCWA. An equivalent notion is introduced in [7] and denoted as iterated closed world assumption. We prefer the name stratified closed world assumption in order to distinguish it from the hierarchical closed world assumption (to be introduced later) which is also generated by iterating the CWA in a suitable way. Definition 4.2. Let T be a ....

....# RA (a) ## a # A, 11) for all a # U L . Hence, we have shown that the relation A is an element of Def L (SCWA(T,L) for some stratified program of height m. # Remark 4.2. This theorem can also be obtained by combining results of Apt and Blair [4] and Gelfond, Przymusinska and Przymusinski [7]. However, our approach is conceptually di#erent and develops the definability theory of stratified programs from the more general point of view of inductive definability. We think that this provides a more perspicuous approach to stratified programs and reveals the close connections between ....

Gelfond, M., Przymusinska, H., and Przymusinski, T., On the Relationship between Circumscription and Negation as Failure, Artificial Intelligence, 38:75-- 94 (1989).

....by the rule q j not q . In this sense, we can see that fixed predicates play the same role as abducible predicates in abductive logic programming. In classical logic programming, every predicate is usually minimized under the closed world reasoning. Fixed predicates are also considered in ECWA [24], which is equivalent to circumscription under some conditions. We will show that ECWA without varying predicates can be simply computed through GEDPs. From the viewpoint of nonmonotonic reasoning, among many nonmonotonic formalisms, Moore s autoepistemic logic can express a stable expansion ....

....lies; so we conclude that it does not fly unless it is a bird. In classical logic programming, every predicate is usually minimized in a PDP by GCWA [48] in which the answer sets of the program are exactly the minimal Herbrand models. An exception can be seen in ECWA proposed by Gelfond et al. [24], which is equivalent to circumscription in the existence of the unique name and domain closure assumptions. We now formalize ECWA for PDPs without varying predicates. Let T be a PDP consisting of rules of the form A 1 j . j A k B 1 ; Bm (k; m 0) where A i s and B j s are ....

Gelfond, M., Przymusinska, H., and Przymusinski, T., On the Relationship between Circumscription and Negation as Failure, Artificial Intelligence 38:75--94 (1989).

....non Horn clauses, the least Herbrand model does not exist in general and some preferred models are often introduced as the intended meaning of the database. For example, supported models [ABW88] or perfect models [Prz88] are the intended models for stratified databases, and the iterated CWA (ICWA) [GPP89] provides negative information as the complement of such a model. These results are extended to the extended CWA (ECWA) GPP89] which is also relating to the circumscription [Mc80] in AI. By contrast, the so called disjunctive databases are indefinite databases which usually have multiple ....

....meaning of the database. For example, supported models [ABW88] or perfect models [Prz88] are the intended models for stratified databases, and the iterated CWA (ICWA) GPP89] provides negative information as the complement of such a model. These results are extended to the extended CWA (ECWA) [GPP89], which is also relating to the circumscription [Mc80] in AI. By contrast, the so called disjunctive databases are indefinite databases which usually have multiple minimal models. For inferring negation from 3 This is a revised version of the paper which is in the Proceedings of the 1st ....

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Gelfond, M., Przymusinska, H. and Przymusinski, T. C., On the Relationship between Circumscription and Negation as Failure, Artificial Intelligence 38, 75--94, 1989.

....Spencer s clause trees 3 [HS95a] uses two stages. In the first stage, a set of Access Clause Trees is built from the intensional database. In the second, the Access Clause Trees are resolved against the extensional database, giving the answers. 2 Disjunctive Stratified Databases Definition 1 [Prz88, GPP89]. Let S be the set of predicate symbols of a disjunctive normal database DB, such that S is partitioned into sets S 1 ; S r , called strata. Let Stratum(a) i if and only if the predicate symbol of the atom a is in Stratum(S i ) Then DB is called a disjunctive stratified database (DSDB) ....

M. Gelfond, H. Przymusinska, and T. C. Przymusinski. On the relationship between circumscription and negation as failure. AI, 38:75--94, 1989.

....can be circumscribed using Lifschitz pointwise circumscription [15, 16] this gives a first order (possibly infinite) formula equivalent to the circumscription of P in W . Several works have highlighted the importance of consequence finding for answering queries about a circumscriptive theory [6, 23, 8, 7, 10]. Consequence finding is the task of computing a set of formulas entailed by a given set of premises and verifying certain properties. In order to answer circumscriptive queries, the consequences of interest are formulae that contain no predicate in R and only positive occurrences of predicates ....

