Well-Founded Semantics Coincides with Three-Valued Stable Semantics1

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.

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

2006 ◽  
Vol 6 (1-2) ◽  
pp. 169-212 ◽  
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.

scholarly journals On the Equivalence Between Abstract Dialectical Frameworks and Logic Programs

2019 ◽  
Vol 19 (5-6) ◽  
pp. 941-956
Author(s):  
JOÃO ALCÂNTARA ◽  
SAMY SÁ ◽  
JUAN ACOSTA-GUADARRAMA
Keyword(s):  
Logic Program ◽  
Logic Programs ◽  
Different Types ◽  
Stable Semantics ◽  
Acceptance Condition ◽  
Normal Logic ◽  
Normal Logic Program ◽  
Stable Models

AbstractAbstract Dialectical Frameworks (ADFs) are argumentation frameworks where each node is associated with an acceptance condition. This allows us to model different types of dependencies as supports and attacks. Previous studies provided a translation from Normal Logic Programs (NLPs) to ADFs and proved the stable models semantics for a normal logic program has an equivalent semantics to that of the corresponding ADF. However, these studies failed in identifying a semantics for ADFs equivalent to a three-valued semantics (as partial stable models and well-founded models) for NLPs. In this work, we focus on a fragment of ADFs, called Attacking Dialectical Frameworks (ADF+s), and provide a translation from NLPs to ADF+s robust enough to guarantee the equivalence between partial stable models, well-founded models, regular models, stable models semantics for NLPs and respectively complete models, grounded models, preferred models, stable models for ADFs. In addition, we define a new semantics for ADF+s, called L-stable, and show it is equivalent to the L-stable semantics for NLPs.

scholarly journals Justifications for programs with disjunctive and causal-choice rules

2016 ◽  
Vol 16 (5-6) ◽  
pp. 587-603 ◽  
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.

Disjunctive logic programs with existential quantification in rule heads

2013 ◽  
Vol 13 (4-5) ◽  
pp. 563-578 ◽  
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.

scholarly journals Extending Well-Founded Semantics with Clark’s Completion for Disjunctive Logic Programs

2018 ◽  
Vol 2018 ◽  
pp. 1-10
Author(s):  
Juan Carlos Nieves ◽  
Mauricio Osorio
Keyword(s):  
Aggregation Operator ◽  
Stable Model ◽  
Logic Programs ◽  
Complete Lattices ◽  
Transformation Rules ◽  
Disjunctive Program ◽  
Stable Semantics ◽  
Disjunctive Logic Programs ◽  
Disjunctive Programs ◽  
Explicit Negation

In this paper, we introduce new semantics (that we call D3-WFS-DCOMP) and compare it with the stable semantics (STABLE). For normal programs, this semantics is based onsuitableintegration of the well-founded semantics (WFS) and the Clark’s completion. D3-WFS-DCOM has the following appealing properties: First, it agrees with STABLE in the sense that it never defines a nonminimal model or a nonminimal supported model. Second, for normal programs it extends WFS. Third, every stable model of a disjunctive programPis a D3-WFS-DCOM model ofP. Fourth, it is constructed using transformation rules accepted by STABLE. We also introduce second semantics that we call D2-WFS-DCOMP. We show that D2-WFS-DCOMP is equivalent to D3-WFS-DCOMP for normal programs but this is not the case for disjunctive programs. We also introduce third new semantics that supports the use of implicit disjunctions. We illustrate how these semantics can be extended to programs including explicit negation, default negation in the head of a clause, and aluboperator, which is a generalization of the aggregation operatorsetofover arbitrary complete lattices.

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

2009 ◽  
Vol 9 (4) ◽  
pp. 529-564 ◽  
Author(s):  
YI-DONG SHEN ◽  
JIA-HUAI YOU ◽  
LI-YAN YUAN
Keyword(s):  
The Body ◽  
Stable Model ◽  
Logic Programs ◽  
Abstract Representation ◽  
Disjunctive Logic Programs ◽  
The One ◽  
Answer Sets ◽  
Semantics Of Logic Programs ◽  
Natural Way ◽  
Stable Models

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.

scholarly journals Well-founded and stable semantics of logic programs with aggregates

2007 ◽  
Vol 7 (3) ◽  
pp. 301-353 ◽  
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.

scholarly journals Knowledge Forgetting in Answer Set Programming

2014 ◽  
Vol 50 ◽  
pp. 31-70 ◽  
Author(s):  
Y. Wang ◽  
Y. Zhang ◽  
Y. Zhou ◽  
M. Zhang
Keyword(s):  
Logic Program ◽  
Answer Set Programming ◽  
Irrelevant Information ◽  
Semantic Knowledge ◽  
Decision Problems ◽  
Stable Model ◽  
Logic Programs ◽  
Knowledge Based ◽  
Substitution Principle ◽  
Answer Set

The ability of discarding or hiding irrelevant information has been recognized as an important feature for knowledge based systems, including answer set programming. The notion of strong equivalence in answer set programming plays an important role for different problems as it gives rise to a substitution principle and amounts to knowledge equivalence of logic programs. In this paper, we uniformly propose a semantic knowledge forgetting, called HT- and FLP-forgetting, for logic programs under stable model and FLP-stable model semantics, respectively. Our proposed knowledge forgetting discards exactly the knowledge of a logic program which is relevant to forgotten variables. Thus it preserves strong equivalence in the sense that strongly equivalent logic programs will remain strongly equivalent after forgetting the same variables. We show that this semantic forgetting result is always expressible; and we prove a representation theorem stating that the HT- and FLP-forgetting can be precisely characterized by Zhang-Zhou's four forgetting postulates under the HT- and FLP-model semantics, respectively. We also reveal underlying connections between the proposed forgetting and the forgetting of propositional logic, and provide complexity results for decision problems in relation to the forgetting. An application of the proposed forgetting is also considered in a conflict solving scenario.

scholarly journals Modularity Aspects of Disjunctive Stable Models

2009 ◽  
Vol 35 ◽  
pp. 813-857 ◽  
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.

scholarly journals Defining the Semantics of Abstract Argumentation Frameworks through Logic Programs and Partial Stable Models (Extended Abstract)

2021 ◽  
Author(s):  
Gianvincenzo Alfano ◽  
Sergio Greco ◽  
Francesco Parisi ◽  
Irina Trubitsyna
Keyword(s):  
Logic Program ◽  
Logic Programs ◽  
Argumentation Framework ◽  
Abstract Argumentation ◽  
Stable Models

Extensions of Dung’s Argumentation Framework (AF) include the class of Recursive Bipolar AFs (Rec-BAFs), i.e. AFs with recursive attacks and supports. We show that a Rec-BAF \Delta can be translated into a logic program P_\Delta so that the extensions of \Delta under different semantics coincide with subsets of the partial stable models of P_\Delta.

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