Characterizations of stable model semantics for logic programs with arbitrary constraint atoms

AbstractThis paper studies the stable model semantics of logic programs with (abstract) constraint atoms and their properties. We introduce a succinct abstract representation of these constraint atoms in which a constraint atom is represented compactly. We show two applications. First, under this representation of constraint atoms, we generalize the Gelfond–Lifschitz transformation and apply it to define stable models (also called answer sets) for logic programs with arbitrary constraint atoms. The resulting semantics turns out to coincide with the one defined by Son et al. (2007), which is based on a fixpoint approach. One advantage of our approach is that it can be applied, in a natural way, to define stable models for disjunctive logic programs with constraint atoms, which may appear in the disjunctive head as well as in the body of a rule. As a result, our approach to the stable model semantics for logic programs with constraint atoms generalizes a number of previous approaches. Second, we show that our abstract representation of constraint atoms provides a means to characterize dependencies of atoms in a program with constraint atoms, so that some standard characterizations and properties relying on these dependencies in the past for logic programs with ordinary atoms can be extended to logic programs with constraint atoms.

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On the existence of stable models of non-stratified logic programs

Theory and Practice of Logic Programming ◽

10.1017/s1471068405002589 ◽

2006 ◽

Vol 6 (1-2) ◽

pp. 169-212 ◽

Cited By ~ 14

Author(s):

STEFANIA COSTANTINI

Keyword(s):

Canonical Form ◽

Logic Program ◽

Practical Importance ◽

The Body ◽

Fundamental Result ◽

Stable Model ◽

Logic Programs ◽

Truth Value ◽

Answer Sets ◽

Stable Models

In this paper we analyze the relationship between cyclic definitions and consistency in Gelfond-Lifschitz's answer sets semantics (originally defined as ‘stable model semantics’). This paper introduces a fundamental result, which is relevant for Answer Set programming, and planning. For the first time since the definition of the stable model semantics, the class of logic programs for which a stable model exists is given a syntactic characterization. This condition may have a practical importance both for defining new algorithms for checking consistency and computing answer sets, and for improving the existing systems. The approach of this paper is to introduce a new canonical form (to which any logic program can be reduced to), to focus the attention on cyclic dependencies. The technical result is then given in terms of programs in canonical form (canonical programs), without loss of generality: the stable models of any general logic program coincide (up to the language) to those of the corresponding canonical program. The result is based on identifying the cycles contained in the program, showing that stable models of the overall program are composed of stable models of suitable sub-programs, corresponding to the cycles, and on defining the Cycle Graph. Each vertex of this graph corresponds to one cycle, and each edge corresponds to one handle, which is a literal containing an atom that, occurring in both cycles, actually determines a connection between them. In fact, the truth value of the handle in the cycle where it appears as the head of a rule, influences the truth value of the atoms of the cycle(s) where it occurs in the body. We can therefore introduce the concept of a handle path, connecting different cycles. Cycles can be even, if they consist of an even number of rules, or vice versa they can be odd. Problems for consistency, as it is well-known, originate in the odd cycles. If for every odd cycle we can find a handle path with certain properties, then the existence of stable model is guaranteed. We will show that based on this results new classes of consistent programs can be defined, and that cycles and cycle graphs can be generalized to components and component graphs.

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Trichotomy and dichotomy results on the complexity of reasoning with disjunctive logic programs

Theory and Practice of Logic Programming ◽

10.1017/s1471068410000463 ◽

2010 ◽

Vol 11 (6) ◽

pp. 881-904 ◽

Cited By ~ 6

Author(s):

MIROSŁAW TRUSZCZYŃSKI

Keyword(s):

