## Twelve Definitions of a Stable Model

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Citations: | 19 - 1 self |

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@MISC{Lifschitz_twelvedefinitions,

author = {Vladimir Lifschitz},

title = {Twelve Definitions of a Stable Model},

year = {}

}

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### Abstract

This is a review of some of the definitions of the concept of a stable model that have been proposed in the literature. These definitions are equivalent to each other, at least when applied to traditional Prologstyle programs, but there are reasons why each of them is valuable and interesting. A new characterization of stable models can suggest an alternative picture of the intuitive meaning of logic programs; or it can lead to new algorithms for generating stable models; or it can work better than others when we turn to generalizations of the traditional syntax that are important from the perspective of answer set programming; or it can be more convenient for use in proofs; or it can be interesting simply because it demonstrates a relationship between seemingly unrelated ideas.

### Citations

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Citation Context ...d-there by (Y, Y ) but is not satisfied by (X, Y ) for any proper subset X of Y . Pearce showed that a set M of atoms is a stable model 4 See [44] for a detailed description of the situation calculus =-=[33]-=- as developed by the Toronto school.of a program Π iff (M, M) is an equilibrium model of the set of propositional formulas (10) corresponding to the grounded rules of Π. This Definition J is importan... |

1520 | The stable model semantics for logic programming
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Citation Context ...these two approaches to semantics of traditional programs follows from the fact that each of them is equivalent to Definition C of a stable model reviewed in the next section. This was established in =-=[14]-=- for the autoepistemic semantics and in [29] for the default logic approach. 5 Definition C, in Terms of the Reduct Definitions A and B are easy to understand – assuming that one is familiar with form... |

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Citation Context ...re syntactic transformations that make a formula stronger. The relationship between circumscription and program completion was investigated in [43].3.2 Default Logic A default theory in the sense of =-=[42]-=- is characterized by a set W of “axioms” – first-order sentences, and a set D of “defaults” – expressions of the form F : M G1, . . . , M Gn , (9) H where F, G1, . . . , Gn, H are first-order formulas... |

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Citation Context ...Completion According to [47], a logic program without negation represents the least (and so, the only minimal) Herbrand model of the corresponding set of Horn clauses. On the other hand, according to =-=[4]-=-, a logic program represents a certain set of first-order formulas, called the program’s completion. These two ideas are closely related to each other, but not equivalent. Take, for instance, the prog... |

859 | The well-founded semantics for general logic programs
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(Show Context)
Citation Context ...s fixpoint doesn’t depend on the choice of a stratification. A really good semantics should be a little easier to define. The stable model semantics, as well as the well-founded semantics proposed in =-=[49, 50]-=-, can be seen as an attempt to generalize and simplify the iterated fixpoint semantics of stratified programs. 3 Nonmonotonic Reasoning Many events in the history of research on stable models can be o... |

831 |
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Citation Context ...act q(b) is added to the program then this conclusion will be retracted. As to the stable model semantics, three nonmonotonic formalisms are particularly relevant. 3.1 Circumscription Circumscription =-=[31, 32]-=- is a syntactic transformation that turns a first-order sentence F into the conjunction of F with another formula, which expresses a minimality condition (the exact form of that condition depends on t... |

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Citation Context ...cterization of stable models (Sect. 12). This is an extended version of the conference paper [20]. 2 Minimal Models, Completion, and Stratified Programs 2.1 Minimal Models vs. Completion According to =-=[47]-=-, a logic program without negation represents the least (and so, the only minimal) Herbrand model of the corresponding set of Horn clauses. On the other hand, according to [4], a logic program represe... |

623 |
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Citation Context ...ects a “good” minimal model satisfying the program’s completion.Such a semantics was proposed in two papers presented at the 1986 Workshop on Foundations of Deductive Databases and Logic Programming =-=[1, 48]-=-. That approach was not without defects, however. First, it is limited to programs in which recursion and negation “don’t mix.” Such programs are called stratified. Unfortunately, some useful Prolog p... |

492 | Applications of circumscription to formalizing common sense knowledge
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Citation Context ...act q(b) is added to the program then this conclusion will be retracted. As to the stable model semantics, three nonmonotonic formalisms are particularly relevant. 3.1 Circumscription Circumscription =-=[31, 32]-=- is a syntactic transformation that turns a first-order sentence F into the conjunction of F with another formula, which expresses a minimality condition (the exact form of that condition depends on t... |