.... form of completeness: the models of MARG( Gamma; r) have to be those of Th( Gamma) L r (in general, Gamma] #r [Th( Gamma) L r ] but the converse does not always hold) Several works have highlighted the importance of consequence finding for answering queries about a nonmonotonic theory [6, 23, 8, 7, 10]. Consequence finding is the task of computing a set of formulas entailed by a given set of premises and verifying certain properties. In order to answer queries about, for example, a circumscriptive theory, the consequences of interest are formulae that contain no predicate from a set R and only ....

Gelfond, M., Przymusi'nska, H., and Przymusi'nski, T. On the relationship between circumscription and negation as failure. Artificial Intelligence, 38(1):75--94, 1989.

....can be circumscribed using Lifschitz pointwise circumscription [12, 13] this gives a first order (possibly infinite) formula equivalent to the circumscription of P in W . Several works have highlighted the importance of consequence finding for answering queries about a circumscriptive theory [4, 20, 6, 5, 8]. Consequence finding is the task of computing a set of formulas entailed by a given set of premises and verifying certain properties. In order to answer circumscriptive queries, the consequences of interest are formulae that contain no predicate in R and only positive occurrences of predicates ....

Gelfond, M., Przymusi'nska, H., and Przymusi'nski, T. On the relationship between circumscription and negation as failure. Artificial Intelligence, 38(1):75--94, 1989.

....about the inputs that later may be adjusted by other (higher level) layers. These assumptions take the form of the Closed World Assumption (CWA) by minimizing a predicate in the layer s input language (Extended CWA, a generalization of CWA, was shown to be equivalent to circumscription [ Gelfond et al. 1989 ] More formally, let Layer i be the theory of layer i, and C i , a set of predicates in L(Layer i ) for which we wish to assert CWA. Then, subsumption is achieved for layer i by using the parallel circumscription policy Circ[Layer i ; C i ; L(Layer i ) When implemented, this formula ....

M. Gelfond, H. Przymusinska, and T. C. Przymusinski. On the relationship between circumscription and negation as failure. Artificial Intelligence, 38(1):75--94, February 1989.

....assumptions about the inputs that later may be adjusted by other (higher level) layers. These assumptions take the form of the Closed World Assumption (CWA) by minimizing a predicate in the layer s input language (Extended CWA, a generalization of CWA, was shown to be equivalent to circumscription [Gelfond et al. 1989] ) During each cycle of any particular layer, before applying any default assumptions, we include those sentences that were asserted in the layer s latches. Then, the layer s theorem prover can try to prove the appropriate goal for that layer. This grants higher layers the right to override ....

Michael Gelfond, Halina Przymusinska, and Teodor C. Przymusinski. On the relationship between circumscription and negation as failure. Artificial Intelligence, 38(1):75--94, February 1989.

....first statement says that if we believe both in F and in F G then we should also believe in G. The second inference rule says that if a formula F is derivable then it is believed (necessitation) The intended meaning of belief atoms BF is based on the principle of predicate minimization (see [24,19] and [23] BF holds if F is minimally entailed or, equivalently: BF holds if F is true in all minimal models. In order to make this intended meaning precise we first have to define what we mean by a minimal model of a belief theory. Definition 17 (Minimal Models [31] A model M of a belief ....

Michael Gelfond, Halina Przymusinska, and Teodor Przymusinski. On the Relationship between Circumscription and Negation as Failure. Artificial Intelligence, 38:75--94, 1989.

....size in logic L 2 . Translations of this kind have recently been investigated in the AI literature, specifically in the context of logics for common sense and nonmonotonic reasoning. As an example, several researchers investigated the feasibility of representing propositional circumscription [20] or propositional default theories [3] as purely propositional formulae. In some sense this is also a form of knowledge compilation, because reasoning in such non monotonic formalisms is typically a problem complete for the second level of the polynomial hierarchy. Let T and # be propositional ....

....p 2 complete. The proof (see appendix) makes use of the size of the varying part. Since inference in circumscription is # p 2 complete, some researchers proposed to use reasoning methods apt for NP complete problems, with some preprocessing. Namely, Gelfond, Przymuszinski and Przymuszinska [20] proposed to: 22 Please write authorrunninghead Author Name(s) in file 1. compute so called free for negation formulae, and add them to the knowledge base x; 2. use standard propositional theorem proving for the augmented knowledge base. In another approach, Nerode et al. 33] ....

