Lifschitz, V., and Woo, T. 1992. Answer sets in general nonmonotonic reasoning (preliminary report). In Proc. of the 3rd Int. Conf. on the Principles of Knowledge Representation and Reasoning (KR'92), 603--614. Morgan Kaufmann, Los Altos.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... be any propositional language (not containing the predicate assert 1) The extended language assert is defined inductively as follows: All propositional atoms in are propositional atoms in assert ; If each of L 0 , L n A well known extension to normal logic programs [10]. is a literal in assert (i.e. a propositional atom A or its default negation not A) then L 0 L 1 , L n is a generalized logic program rule over assert ; If R is a rule over assert then assert(R) is a propositional atom of assert ; Nothing else is a propositional ....

....as in DLP, where the most recent rules are set in force, and previous rules are valid (by inertia) insofar as possible, i.e. they are kept for as long as they do not conflict with more recent ones. In DLP, default negation is treated as in stable models of normal [7] and generalized programs [10]. Formally, a dynamic logic program is a sequence P 1 # # P n (also denoted P, where is a set of generalized logic programs indexed by 1, n) and its semantic is determined by let S, and let M be a set of propositional atoms of L. Then: Default s (M) not . ....

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V. Lifschitz and T. Woo. Answer sets in general non-monotonic reasoning (preliminary report). In KR'92. Morgan-Kaufmann, 1992.

....when describing interpretations and models. We say that a (2 valued) interpretation M of L is a stable model of a generalized logic program P if M = least(P [ M ) The class of generalized logic programs can be viewed as a special case of yet broader classes of programs, introduced earlier in [14], and, for the special case of normal programs, their semantics coincides with the stable models semantics [10] 3 Weighted Directed Acyclic Graphs A directed graph D = V; E) is a pair comprised of a nite set V of vertices and a nite set E of ordered pairs (v; w) called edges, where v and w ....

V. Lifschitz and T. Woo. Answer sets in general non-monotonic reasoning (preliminary report). In B. Nebel, C. Rich, and W. Swartout, editors, KR'92. MorganKaufmann, 1992.

....Ln g, by body (r) the set of all objective atoms in body(r) and by body (r) the set of all default atoms in body(r) We refer to body (r) as the prerequisites of r. Whenever a literal L is of the form not A, not L stands for the objective atom A. The semantics of generalized logic programs [12, 13] is de ned as a generalization of the stable model semantics [10] Here we use the de nition that appeared in [2] proven there equivalent to [12, 13] De nition 2 (Default assumptions) Let M be an interpretation of P . Then: Default(P; M) fnot A j6 9r 2 P : head(r) A and M j= body(r)g: ....

....prerequisites of r. Whenever a literal L is of the form not A, not L stands for the objective atom A. The semantics of generalized logic programs [12, 13] is de ned as a generalization of the stable model semantics [10] Here we use the de nition that appeared in [2] proven there equivalent to [12, 13]) De nition 2 (Default assumptions) Let M be an interpretation of P . Then: Default(P; M) fnot A j6 9r 2 P : head(r) A and M j= body(r)g: De nition 3 (Stable Models of Generalized Programs) Let P be a generalized logic program and M an interpretation of P . M is a (regular) stable model ....

V. Lifschitz and T. Woo. Answer sets in general non-monotonic reasoning (preliminary report). In B. Nebel, C. Rich, and W. Swartout (eds.) KR'92. MorganKaufmann, 1992.

....0 , by body(r) the set of literals fL 1 ; Ln g, by body (r) the set of all objective atoms in body(r) and by body (r) the set of all default atoms in body(r) Whenever a literal L is of the form not A, not L stands for the objective atom A. The semantics of generalized logic programs [7, 8] is de ned as a generalization of the stable model semantics [6] Here we use the de nition that appeared in [2] proven there equivalent to [7, 8] In the remainder of the paper, by (2valued) interpretation M of LK we mean any set of propositional variables from LK such that for any A in LK ....

....atoms in body(r) Whenever a literal L is of the form not A, not L stands for the objective atom A. The semantics of generalized logic programs [7, 8] is de ned as a generalization of the stable model semantics [6] Here we use the de nition that appeared in [2] proven there equivalent to [7, 8]) In the remainder of the paper, by (2valued) interpretation M of LK we mean any set of propositional variables from LK such that for any A in LK precisely one A or not A belongs to M . De nition 2 (Default assumptions) Let P be a set of generalized rules and Default(P; M) fnot A j6 9r 2 P ....