447 |
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Citation Context ...ults D. In Sect. 4 we will see that one of the earliest incarnations of the stable model semantics was based on treating rules as defaults in the sense of Reiter. 3.3 Autoepistemic Logic According to =-=[36]-=-, autoepistemic logic “is intended to model the beliefs of an agent reflecting upon his own beliefs.” Moore’s definition of propositional autoepistemic logic builds on the ideas of [35] and [34]. Form... |

431 |
Knowledge in action : logical foundations for specifying and implementing dynamical systems
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Citation Context ...tional formula F if F is satisfied in the logic of here-and-there by (Y, Y ) but is not satisfied by (X, Y ) for any proper subset X of Y . Pearce showed that a set M of atoms is a stable model 4 See =-=[44]-=- for a detailed description of the situation calculus [33] as developed by the Toronto school.of a program Π iff (M, M) is an equilibrium model of the set of propositional formulas (10) corresponding... |

340 |
Logic programs with classical negation
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Citation Context ... of finitely many rules of type (1), although generalizations of this syntactic form are mentioned several times in the second half of the paper. Some work on the stable model semantics, for instance =-=[15, 23, 39, 2]-=-, is not discussed here because it is about extending, rather than modifying, the definitions proposed earlier; this kind of work does not tell us much new about stable models of traditional programs.... |

330 | Logic programs with stable model semantics as a constraint programming paradigm
- Niemelä
- 1999
(Show Context)
Citation Context ...etween seemingly unrelated ideas. 1 Introduction Stable models of logic programs have been studied by many researchers, mainly because of their role in the foundations of answer set programming (ASP) =-=[30, 37]-=-. This programming paradigm provides a declarative approach to solving combinatorial search problems, and it has found applications in several areas of science and technology [21]. In ASP, a search pr... |

251 | Stable models and an alternative logic programming paradigm. CoRR cs.LO/9809032
- Marek, Truszczyński
- 1998
(Show Context)
Citation Context ...etween seemingly unrelated ideas. 1 Introduction Stable models of logic programs have been studied by many researchers, mainly because of their role in the foundations of answer set programming (ASP) =-=[30, 37]-=-. This programming paradigm provides a declarative approach to solving combinatorial search problems, and it has found applications in several areas of science and technology [21]. In ASP, a search pr... |

228 |
Non-monotonic logic I
- MCDERMOTT, DOYLE
- 1980
(Show Context)
Citation Context ...c According to [36], autoepistemic logic “is intended to model the beliefs of an agent reflecting upon his own beliefs.” Moore’s definition of propositional autoepistemic logic builds on the ideas of =-=[35]-=- and [34]. Formulas of this logic are constructed from atoms using propositional connectives and the modal operator L (“is believed”). Its semantics specifies, for any set A of formulas (“axioms”), wh... |

220 |
Unfounded sets and wellfounded semantics for general logic programs
- Gelder, Ross, et al.
- 1988
(Show Context)
Citation Context ...s fixpoint doesn’t depend on the choice of a stratification. A really good semantics should be a little easier to define. The stable model semantics, as well as the well-founded semantics proposed in =-=[49, 50]-=-, can be seen as an attempt to generalize and simplify the iterated fixpoint semantics of stratified programs. 3 Nonmonotonic Reasoning Many events in the history of research on stable models can be o... |

203 | ASSAT: Computing answer sets of a logic program by SAT solvers
- Lin, Zhao
- 2004
(Show Context)
Citation Context ...ogram Π iff (i) M satisfies Π, 1 and (ii) no non-empty subset of M is unfounded for Π with respect to M. 2 This Definition D can be refined using the concept of a loop, introduced many years later in =-=[27]-=-. A loop of a grounded program Π is a non-empty set L of ground atoms such that any elements A, A ′ of L can be connected by a chain A = A1, A2, . . . , Ak−1, Ak = A ′ (k > 1) of elements of L satisfy... |

199 | Strongly equivalent logic programs
- Lifschitz, Pearce, et al.
- 2001
(Show Context)
Citation Context ... viewed as abbreviations for propositional formulas [11]. Second, Definition J is a key to the theorem about the relationship between the concept of strong equivalence and the logic of here-and-there =-=[22]-=-. 11 Definitions K and L, in Terms of Modified Reducts In [7] the definition of the reduct reproduced in Sect. 5 is modified by including the positive members of the body, along with negative members,... |

131 | Stable models and non-determinism in logic programs with negation - Saccá, Zaniolo - 1990 |

