scholarly journals Characterizing and extending answer set semantics using possibility theory

2014 ◽  
Vol 15 (1) ◽  
pp. 79-116 ◽  
Author(s):  
KIM BAUTERS ◽  
STEVEN SCHOCKAERT ◽  
MARTINE DE COCK ◽  
DIRK VERMEIR
Keyword(s):  
Possibility Theory ◽  
Combinatorial Problems ◽  
The Body ◽  
Uncertain Information ◽  
Possibilistic Logic ◽  
Possibility Distributions ◽  
Answer Set Semantics ◽  
Answer Sets ◽  
Answer Set

AbstractAnswer Set Programming (ASP) is a popular framework for modelling combinatorial problems. However, ASP cannot be used easily for reasoning about uncertain information. Possibilistic ASP (PASP) is an extension of ASP that combines possibilistic logic and ASP. In PASP a weight is associated with each rule, whereas this weight is interpreted as the certainty with which the conclusion can be established when the body is known to hold. As such, it allows us to model and reason about uncertain information in an intuitive way. In this paper we present new semantics for PASP in which rules are interpreted as constraints on possibility distributions. Special models of these constraints are then identified as possibilistic answer sets. In addition, since ASP is a special case of PASP in which all the rules are entirely certain, we obtain a new characterization of ASP in terms of constraints on possibility distributions. This allows us to uncover a new form of disjunction, called weak disjunction, that has not been previously considered in the literature. In addition to introducing and motivating the semantics of weak disjunction, we also pinpoint its computational complexity. In particular, while the complexity of most reasoning tasks coincides with standard disjunctive ASP, we find that brave reasoning for programs with weak disjunctions is easier.

scholarly journals Applications of intuitionistic logic in Answer Set Programming

2004 ◽  
Vol 4 (3) ◽  
pp. 325-354 ◽  
Author(s):  
MAURICIO OSORIO ◽  
JUAN A. NAVARRO ◽  
JOSÉ ARRAZOLA
Keyword(s):  
Recent Result ◽  
Intuitionistic Logic ◽  
Answer Set Programming ◽  
Minimal Models ◽  
Logic Programs ◽  
Intermediate Logics ◽  
Answer Set Semantics ◽  
Answer Sets ◽  
Answer Set

We present some applications of intermediate logics in the field of Answer Set Programming (ASP). A brief, but comprehensive introduction to the answer set semantics, intuitionistic and other intermediate logics is given. Some equivalence notions and their applications are discussed. Some results on intermediate logics are shown, and applied later to prove properties of answer sets. A characterization of answer sets for logic programs with nested expressions is provided in terms of intuitionistic provability, generalizing a recent result given by Pearce. It is known that the answer set semantics for logic programs with nested expressions may select non-minimal models. Minimal models can be very important in some applications, therefore we studied them; in particular we obtain a characterization, in terms of intuitionistic logic, of answer sets which are also minimal models. We show that the logic G3 characterizes the notion of strong equivalence between programs under the semantic induced by these models. Finally we discuss possible applications and consequences of our results. They clearly state interesting links between ASP and intermediate logics, which might bring research in these two areas together.

scholarly journals Normal forms for answer sets programming

2005 ◽  
Vol 5 (6) ◽  
pp. 747-760 ◽  
Author(s):  
STEFANIA COSTANTINI ◽  
ALESSANDRO PROVETTI
Keyword(s):  
Normal Form ◽  
Static Analysis ◽  
Normal Forms ◽  
The Body ◽  
Logic Programs ◽  
Answer Set Semantics ◽  
Answer Sets ◽  
Kernel Form ◽  
Cycle Graph

Normal forms for logic programs under stable/answer set semantics are introduced. We argue that these forms can simplify the study of program properties, mainly consistency. The first normal form, called the kernel of the program, is useful for studying existence and number of answer sets. A kernel program is composed of the atoms which are undefined in the Well-founded semantics, which are those that directly affect the existence of answer sets. The body of rules is composed of negative literals only. Thus, the kernel form tends to be significantly more compact than other formulations. Also, it is possible to check consistency of kernel programs in terms of colorings of the Extended Dependency Graph program representation which we previously developed. The second normal form is called 3-kernel. A 3-kernel program is composed of the atoms which are undefined in the Well-founded semantics. Rules in 3-kernel programs have at most two conditions, and each rule either belongs to a cycle, or defines a connection between cycles. 3-kernel programs may have positive conditions. The 3-kernel normal form is very useful for the static analysis of program consistency, i.e. the syntactic characterization of existence of answer sets. This result can be obtained thanks to a novel graph-like representation of programs, called Cycle Graph which presented in the companion article Costantini (2004b).

scholarly journals Paracoherent Answer Set Semantics meets Argumentation Frameworks

2019 ◽  
Vol 19 (5-6) ◽  
pp. 688-704
Author(s):  
GIOVANNI AMENDOLA ◽  
FRANCESCO RICCA
Keyword(s):  
Logic Programming ◽  
Logic Program ◽  
Great Success ◽  
Argumentation Framework ◽  
Computational Costs ◽  
Abstract Argumentation ◽  
Answer Set Semantics ◽  
Answer Sets ◽  
Answer Set