V. Lifschitz and T. Woo. Answer sets in general non-monotonic reasoning (preliminary report). In B. Nebel et al. (eds.), KR'92. Morgan-Kaufmann, 1992.

.... Revision programming was introduced and studied in [MT95, MT98] The formalism was shown to be closely related to logic programming with stable model semantics [MT98, PT97] In [MPT99] a simple correspondence of revision programming with the general logic programming system of Lifschitz and Woo [LW92] was discovered. Roots of another recent formalism of dynamic logic programming [ALP 98] can also be traced back to revision programming. Unannotated) revision rules come in two forms of in rules and out rules: in(a) in(a 1 ) in(a m ) out(b 1 ) out(b n ) 1) out(a) ....

V. Lifschitz and T.Y.C. Woo. Answer sets in general nonmonotonic reasoning. In Proceedings of the 3rd international conference on principles of knowledge representation and reasoning, KR '92, pages 603--614, San Mateo, CA, Morgan Kaufmann, 1992.

....(DLP) 1] where the most recent rules are put in force, and the previous rules are valid (by inertia) as far as possible, i.e. they are kept for as long as they do not conflict with more recent ones. In DLP, default negation is treated as in stable models of normal [8] and generalized programs [12]. Formally, a dynamic logic program is a sequence P 1 . P n (also denoted P, where is a set of generalized logic programs indexed by 1, n) and its semantic is determined by : L, let s S, and let M be a set of propositional atoms of L. Then: Default s (M) ....

V. Lifschitz and T. Woo. Answer sets in general non-monotonic reasoning (preliminary report). In B. Nebel, C. Rich, and W. Swartout, editors, KR'92. MorganKaufmann, 1992.

....a gen eralized logic program P if r(M) least (r(P) U r(M ) where r( univocally renames every default literal not A in a program or model into new atoms, say not A. The class of generalized logic programs can be viewed as a special case of yet broader classes of programs, introduced earlier in [14], and, for the special case of normal programs, their semantics coincides with the stable models semantics [8] Dynamic Logic Programming Next we recall the semantics of dynamic logic programming [2] A dynamic logic program is a sequence 7o . also denoted by ( 7) where 7 ) Ps: s is ....

V. Lifschitz and T. Woo. Answer sets in general non-monotonic reasoning (preliminary report). In Procs. of KR-92. Morgan-Kaufmann, 1992.

....a generalized logic program P if r(M) least (r(P) U r(M ) where r( univocally renames every default literal not A in a program or model into new atoms, say not A. The class of generalized logic programs can be viewed as a special case of yet broader classes of programs, introduced earlier in [13], and, for the special case of normal programs, their semantics coincides with the stable models semantics [7] In [2] the reader can find the motivation for the usage of generalized logic programs, instead of using simple denials as a result of freely moving the head not s into the body. 2.2 ....

V. Lifschitz and T. Woo. Answer sets in general non-monotonic reasoning (preliminary report). In Procs. of KR-92. Morgan-Kaufmann, 1992.

....then this deletion causes A to be false. In the updates setting, the CWA must be explicitly encoded from the start, by making all not A false in the initial program being updated. That is, the two concepts, deletion and CWA, are orthogonal and must be separately incorporated. In the stable models [26, 32] and well founded semantics [12] of single generalized programs, the CWA is adopted ab initio, and default negation in the heads is con ated with non provability because there is no updating and thus no deletion. Note however that, unlike with single generalized programs (cf. 26] in updates the ....

....body literal in the denial would be moved to the head. Example 12 shows just that. We now recall the semantics of single generalized logic programs. The class of generalized logic programs can be viewed as a special case of yet broader classes of programs, introduced earlier in [26] and in [32]. As shown in [4] their semantics coincides with the stable models semantics [19] for the special case of normal programs. Moreover, the semantics also coincides with the one in [32] and, consequently, with the one in [26] when the latter is restricted to the language of generalized programs. ....