131 |
Negation as Failure Using Tight Derivations for General Logic Programs
- Gelder
- 1989
(Show Context)
Citation Context ...ects a “good” minimal model satisfying the program’s completion.Such a semantics was proposed in two papers presented at the 1986 Workshop on Foundations of Deductive Databases and Logic Programming =-=[1, 48]-=-. That approach was not without defects, however. First, it is limited to programs in which recursion and negation “don’t mix.” Such programs are called stratified. Unfortunately, some useful Prolog p... |

113 | Nested expressions in logic programs
- Lifschitz, Tang, et al.
- 1999
(Show Context)
Citation Context ... of finitely many rules of type (1), although generalizations of this syntactic form are mentioned several times in the second half of the paper. Some work on the stable model semantics, for instance =-=[15, 23, 39, 2]-=-, is not discussed here because it is about extending, rather than modifying, the definitions proposed earlier; this kind of work does not tell us much new about stable models of traditional programs.... |

110 |
On strati autoepistemic theories
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- 1987
(Show Context)
Citation Context ...e formalisms turned out to be difficult. Discussing technical work on that topic is beyond the scope of this paper. 4 Definitions A and B, in Terms of Translations into Nonmonotonic Logic The idea of =-=[13]-=- is to think of the expression not A in a logic program as synonymous with the autoepistemic formula ¬LA (“A is not believed”). Since autoepistemic logic is propositional, the program needs to be grou... |

98 |
Nonnaonotonic Logic II: Non- monotonic Modal Theories
- McDermott
- 1982
(Show Context)
Citation Context ...ng to [36], autoepistemic logic “is intended to model the beliefs of an agent reflecting upon his own beliefs.” Moore’s definition of propositional autoepistemic logic builds on the ideas of [35] and =-=[34]-=-. Formulas of this logic are constructed from atoms using propositional connectives and the modal operator L (“is believed”). Its semantics specifies, for any set A of formulas (“axioms”), which sets ... |

93 | G.: Recursive aggregates in disjunctive logic programs: Semantics and complexity - Faber, Leone, et al. - 2004 |

72 | Answer sets for propositional theories
- Ferraris
- 2005
(Show Context)
Citation Context ...ense of Sect. 5, the modified reduct of a program has several minimal models. According to Definition K, M is a stable model of Π if M is a minimal model of the modified reduct of Π relative to M. In =-=[9]-=- the definition of the reduct is modified in a different way. The reduct of a program Π in the sense of Ferraris is obtained from the formulas (10) corresponding to the grounding rules of Π by replaci... |

70 |
Die formalen Regeln der intuitionistischen Logik, Sitzungsberichte der Preuischen Akademie der Wissenschaften, Phys.-mathem. Klasse
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- 1930
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Citation Context ...sor state axioms, which is standard in applications of the situation calculus. 10 Definition J, in Terms of Equilibrium Logic The logic of here-and-there, going back to the early days of modern logic =-=[16]-=-, is a modification of classical propositional logic in which propositional interpretations in the usual sense – assignments, or sets of atoms – are replaced by pairs (X, Y ) of sets of atoms such tha... |

69 | V.: A new perspective on stable models
- Ferraris, Lee, et al.
- 2007
(Show Context)
Citation Context ... semantics turned out to be equivalent to the generalization based on equilibrium logic that was mentioned at the end of Sect. 10. 12 Definition M, in Terms of Modified Circumscription The authors of =-=[10]-=- defined a modification of circumscription called the stable model operator, SM. According to their Definition M, an Herbrand interpretation is a stable model of Π iff it satisfies SM[F ] for the conj... |

65 | Logic Programs with ConsistencyRestoring Rules
- Balduccini, Gelfond
- 2003
(Show Context)
Citation Context ... of finitely many rules of type (1), although generalizations of this syntactic form are mentioned several times in the second half of the paper. Some work on the stable model semantics, for instance =-=[15, 23, 39, 2]-=-, is not discussed here because it is about extending, rather than modifying, the definitions proposed earlier; this kind of work does not tell us much new about stable models of traditional programs.... |

56 |
On the declarative semantics of logic programs with negation
- Lifschitz
- 1988
(Show Context)
Citation Context ...d-order quantifiers. Circumscription generalizes the concept of a minimal model from [47]. The iterated fixpoint semantics of stratified programs can be characterized in terms of circumscription also =-=[19]-=-. On the other hand, circumscription is similar to program completion in the sense that both are syntactic transformations that make a formula stronger. The relationship between circumscription and pr... |