M. Gelfond, H. Przymusinska, and T. Przymusinsky. On the relationship between circumscription and negation as failure. Artificial Intelligence, 38:49--73, 1989.

....Also in other application domains, like model based diagnosis, configuration or decision making, preferences play a fundamental role and their relevance is well recognized. Prioritized versions for most of the existing nonmonotonic formalisms have been proposed, e.g. prioritized circumscription [16], hierarchic autoepistemic logic [19] prioritized default logic [24, 5, 1] prioritized theory revision [3, 25] or prioritized abduction [11] Somewhat surprisingly, preferences have received less attention in logic programming. This may be explained by the fact that for a long period, logic ....

M. Gelfond, H. Przymusinska, and T. Przymusinski. On the Relationship Between Circumscription and Negation as Failure. Artificial Intelligence, 38:75--94, 1989.

....short, is usually employed to infer negative information from a disjunctive logic program (see [LMR92] for more discussion and details) GCWA allows one to assume an atom to be false, if it doesn t appear in any minimal model of the program. This has been independently extended for sentences in [YH85, GPP89]. All these versions are not fundamentally different, and for the purpose of this paper, we simply refer to the following definition of closed world assumption in disjunctive logic programming. Definition 2.1 (Minimal Model) Let D be a disjunctive logic program. A model M of D is a minimal model ....

Michael Gelfond, Halina Przymusinska, and Teodor Przymusinski. On the relationship between circumscription and negation as failure. Artificial Intelligence, 38:75--94, 1989.

....indicates, non monotonic knowledge bases must be equipped with a nonmonotonic semantics. Intuitively this means that we need to provide a meaning to the default negation atoms Not F . We want the intended meaning of default atoms Not F to be based on the principle of predicate minimization (see [Min82, GPP89] and [McC80] Not F holds if F is minimally entailed or, equivalently: Not F holds if F is false in all minimal models. In order to make this intended meaning precise we first have to define what we mean by a minimal model of a knowledge base. Definition 2.4 (Minimal Models [Prz97] A model ....

Michael Gelfond, Halina Przymusinska, and Teodor Przymusinski. On the Relationship between Circumscription and Negation as Failure. Artificial Intelligence, 38:75--94, 1989.

....knowledge bases must be equipped with a non monotonic semantics. Intuitively this means that we need to provide a meaning to the default negation atoms Not F . We want the intended meaning of default atoms Not F to be based on the principle of predicate minimization (see [Minker 1982; Gelfond et al. 1989] and [McCarthy 1980] Not F holds if :F is minimally entailed or, equivalently: Not F holds if F is false in all minimal models. 8 Delta S. Brass and J. Dix and T. C. Przymusinski In order to make this intended meaning precise we first have to define what we mean by a minimal model of a ....

Gelfond, M., Przymusinska, H., and Przymusinski, T. 1989. On the Relationship between Circumscription and Negation as Failure. Artificial Intelligence 38, 75--94.

....models which says that negation of C can be assumed by default if and only if C is false in all minimal models of the theory. In other words, we base the meaning of negation by default on the principle of predicate minimization or, equivalently, on the Generalized Closed World Assumption GCWA [Min82, GPP89], or, Circumscription CIRC [McC80] While our choice of the default formalism is very natural and seems to closely correspond to the intuitive meaning of negation in logic programs and deductive databases, other non monotonic default formalisms can be used in place of predicate minimization, ....

M. Gelfond, H. Przymusinska, and T. Przymusinski. On the relationship between circumscription and negation as failure. Journal of Artificial Intelligence, 38:75--94, 1989. 27

....of T then we write: T j= min F and say that F is minimally entailed 3 by T . The intended meaning of belief atoms B el F is based on predicate minimization: B el F j F is minimally entailed j F is true in all minimal models. Accordingly, beliefs in AELB can be called minimal beliefs (see [Min82, GPP89] and [McC80] As in Moore s Autoepistemic Logic, also in the Autoepistemic Logic of Beliefs the intended meaning of belief atoms is implemented by defining plausible sets of beliefs that an ideally rational and introspective agent may hold, given a set of premises T . Definition 2.5 (Static ....

Michael Gelfond, Halina Przymusinski, and Teodor C. Przymusinski. On the relationship between circumscription and negation as failure. Journal of Artificial Intelligence, 38(1):75--94, February 1989.

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Michael Gelfond, Halina Przymusinska, and Teodor Przymusinski. On the relationship between circumscription and negation as failure. Artificial Intelligence, 38(1):75--94, 1989.

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