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V. Lifschitz and T. Woo. Answer sets in general non-monotonic reasoning (preliminary report). In B. Nebel, C. Rich, and W. Swartout, editors, KR'92. MorganKaufmann, 1992.

.... Revision programming was introduced and studied in [MT95, MT98] The formalism was shown to be closely related to logic programming with stable model semantics [MT98, PT97] In [MPT99] a simple correspondence of revision programming with the general logic programming system of Lifschitz and Woo [LW92] was discovered. Roots of another recent formalism of dynamic logic programming [ALP 98] can also be traced back to revision programming. Unannotated) revision rules come in two forms of in rules and out rules: in(a) in(a 1 ) in(a m ) out(b 1 ) out(b n ) 1) and ....

V. Lifschitz and T.Y.C. Woo. Answer sets in general nonmonotonic reasoning. In Proceedings of the 3rd international conference on principles of knowledge representation and reasoning, KR '92, pages 603--614, San Mateo, CA, Morgan Kaufmann, 1992.

....constraints. The main semantics of such disjunctive programs are built on the paradigm of minimal models [18] Hence, even simple choice rules cannot be translated to disjunctive rules without introducing new atoms. However, answer sets of general extended disjunctive programs, introduced in [37] as a subclass of the logic of minimal belief and negation as failure, allow non minimal models [53] In this class of programs negative literals can be used in the heads of the rules as in weight constraint rules and choice rules can be encoded without new atoms. Negative head literals are ....

V. Lifschitz and T.Y.C. Woo. Answer sets in general nonmonotonic reasoning (preliminary report). In Proceedings of the 3rd International Conference on Principles of Knowledge Representation and Reasoning, pages 603-614, Cambridge, MA, USA, October 1992. Morgan Kaufmann Publishers.

.... autoepistemic logic (see below) can be seen as a (non modal) reformulation of the corresponding results from [Lifschitz 1994] and [Lin and Shoham 1992] In addition, this establishes the correspondence between expansions and answer sets of logic programs of a most general kind (see below; cf. also [Lifschitz and Woo 1992, Inoue and Sakama 1994] 3.2 Modal Nonmonotonic Logics Recall that the semantic requirement corresponding to the Consistency rule is that, for any bistate (U; V ) V U . In this case, our bicomponent models can be seen as a proper generalization of the minimal model semantics for modal ....

V. Lifschitz and T. Woo (1992) Answer sets in general nonmonotonic reasoning (preliminary report). In Proc. Third Int. Conf. on Principles of Knowledge Representation and Reasoning, KR`92, Morgan Kaufmann, 1992, pages 603-614.

....atom. There are several semantics of ELPs [57] one of the most important is the concept of an answer set [77] discussed below, which generalize the concept of a stable model for NLPs [76] Similar definitions for GLPs and other classes of programs can be found in the literature (see, e.g. [108]) A (partial) interpretation is a set I of classical ground literals, i.e. literals without not, such that no opposite literals A; A belong to I simultaneously. A classical literal A (resp. A) is true in I , if A (resp. A) belongs to I , while a default literal notA (resp. not :A) is true ....

V. Lifschitz and T. Woo. Answer Sets in General Nonmonotonic Reasoning (Preliminary Report). In B. Nebel, C. Rich, and W. Swartout, editors, Proceedings of the Third International Conference on Principles of Knowledge Representation and Reasoning (KR'92), pages 603--614. Morgan Kaufmann, Oct. 1992.

....program in every possible way. Intuitively, the rule (1) is read as: if all L l 1 , Lm are believed and all Lm 1 , L n are disbelieved, then either some L i (1 # i # k) should be believed or some L j (k 1 # j # l) should be disbelieved. The class of GEDPs is introduced in [37,26] as a subclass of minimal belief and negation as failure (MBNF) 38] GEDPs are a fairly general class of existing LP languages in the sense that it includes the so called normal , disjunctive and extended logic programs. Moreover, it can also express the class of abductive logic programs, which ....

....An answer set S of a GEDP P is minimal if there is no other answer set S # of P s.t. S # # S. The set of all answer sets of P is written as AS P . The above definition of answer sets reduces to that of Gelfond and Lifschitz [20] in an EDP. Note that every answer set of any EDP is minimal [20,37], but the minimality of answer sets no longer holds for GEDPs. For example, suppose a program with the single rule L not L #, 4 saying, L is true or not. Then, it has two answer sets L and #. 2.2 Prioritized logic programs Next we introduce a prioritization mechanism to a program. ....