52 |
Weight constraints as nested expressions. Theory and Practice of Logic Programming
- Ferraris, Lifschitz
(Show Context)
Citation Context ... of answer set programming, because many “nonstandard” constructs commonly used in ASP programs, such as choice rules and weight constraints, can be viewed as abbreviations for propositional formulas =-=[11]-=-. Second, Definition J is a key to the theorem about the relationship between the concept of strong equivalence and the logic of here-and-there [22]. 11 Definitions K and L, in Terms of Modified Reduc... |

48 |
Encodings for equilibrium logic and logic programs with nested expressions
- Pearce, Tompits, et al.
- 2001
(Show Context)
Citation Context ... any references to grounding [18]. Among the other definitions of a stable model discussed in this paper, Definition J, based on equilibrium logic, is the closest relative of Definition M. Indeed, in =-=[41]-=- the semantics of equilibrium logic is expressed by quantified Boolean formulas, and we can say that Definition M eliminated the need to ground the program using the fact that the approach of that pap... |

45 |
A rational reconstruction of nonmonotonic truth maintenance systems
- Elkan
- 1990
(Show Context)
Citation Context ... iff every element A of M is supported, in the sense that the program contains a rule (1) such that A0 = A, A1, . . . , Am ∈ M, Am+1, . . . , An ∈ M. (17) A stronger form of this condition, given in =-=[6]-=-, characterizes the class of stable models. According to his Definition G, an Herbrand model M of a grounded program is stable if there exists a well-ordering ≤ of M such that every element A of M is ... |

44 | What is answer set programming
- Lifschitz
(Show Context)
Citation Context ...rogramming (ASP) [30, 37]. This programming paradigm provides a declarative approach to solving combinatorial search problems, and it has found applications in several areas of science and technology =-=[21]-=-. In ASP, a search problem is reduced to computing stable models, and programs for generating stable models (“answer set solvers”) are used to perform search. This paper is a review of some of the def... |

42 |
Minimalism subsumes default logic and circumscription
- Bidoit, Froidevaux
- 1987
(Show Context)
Citation Context ...ne other feature of Definition A that makes it attractive is the simplicity of the underlying intuition: negation as failure expresses the absence of belief.The “default logic semantics” proposed in =-=[3]-=- is translational as well; it interprets logic programs as default theories. The head A0 of a rule (1) turns into the conclusion of the default, the conjunction A1 ∧ · · · ∧ Am of the positive members... |

35 |
A new fixpoint semantics for general logic programs compared with the well-founded and the stable model semantics
- FAGES
- 1991
(Show Context)
Citation Context ...classical logic that use auxiliary predicates. Programs that have no loops, such as (11), are called tight. The stable models of a tight program are identical to the Herbrand models of its completion =-=[8]-=-. Definition H [51] is based on a process of “tightening” that makes an arbitrary traditional program tight. This process uses two auxiliary symbols: the object constant 0 and the unary function const... |

35 |
A model-theoretic counterpart of loop formulas
- Lee
(Show Context)
Citation Context ...ogram (2) has 3 loops: p(a, b). p(a, a) ← p(a, a). p(a, b) ← p(b, a). p(b, a) ← p(a, b). p(b, b) ← p(b, b). (14) {p(a, a)}, {p(b, b)}, {p(a, b), p(b, a)}. (15) Program (11) has no loops. According to =-=[17]-=-, if we require in condition (i) above that M satisfy the completion of the program, rather than the program itself, then it will be possible to relax condition (ii) and require only that no loop cont... |

34 |
A new logical characterization of stable models and answer sets
- Pearce
- 1997
(Show Context)
Citation Context ... X ⊆ Y . (We think of X as the set of the atoms that are true “here”, and Y as the set of the atoms that are true “there.”) The semantics of this logic defines when (X, Y ) satisfies a formula F . In =-=[40]-=-, the logic of here-and-there was used as a starting point for defining a nonmonotonic logic closely related to stable models. According to Pearce, a pair (Y, Y ) is an equilibrium model of a proposit... |

34 |
Circumscription implies predicate completion (sometimes
- Reiter
- 1982
(Show Context)
Citation Context ...is similar to program completion in the sense that both are syntactic transformations that make a formula stronger. The relationship between circumscription and program completion was investigated in =-=[43]-=-.3.2 Default Logic A default theory in the sense of [42] is characterized by a set W of “axioms” – first-order sentences, and a set D of “defaults” – expressions of the form F : M G1, . . . , M Gn , ... |