The Other ◽

Logic Programs ◽

Finite Representation ◽

Potential Interest ◽

Disjunctive Logic Programs ◽

Problem Instances ◽

The One ◽

Answer Sets ◽

Disjunctive Programs ◽

Credulous Reasoning

AbstractWe present trichotomy results characterizing the complexity of reasoning with disjunctive logic programs. To this end, we introduce a certain definition schema for classes of programs based on a set of allowed arities of rules. We show that each such class of programs has a finite representation, and for each of the classes definable in the schema, we characterize the complexity of the existence of an answer set problem. Next, we derive similar characterizations of the complexity of skeptical and credulous reasoning with disjunctive logic programs. Such results are of potential interest. On the one hand, they reveal some reasons responsible for the hardness of computing answer sets. On the other hand, they identify classes of problem instances, for which the problem is “easy” (in P) or “easier than in general” (in NP). We obtain similar results for the complexity of reasoning with disjunctive programs under the supported-model semantics.

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Justifications for programs with disjunctive and causal-choice rules

Theory and Practice of Logic Programming ◽

10.1017/s1471068416000454 ◽

2016 ◽

Vol 16 (5-6) ◽

pp. 587-603 ◽

Cited By ~ 2

Author(s):

PEDRO CABALAR ◽

JORGE FANDINNO

Keyword(s):

Choice Rule ◽

Algebraic Expression ◽

Stable Model ◽

Logic Programs ◽

Causal Information ◽

Disjunctive Logic Programs ◽

New Type ◽

Algebraic Operations ◽

Disjunctive Programs ◽

Stable Models

AbstractIn this paper, we study an extension of the stable model semantics for disjunctive logic programs where each true atom in a model is associated with an algebraic expression (in terms of rule labels) that represents its justifications. As in our previous work for non-disjunctive programs, these justifications are obtained in a purely semantic way, by algebraic operations (product, addition and application) on a lattice of causal values. Our new definition extends the concept ofcausal stable modelto disjunctive logic programs and satisfies that each (standard) stable model corresponds to a disjoint class of causal stable models sharing the same truth assignments, but possibly varying the obtained explanations. We provide a pair of illustrative examples showing the behaviour of the new semantics and discuss the need of introducing a new type of rule, which we callcausal-choice. This type of rule intuitively captures the idea of “Amay causeB” and, when causal information is disregarded, amounts to a usual choice rule under the standard stable model semantics.

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Contextual hypotheses and semantics of logic programs

Theory and Practice of Logic Programming ◽

10.1017/s1471068411000378 ◽

2011 ◽

Vol 12 (6) ◽

pp. 843-887 ◽

Cited By ~ 2

Author(s):

ÉRIC A. MARTIN

Keyword(s):

Logic Programming ◽

The Body ◽

Stable Model ◽

Logic Programs ◽

Hypothetical Reasoning ◽

Unique Form ◽

Classical Negation ◽

Unified View ◽

Answer Set Semantics ◽

Semantics Of Logic Programs

AbstractLogic programming has developed as a rich field, built over a logical substratum whose main constituent is a nonclassical form of negation, sometimes coexisting with classical negation. The field has seen the advent of a number of alternative semantics, with Kripke–Kleene semantics, the well-founded semantics, the stable model semantics, and the answer-set semantics standing out as the most successful. We show that all aforementioned semantics are particular cases of a generic semantics, in a framework where classical negation is the unique form of negation and where the literals in the bodies of the rules can be ‘marked’ to indicate that they can be the targets of hypotheses. A particular semantics then amounts to choosing a particular marking scheme and choosing a particular set of hypotheses. When a literal belongs to the chosen set of hypotheses, all marked occurrences of that literal in the body of a rule are assumed to be true, whereas the occurrences of that literal that have not been marked in the body of the rule are to be derived in order to contribute to the firing of the rule. Hence the notion of hypothetical reasoning that is presented in this framework is not based on making global assumptions, but more subtly on making local, contextual assumptions, taking effect as indicated by the chosen marking scheme on the basis of the chosen set of hypotheses. Our approach offers a unified view on the various semantics proposed in logic programming, classical in that only classical negation is used, and links the semantics of logic programs to mechanisms that endow rule-based systems with the power to harness hypothetical reasoning.