V. Lifschitz and T. Y. C. Woo, Answer sets in general nonmonotonic reasoning (preliminary report), in: Proceedings of the 3rd International Conference on Principles of Knowledge Representation and Reasoning, (Morgan Kaufmann, Los Altos, CA, 1992) 603--614.

....That is, the two concepts, deletion and CWA, are orthogonal and must be separately incorporated. Thus, in general, using logic programs extended with explicit negation [10] wouldn t be adequate, because explicitly negated heads express the negated is false, not just deleted. In the stable models [19,14] and well founded semantics [6] of single generalized programs, the CWA is adopted ab initio, and default negation in the heads is con ated with non provability because there is no updating and thus no deletion. Note however that, unlike with single generalized programs (cf. 14] in updates the ....

....would be moved to the head. Example 9 shows just that. In this section we recall the semantics of single generalized logic programs, as de ned in [1,2] The class of generalized logic programs can be viewed as a special case of yet broader classes of programs, introduced earlier in [14] and in [19]. As shown in [2] their semantics coincides with the stable models semantics [9] for the special case of normal programs. Moreover, the semantics also coincides with the one in [19] and, consequently, with the one in [14] when the latter is restricted to the language of generalized programs. ....

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V. Lifschitz and T. Woo. Answer sets in general non-monotonic reasoning (preliminary report). In B. Nebel, C. Rich, and W. Swartout, editors, KR'92. Morgan-Kaufmann, 1992.

.... Programs) A model M is a stable model of the generalized program P i M = least(P [ Default(P; M) It is easy to check that, for normal programs, this de nition is equivalent to the original de nition of stable models [18] As shown in [4] it also coincides with the semantics presented in [29] when the latter is restricted to the language of generalized programs. In DLP, sequences of generalized programs P 1 : Pn are given. As said before, intuitively a sequence may be viewed as the result of, starting with program P 1 , updating it with program P 2 , and updating it ....

V. Lifschitz and T. Woo. Answer sets in general non-monotonic reasoning (preliminary report). In B. Nebel, C. Rich, and W. Swartout, editors, KR'92. MorganKaufmann, 1992.

....the reader can nd the motivation for their introduction, instead of using simple denials by freely moving the head not s into the body. The class of generalized logic programs can be viewed as a special case of yet broader classes of programs, introduced earlier in [Inoue and Sakama, 1998] and in [Lifschitz and Woo, 1992] , and, for the special case of normal programs, their semantics coincides with the stable models semantics [Gelfond and Lifschitz, 1988] For our purposes, it will be convenient to syntactically represent generalized logic programs as propositional Horn theories. In particular, we will ....

V. Lifschitz and T. Woo. Answer sets in general non-monotonic reasoning (preliminary report) . In Procs. of (KR-92). Morgan-Kaufmann, 1992.

....in [7] suffered from some drawbacks and was not sufficiently general. We begin in Section 2 by defining the language of generalized logic programs, which allow default negation in rule heads. We describe stable model semantics of such programs as a special case of the approach proposed earlier in [8]. In Section 3 we define the program update P Phi U of the initial program P by the updating program U . In Section 4 we provide a complete characterization of the semantics of program updates P Phi U and in Section 5 we study their basic properties. In Section 6 we introduce the notion of ....

....In this section we introduce generalized logic programs and extend the stable model semantics of normal logic programs [3] to this broader class of programs 6 . The class of generalized logic programs can be viewed as a special case of a yet broader class of programs introduced earlier in [8]. While our definition is different and seems to be simpler than the one used in [8] when restricted to the language that we are considering, the two definitions can be shown to be equivalent 7 . It will be convenient to syntactically represent generalized logic programs as propositional Horn ....