26 | A reductive semantics for counting and choice in answer set programming
- Lee, Lifschitz, et al.
- 2008
(Show Context)
Citation Context ...y to formulas of an arbitrarily complex logical form. This fact makes it possible to give a semantics for an ASP language with choice rules and the count aggregate without any references to grounding =-=[18]-=-. Among the other definitions of a stable model discussed in this paper, Definition J, based on equilibrium logic, is the closest relative of Definition M. Indeed, in [41] the semantics of equilibrium... |

25 | A Study of Nonmonotonic Reasoning - Lin - 1991 |

17 |
consistent and computable completions for unrestricted logic programs
- Tight
- 1993
(Show Context)
Citation Context ...at use auxiliary predicates. Programs that have no loops, such as (11), are called tight. The stable models of a tight program are identical to the Herbrand models of its completion [8]. Definition H =-=[51]-=- is based on a process of “tightening” that makes an arbitrary traditional program tight. This process uses two auxiliary symbols: the object constant 0 and the unary function constant s (“successor”)... |

14 | Rules as actions: A situation calculus semantics for logic programs
- Lin, Reiter
- 1997
(Show Context)
Citation Context ...te that an occurrence of a predicate does not get an additional numeric argument if it is negated). Finally, an atom holds if it can be established by some number N of rule applications. Definition I =-=[25]-=- treats a rule in a logic program as an abbreviated description of the effect of an action – the action of “applying” that rule – in the situation calculus. 4 For instance, if the action corresponding... |

13 |
From answer set logic programming to circumscription via logic of GK
- Lin, Zhou
(Show Context)
Citation Context ...that paper can be easily extended from propositional formulas to first-order formulas. A characterization of stable models that involves grounding but is otherwise similar to Definition M is given in =-=[28]-=-. It has emerged from research on the nonmonotonic logic of knowledge and justified assumptions [26]. 13 Conclusion Research on stable models has brought us many pleasant surprises. At the time when t... |

12 |
The justification of negation as failure
- Fine
- 1989
(Show Context)
Citation Context ... p(a). p(b). q(a). r(b) ← p(b). (13) The minimal model of this program is the set (7) that we started with; consequently, that set is a stable model of (6). Definition C was independently invented in =-=[12]-=-.6 Definitions D and E, in Terms of Unfounded Sets and Loop Formulas From [45] we learned that stable models can be characterized in terms of the concept of an unfounded set, which was introduced in ... |

8 |
and Miroslaw Truszczyński. Stable semantics for logic programs and default theories
- Marek
- 1989
(Show Context)
Citation Context ...onal programs follows from the fact that each of them is equivalent to Definition C of a stable model reviewed in the next section. This was established in [14] for the autoepistemic semantics and in =-=[29]-=- for the default logic approach. 5 Definition C, in Terms of the Reduct Definitions A and B are easy to understand – assuming that one is familiar with formal nonmonotonic reasoning. Can we make these... |

8 |
Gebser M., Truszczynski.: The Second Answer Set Programming Competition
- Denecker, Vennekens, et al.
(Show Context)
Citation Context ...s led to the creation of systems for generating stable models that use SAT solvers for search (“SAT-based answer set programming”). Several systems of this kind performed well in a recent competition =-=[5]-=-. 7 Definition F, in Terms of Circumscription We saw in Sect. 4 that a logic program can be viewed as shorthand for an autoepistemic theory or a default theory. The characterization of stable models d... |

6 | Stable models and difference logic
- Niemelä
- 2008
(Show Context)
Citation Context ...sitional logic that includes variables for integers and atomic formulas of the form x − y ≥ c. This observation suggests the posibility of using solvers for difference logic to generate stable models =-=[38]-=-. 9 Definitions H and I, in Terms of Tightening and the Situation Calculus We will talk now about two characterizations of stable models that are based, like Definition F, on translations into classic... |

3 | Gayathri Namasivayam, André Neumann, Torsten Schaub, and Miroslaw Truszczyński. The first answer set programming system competition - Gebser, Liu - 2007 |

3 |
The logic of two-person games
- Emden, Clark
- 1984
(Show Context)
Citation Context ...tely, some useful Prolog programs do not satisfy this condition. For instance, we can say that a position in a two-person game is winning if there exists a move from it to a non-winning position (cf. =-=[46]-=-). This rule is not stratified: it recursively defines winning in terms of non-winning. A really good semantics should be applicable to rules like this. Second, the definition of the semantics of stra... |