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Disjunctive logic programs with existential quantification in rule heads

Theory and Practice of Logic Programming ◽

10.1017/s1471068413000355 ◽

2013 ◽

Vol 13 (4-5) ◽

pp. 563-578 ◽

Cited By ~ 1

Author(s):

JIA-HUAI YOU ◽

HENG ZHANG ◽

YAN ZHANG

Keyword(s):

Stable Model ◽

Logic Programs ◽

Disjunctive Logic Programs ◽

Decidable Fragment ◽

Definition Of ◽

Existential Quantification ◽

Disjunctive Programs ◽

Second Order Logic ◽

Stable Models

AbstractWe consider disjunctive logic programs without function symbols but with existential quantification in rule heads, under the semantics of general stable models. There are at least two interesting prospects in these programs. The first is that a program can be made more succinct by using existential variables, and the second is on the potential in representing defeasible ontological knowledge by these logic programs. This paper studies some of the properties of these programs. First, we show a simple yet intuitive definition of stable models for these programs that does not resort to second-order logic. Second, the stable models of these programs can be characterized by an extension of progression for disjunctive programs, which provides a native characterization of justification for stable models. We then study the decidability issue. While the stable model existence problem for safe disjunctive programs is decidable, with existential quantification allowed in rule heads the problem becomes undecidable. We identify an interesting decidable fragment by exploring a new notion of stratification over existential quantification.

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Well-Founded Semantics Coincides with Three-Valued Stable Semantics1

Fundamenta Informaticae ◽

10.3233/fi-1990-13404 ◽

1990 ◽

Vol 13 (4) ◽

pp. 445-463

Author(s):

Teodor Przymusinski

Keyword(s):

Logic Program ◽

Natural Extension ◽

Natural Generalization ◽

Stable Model ◽

Logic Programs ◽

Default Theory ◽

Recent Approach ◽

Stable Semantics ◽

Disjunctive Logic Programs ◽

Stable Models

We introduce 3-valued stable models which are a natural generalization of standard (2-valued) stable models. We show that every logic program P has at least one 3-valued stable model and that the well-founded model of any program P [Van Gelder et al., 1990] coincides with the smallest 3-valued stable model of P. We conclude that the well-founded semantics of an arbitrary logic program coincides with the 3-valued stable model semantics. The 3-valued stable semantics is closely related to non-monotonic formalisms in AI. Namely, every program P can be translated into a suitable autoepistemic (resp. default) theory P ˆ so that the 3-valued stable semantics of P coincides with the (3-valued) autoepistemic (resp. default) semantics of P ˆ. Similar results hold for circumscription and CWA. Moreover, it can be shown that the 3-valued stable semantics has a natural extension to the class of all disjunctive logic programs and deductive databases. Finally, following upon the recent approach developed by Gelfond and Lifschitz, we extend all of our results to more general logic programs which, in addition to the use of negation as failure, permit the use of classical negation.

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Well-founded and stable semantics of logic programs with aggregates

Theory and Practice of Logic Programming ◽

10.1017/s1471068406002973 ◽

2007 ◽

Vol 7 (3) ◽

pp. 301-353 ◽

Cited By ~ 57

Author(s):

NIKOLAY PELOV ◽

MARC DENECKER ◽

MAURICE BRUYNOOGHE

Keyword(s):

Logic Programming ◽

Second Order ◽

Logic Programs ◽

Consequence Operator ◽

Trade Offs ◽

Stable Semantics ◽

Consequence Operators ◽

Semantics Of Logic Programs ◽

Stable Models

AbstractIn this paper, we present a framework for the semantics and the computation of aggregates in the context of logic programming. In our study, an aggregate can be an arbitrary interpreted second order predicate or function. We define extensions of the Kripke-Kleene, the well-founded and the stable semantics for aggregate programs. The semantics is based on the concept of a three-valuedimmediate consequence operatorof an aggregate program. Such an operatorapproximatesthe standard two-valued immediate consequence operator of the program, and induces a unique Kripke-Kleene model, a unique well-founded model and a collection of stable models. We study different ways of defining such operators and thus obtain a framework of semantics, offering different trade-offs betweenprecisionandtractability. In particular, we investigate conditions on the operator that guarantee that the computation of the three types of semantics remains on the same level as for logic programs without aggregates. Other results show that, in practice, even efficient three-valued immediate consequence operators which are very low in the precision hierarchy, still provide optimal precision.