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V. Lifschitz and T. Woo. Answer sets in general non-monotonic reasoning (preliminary report). In B. Nebel, C. Rich and W. Swartout, editors, Principles of Knowledge Representation and Reasoning, Proceedings of the Third International Conference (KR92), pages 603-614. Morgan-Kaufmann, 1992

.... Programs) A model M is a stable model of the generalized program P i M = least(P [ Default(P; M) It is easy to check that, for normal programs, this de nition is equivalent to the original de nition of stable models [9] As shown in [1] it also coincides with the semantics presented in [12] when the latter is restricted to the language of generalized programs. In DLP, sequences of generalized programs P 1 : Pn are given. Intuitively a sequence may be viewed as the result of, starting with program P 1 , updating it with program P 2 , and updating it with program Pn . ....

V. Lifschitz and T. Woo. Answer sets in general non-monotonic reasoning (preliminary report). In B. Nebel, C. Rich, and W. Swartout, editors, KR'92. MorganKaufmann, 1992.

....atoms in LK . De nition 2 (Generalized Logic Program) A generalized logic program P in the language LK is a (possibly in nite) set of propositional rules of the 2 The class of generalized logic programs can be viewed as a special case of a yet broader class of programs introduced earlier in [15]. 5 form L L 1 ; L n where L; L 1 ; L n are literals. If none of the literals appearing in heads of rules of P are default ones, then we say that the logic program P is normal. By a (2 valued) interpretation M of LK we mean any set of atoms from LK satisfying the condition ....

....literal and P [ M Lg As proven in [1] the class of stable models of generalized logic programs extends the class of stable models of normal programs [7] in the sense that, for the special case of normal programs, both semantics coincide. Moreover, the semantics also coincides with the one in [15] when the latter is restricted to the language of generalized programs. 3 Language for updates In our update framework, knowledge evolves from one knowledge state to another as a result of update commands stated in object language. Knowledge states KS i represent dynamically evolving states of ....

V. Lifschitz and T. Woo. Answer sets in general non-monotonic reasoning (preliminary report). In B. Nebel, C. Rich, and W. Swartout, editors, KR'92. Morgan-Kaufmann, 1992.

....example, note that, in application to (2) gives the formula :not p not oe Bp : This translation is essentially an extension of the translation from Section 5 of [7] to programs with nested expressions. The translation Pi of any program Pi is an MBNF theory of the special type studied in [9] a theory with protected literals. This means that every occurrence of an atom in an axiom of this theory is a part of an expression of the form BL or not L, where L is a literal. Recall that models of a propositional MBNF theory are defined as the pairs (I; S) where I is an interpretation ....

Vladimir Lifschitz and Thomas Woo. Answer sets in general nonmonotonic reasoning (preliminary report). In Bernhard Nebel, Charles Rich, and William Swartout, editors, Proc. Third Int'l Conf. on Principles of Knowledge Representation and Reasoning, pages 603--614, 1992.

.... n) is an objective atom, a default atom or a project. The following de nition introduces rules that are evaluated bottom up. To emphasize this aspect, we use a di erent notation for them. 1 For further motivation and intuitive reading of logic programs with default negations in the heads see [6]. I Let L 1 ; Ln (n 1) be objective or default atoms. An integrity constraint is an implication of the form: L 1 : Ln ) false and an action rule is an implication of the form: L 1 : Ln ) A where A is a project. Integrity constraints and action rules are called ....

M. Gelfond and V. Lifschitz. Answer sets in general non-monotonic reasoning (preliminary report). In B. Nebel, C. Rich, and W. Swartout, editors, KR'92, pages 603-614. Morgan-Kaufmann, 1992.

....the class of generalized logic programs 4 . Under the 2 valued semantics, the above situation results in an inconsistent update, even though the updated knowledge 4 The class of generalized logic programs can be viewed as a special case of a yet broader class of programs introduced earlier in [GL92] 12 base KB KB 0 does not contain any truly contradictory information, such as A and not A. In this section we extend our approach to the (3 valued) well founded semantics of generalized logic programs. This will enable us to model knowledge updates with non committal or unde ned outcome, as ....

M. Gelfond and V. Lifschitz. Answer sets in general non-monotonic reasoning (preliminary report). In B. Nebel, C. Rich, and W. Swartout, editors, KR'92. Morgan-Kaufmann, 1992.

....0 L 1 : Ln ) j (L 1 : Ln ) A) j (L 1 : Ln ) false) j ( L 1 : Ln ) Objective atoms, default atoms, projects and updates are generically called atoms. 1 For further motivation and intuitive reading of logic programs with default negations in the heads see [7]. I A generalized logic program P is a set of propositional Horn clauses of the form: L 0 L 1 : Ln (n 0) where L 0 is an objective or default atom, and every L i (1 i n) is a project, or an objective or default atom. If all the atoms appearing in heads of clauses of P are ....

M. Gelfond and V. Lifschitz. Answer sets in general non-monotonic reasoning (preliminary report). In B. Nebel, C. Rich, and W. Swartout, editors, KR'92, pages 603-614. Morgan-Kaufmann, 1992.

.... Programs) A model M is a stable model of the generalized program P i : M = least(P [ Default(P; M) It is easy to check that, for normal programs, this de nition is equivalent to the original de nition of stable models [11] As shown in [1] it also coincides with the semantics presented in [13] when the latter is restricted to the language of generalized programs. In DLP sequences of generalized programs P 1 : Pn are given. Intuitively such a sequence may be viewed as the result of, starting with program P 1 , updating it with program P 2 , and updating it with program Pn ....

M. Gelfond and V. Lifschitz. Answer sets in general non-monotonic reasoning (preliminary report). In B. Nebel, C. Rich, and W. Swartout, editors, KR'92. Morgan-Kaufmann, 1992.

....for each propositional symbol A the new symbol A. P is the normal program obtained from the generalized program P through replacing every negative head not A by A. 2 The class of generalized logic programs can be viewed as a special case of a yet broader class of programs introduced earlier in [6]. The definition of the stable models of generalized programs can now be obtained from the stable models of the program P . The idea is quite simple: since P is a normal program, its stable models can be obtained by the usual GelfondLifschitz definition [5] afterwards, all it remains to do is to ....

M. Gelfond and V. Lifschitz. Answer sets in general non-monotonic reasoning (preliminary report). In B. Nebel, C. Rich, and W. Swartout, editors, KR'92, pages 603--614. Morgan-Kaufmann, 1992.

....is a stable expansion of fi( Pi) Moreover, every stable expansion of fi( Pi) can be represented in the above form. 2 There are similar results describing mappings of disjunctive databases into reflexive autoepistemic logic [Sch91] and a logic of minimal belief and negation as failure called MBNF [LW92], LS92] 9.2 Defaults and Logic Programming A default is an expression of the form F G : MH 1 ; MH k ; 22) 65 where F; G; H 1 ; H k (k 0) are quantifier free formulas 22 . F is the consequent of the default, G is its prerequisite, and H 1 ; H k are its ....

....associated with the use of superclassical logics: existence of several natural, but nonequivalent translations from natural language statements into the formalism. Consider for instance Example 4.2. Simple disjunctive statement Matt s left or right hand is broken can be translated in, say, MBNF [LW92], as lh broken rh broken or as B lh broken B rh broken where B is the belief operator of MBNF. Only the second translation (probably the less obvious one) leads to the correct result. There are of course many remaining problems. Even though in many cases application of our techniques led to ....

V. Lifschitz and T. Woo. Answer sets in general nonmonotonic reasoning. In Proc. of the Third Int'l Conf. on Principles of Knowledge Representation and Reasoning, pages 603--614, 1992.

....one can be shifted onto another. The first of these two observations (the possibility to project revision problems onto problems with the empty database) allows us to establish a direct correspondence between revision programming and a version of logic programming proposed by Lifschitz and Woo [LW92] We will refer to this latter system as general disjunctive logic programming or, simply, general logic programming. In general logic programming both disjunction and negation as failure operators are allowed in the heads of rules. In this paper we study the relationship between revision ....

....as logic program clauses. As a consequence to our main result (Theorem 4) we obtain an alternative (and in some respects, simpler) connection between revision programming and logic programming. Namely, we establish a direct correspondence between revision programs and general logic programs of [LW92] 3 Shifting initial databases and programs In this section we will introduce a transformation of revision programs and databases that preserves justified revisions. Our results can be viewed as a generalization of the results from [MT98] on the duality between in and out in revision ....

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V. Lifschitz and T.Y.C. Woo. Answer sets in general nonmonotonic reasoning. In Proceedings of the 3rd international conference on principles of knowledge representation and reasoning, KR '92, pages 603--614, San Mateo, CA, 1992. Morgan Kaufmann.