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Order-consistent programs are cautiously monotonic

Theory and Practice of Logic Programming ◽

10.1017/s1471068401001144 ◽

2001 ◽

Vol 1 (4) ◽

pp. 487-495 ◽

Cited By ~ 2

Author(s):

HUDSON TURNER

Keyword(s):

Stable Model ◽

Logic Programs ◽

Normal Logic ◽

Answer Sets ◽

Answer Set

Some normal logic programs under the answer set (or stable model) semantics lack the appealing property of ‘cautious monotonicity.’ That is, augmenting a program with one of its consequences may cause it to lose another of its consequences. The syntactic condition of ‘order-consistency’ was shown by Fages to guarantee existence of an answer set. This note establishes that order-consistent programs are not only consistent, but cautiously monotonic. From this it follows that they are also ‘cumulative’. That is, augmenting an order-consistent program with some of its consequences does not alter its consequences. In fact, as we show, its answer sets remain unchanged.

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Modularity Aspects of Disjunctive Stable Models

Journal of Artificial Intelligence Research ◽

10.1613/jair.2810 ◽

2009 ◽

Vol 35 ◽

pp. 813-857 ◽

Cited By ~ 35

Author(s):

T. Janhunen ◽

E. Oikarinen ◽

H. Tompits ◽

S. Woltran

Keyword(s):

Programming Languages ◽

Connected Components ◽

Stable Model ◽

Logic Programs ◽

Strongly Connected Components ◽

Verification Method ◽

Strongly Connected ◽

Disjunctive Logic Programs ◽

Mutual Comparison ◽

Output Interface

Practically all programming languages allow the programmer to split a program into several modules which brings along several advantages in software development. In this paper, we are interested in the area of answer-set programming where fully declarative and nonmonotonic languages are applied. In this context, obtaining a modular structure for programs is by no means straightforward since the output of an entire program cannot in general be composed from the output of its components. To better understand the effects of disjunctive information on modularity we restrict the scope of analysis to the case of disjunctive logic programs (DLPs) subject to stable-model semantics. We define the notion of a DLP-function, where a well-defined input/output interface is provided, and establish a novel module theorem which indicates the compositionality of stable-model semantics for DLP-functions. The module theorem extends the well-known splitting-set theorem and enables the decomposition of DLP-functions given their strongly connected components based on positive dependencies induced by rules. In this setting, it is also possible to split shared disjunctive rules among components using a generalized shifting technique. The concept of modular equivalence is introduced for the mutual comparison of DLP-functions using a generalization of a translation-based verification method.

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Logic programming with social features

Theory and Practice of Logic Programming ◽

10.1017/s1471068408003463 ◽

2008 ◽

Vol 8 (5-6) ◽

pp. 643-690 ◽

Cited By ~ 3

Author(s):

FRANCESCO BUCCAFURRI ◽

GIANLUCA CAMINITI

Keyword(s):

Logic Programming ◽

Everyday Life ◽

Mental State ◽

Mental States ◽

Stable Model ◽

Logic Programs ◽

Social Features ◽

Knowledge Representation Language ◽

Representation Language ◽

Semantics Of Logic Programs

AbstractIn everyday life it happens that a person has to reason out what other people think and how they behave, in order to achieve his goals. In other words, an individual may be required to adapt his behavior by reasoning about the others' mental state. In this paper we focus on a knowledge-representation language derived from logic programming which both supports the representation of mental states of individual communities and provides each with the capability of reasoning about others' mental states and acting accordingly. The proposed semantics is shown to be translatable into stable model semantics of logic programs with aggregates.

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