.... Tang and Turner allow both the head and the body of a rule to contain negation as failure, conjunction ( and disjunction ( and these three operators can be nested arbitrarily [Lifschitz et al. 1999] In particular, negation as failure can occur in the head of a rule, as proposed in [Lifschitz and Woo, 1992] . For instance, a; not a (1) 1 is a rule with the empty body. It can be shown to have two answer sets: and fag. The rule a not not a (2) has the same answer sets as (1) According to the second proposal [Niemel a and Simons, 2000] rules are allowed to contain cardinality constraints ....

Vladimir Lifschitz and Thomas Woo. Answer sets in general nonmonotonic reasoning (preliminary report). In Bernhard Nebel, Charles Rich, and William Swartout, editors, Proc. Third Int'l Conf. on Principles of Knowledge Representation and Reasoning, pages 603--614, 1992.

....procedures for EDPs. ffl There are close relationships between the class of GEDPs and existing nonmonotonic formalisms, which are also natural extensions of previously known results [19, 39, 12] 1.1. Historical Background Historically, the class of GEDPs 1 was introduced by Lifschitz and Woo [41] as a subset of the logic of minimal belief and negation as failure (MBNF) MBNF was proposed by Lifschitz [38] as a general nonmonotonic logic that includes the class of logic programs permitting both classical negation and negation as failure. In fact, MBNF is one of the most expressive logics ....

....attractive, and each rule with negation as failure in the head can be regarded as a bisequent [8] that is, a pair of positive and negative beliefs appears in both the antecedent and the succedent of a sequent. The semantics of GEDPs is also clearly defined in terms of the notion of answer sets [41]. A unique feature of GEDPs, which distinguishes them from other traditional logic programs, is that the minimality of answer sets for EDPs [22] does not hold in general. For example, the program consisting of the rule p j not p has two answer sets: one containing p, and the other containing ....

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Lifschitz, V. and Woo, T. Y. C., Answer Sets in General Nonmonotonic Reasoning (Preliminary Report), in: B. Nebel, C. Rich, and W. Swartout (eds.), Principles of Knowledge Representation and Reasoning: Proceedings of the Third International Conference, Morgan Kaufmann, 1992, pp. 603--614.

.... Kunen, 1989 ] SLDNF calculi are further generalized in [ Lifschitz, 1995 ] and [ Lifschitz et al. 1995 ] The notion of an answer set for disjunctive programs without negation as failure in the heads of rules was defined in [ Gelfond and Lifschitz, 1991 ] This last limitation was removed by Lifschitz and Woo [1992]. Inoue and Sakama [1994] related negation as failure in heads to the important area of abductive logic programming [ Kakas et al. 1992 ] Default logic was invented by Reiter [1980] His definition of an extension was, historically, a source of the idea of an answer set; the work by Bidoit ....

Vladimir Lifschitz and Thomas Woo. Answer sets in general nonmonotonic reasoning (preliminary report). In Bernhard Nebel, Charles Rich, 55 and William Swartout, editors, Proc. Third Int'l Conf. on Principles of Knowledge Representation and Reasoning, pages 603--614, 1992.

....so X 2 is an answer set for (4) also. Proposition 1. For a program whose rules have form (5) the answer sets according to the definition of an answer set given above are exactly the consistent answer sets according to the definition from [13] 7 Unlike the definitions of an answer set given in [7,15,13], the definition introduced above does not allow for an inconsistent answer set. All three previous definitions agree, where they overlap, on the question of consistent answer sets, but there is some disagreement among them on inconsistent answer sets. For instance, according to [15] the program ....

....given in [7,15,13] the definition introduced above does not allow for an inconsistent answer set. All three previous definitions agree, where they overlap, on the question of consistent answer sets, but there is some disagreement among them on inconsistent answer sets. For instance, according to [15], the program not p has two answer sets: and the set of all literals. According to [13] it has only the empty answer set. In [7] negation as failure in the head is not considered. As another example, consider the program p ; p : According to [7] and [13] this program has no answer ....

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Vladimir Lifschitz and Thomas Woo. Answer sets in general nonmonotonic reasoning (preliminary report). In Bernhard Nebel, Charles Rich, and William Swartout, editors, Proc. Third Int'l Conf. on Principles of Knowledge Representation and Reasoning, pages 603--614, 1992.

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Lifschitz, V., and Woo, T. 1992. Answer sets in general nonmonotonic reasoning (preliminary report). In Proc. of the 3rd Int. Conf. on the Principles of Knowledge Representation and Reasoning (KR'92), 603--614. Morgan Kaufmann, Los Altos.

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V. Lifschitz and T. Woo. Answer sets in general non-monotonic reasoning. In B. Nebel, C. Rich, and W. Swartout, editors, KR-92, 1992.

No context found.

V. Lifschitz and T. Woo. Answer sets in general non-monotonic reasoning (preliminary report). In B. Nebel, C. Rich, and W. Swartout, editors, KR-92. MorganKaufmann, 1992.

No context found.

V. Lifschitz and T. Woo. Answer sets in general non-monotonic reasoning (preliminary report). In B. Nebel, C. Rich, and W. Swartout, editors, Proceedings of the 3th International Conference on Principles of Knowledge Representation and Reasoning (KR-92). Morgan-Kaufmann, 1992.

No context found.

V. Lifschitz and T. Woo. Answer sets in general non-monotonic reasoning (preliminary report). In B. Nebel, C. Rich, and W. Swartout, editors, KR'92. MorganKaufmann, 1992.

No context found.

V. Lifschitz and T. Woo. Answer sets in general non-monotonic reasoning (preliminary report). In Procs of KR'92. Morgan-Kaufmann, 1992.

No context found.

V. Lifschitz and T. Woo. Answer sets in general non-monotonic reasoning (preliminary report). In KR'92. Morgan-Kaufmann, 1992.

No context found.

V. Lifschitz and T. Woo. Answer sets in general non-monotonic reasoning (preliminary report). In KR'92. Morgan-Kaufmann, 1992.

No context found.

V. Lifschitz and T. Woo. Answer sets in general non-monotonic reasoning (preliminary report). In KR'92. Morgan-Kaufmann, 1992.

No context found.

V. Lifschitz and T. Woo. Answer sets in general non-monotonic reasoning (preliminary report). In B. Nebel, C. Rich, and W. Swartout, editors, Proceedings of the 3th International Conference on Principles of Knowledge Representation and Reasoning (KR-92), pages 603--614. Morgan-Kaufmann, 1992.

No context found.

V. Lifschitz and T. Woo. Answer sets in general nonmonotonic reasoning (preliminary report). In Proceedings of the Third International Conference on Principles of Knowledge Representation and Reasoning, pages 603614, 1992.

No context found.

V. Lifschitz and T. Woo. Answer sets in general non-monotonic reasoning (preliminary report). In B. Nebel, C. Rich, and W. Swartout, editors, Proceedings of the 3th International Conference on Principles of Knowledge Representation and Reasoning (KR-92). Morgan-Kaufmann, 1992.

No context found.

V. Lifschitz and T. Woo. Answer sets in general non-monotonic reasoning (preliminary report). In KR'92. Morgan-Kaufmann, 1992.

No context found.

V. Lifschitz and T. Woo. Answer sets in general non-monotonic reasoning (preliminary report). In Procs. of KR-92. Morgan-Kaufmann, 1992.

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V. Lifschitz and T. Woo. Answer sets in general non-monotonic reasoning (preliminary report). In Procs. of KR-9. Morgan-Kaufmann, 1992.

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V. Lifschitz and T. Woo. Answer sets in general non-monotonic reasoning (preliminary report). In B. Nebel, C. Rich, and W. Swartout, editors, Proceedings of the 3th International Conference on Principles of Knowledge Representation and Reasoning (KR-92). Morgan-Kaufmann, 1992.

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V. Lifschitz and T. Woo (1992) Answer sets in general nonmonotonic reasoning (preliminary report). In Proc. Third Int. Conf. on Principles of Knowledge Representation and Reasoning, KR`92, Morgan Kauman, 1992, pages 603-614.

No context found.

V. Lifschitz and T.C. Woo. Answer Sets in General Nonmonotonic Reasoning. In B. Nebel, C. Rich, and W. Swartout, editors, Proceedings KR'92, Morgan Kaufmann, 1992, pp. 603--614.

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