AbstractIn the last years, abstract argumentation has met with great success in AI, since it has served to capture several non-monotonic logics for AI. Relations between argumentation framework (AF) semantics and logic programming ones are investigating more and more. In particular, great attention has been given to the well-known stable extensions of an AF, that are closely related to the answer sets of a logic program. However, if a framework admits a small incoherent part, no stable extension can be provided. To overcome this shortcoming, two semantics generalizing stable extensions have been studied, namely semi-stable and stage. In this paper, we show that another perspective is possible on incoherent AFs, called paracoherent extensions, as they have a counterpart in paracoherent answer set semantics. We compare this perspective with semi-stable and stage semantics, by showing that computational costs remain unchanged, and moreover an interesting symmetric behaviour is maintained.

scholarly journals A common view on strong, uniform, and other notions of equivalence in answer-set programming

2008 ◽  
Vol 8 (2) ◽  
pp. 217-234 ◽  
Author(s):  
STEFAN WOLTRAN
Keyword(s):  
Syntactic Structure ◽  
Answer Set Programming ◽  
Common View ◽  
Complexity Bounds ◽  
Special Cases ◽  
Implementation Method ◽  
The One ◽  
Answer Set Semantics ◽  
Answer Sets ◽  
Answer Set

AbstractLogic programming under the answer-set semantics nowadays deals with numerous different notions of program equivalence. This is due to the fact that equivalence for substitution (known as strong equivalence) and ordinary equivalence are different concepts. The former holds, given programs P and Q, iff P can be faithfully replaced by Q within any context R, while the latter holds iff P and Q provide the same output, that is, they have the same answer sets. Notions in between strong and ordinary equivalence have been introduced as theoretical tools to compare incomplete programs and are defined by either restricting the syntactic structure of the considered context programs R or by bounding the set $\A$ of atoms allowed to occur in R (relativized equivalence). For the latter approach, different $\A$ yield properly different equivalence notions, in general. For the former approach, however, it turned out that any “reasonable” syntactic restriction to R coincides with either ordinary, strong, or uniform equivalence (for uniform equivalence, the context ranges over arbitrary sets of facts, rather than program rules). In this paper, we propose a parameterization for equivalence notions which takes care of both such kinds of restrictions simultaneously by bounding, on the one hand, the atoms which are allowed to occur in the rule heads of the context and, on the other hand, the atoms which are allowed to occur in the rule bodies of the context. We introduce a general semantical characterization which includes known ones as SE-models (for strong equivalence) or UE-models (for uniform equivalence) as special cases. Moreover, we provide complexity bounds for the problem in question and sketch a possible implementation method making use of dedicated systems for checking ordinary equivalence.

scholarly journals A program-level approach to revising logic programs under the answer set semantics

2010 ◽  
Vol 10 (4-6) ◽  
pp. 565-580 ◽  
Author(s):  
JAMES P. DELGRANDE
Keyword(s):  
Logic Program ◽  
Fundamental Principle ◽  
Logic Programs ◽  
Minimal Set ◽  
Essential Idea ◽  
The One ◽  
Answer Set Semantics ◽  
Answer Sets ◽  
Update Logic ◽  
Answer Set

AbstractAn approach to the revision of logic programs under the answer set semantics is presented. For programs P and Q, the goal is to determine the answer sets that correspond to the revision of P by Q, denoted P * Q. A fundamental principle of classical (AGM) revision, and the one that guides the approach here, is the success postulate. In AGM revision, this stipulates that α ∈ K * α. By analogy with the success postulate, for programs P and Q, this means that the answer sets of Q will in some sense be contained in those of P * Q. The essential idea is that for P * Q, a three-valued answer set for Q, consisting of positive and negative literals, is first determined. The positive literals constitute a regular answer set, while the negated literals make up a minimal set of naf literals required to produce the answer set from Q. These literals are propagated to the program P, along with those rules of Q that are not decided by these literals. The approach differs from work in update logic programs in two main respects. First, we ensure that the revising logic program has higher priority, and so we satisfy the success postulate; second, for the preference implicit in a revision P * Q, the program Q as a whole takes precedence over P, unlike update logic programs, since answer sets of Q are propagated to P. We show that a core group of the AGM postulates are satisfied, as are the postulates that have been proposed for update logic programs.

scholarly journals A general framework for equivalences in Answer-Set Programming by countermodels in the logic of Here-and-There

2011 ◽  
Vol 11 (2-3) ◽  
pp. 171-202 ◽  
Author(s):  
MICHAEL FINK
Keyword(s):  
General Framework ◽  
Answer Set Programming ◽  
Program Optimization ◽  
Logic Programs ◽  
Infinite Domains ◽  
General Semantics ◽  
Answer Set Semantics ◽  
Selection Of ◽  
Answer Set

AbstractDifferent notions of equivalence, such as the prominent notions of strong and uniform equivalence, have been studied in Answer-Set Programming, mainly for the purpose of identifying programs that can serve as substitutes without altering the semantics, for instance in program optimization. Such semantic comparisons are usually characterized by various selections of models in the logic of Here-and-There (HT). For uniform equivalence however, correct characterizations in terms of HT-models can only be obtained for finite theories, respectively programs. In this paper, we show that a selection of countermodels in HT captures uniform equivalence also for infinite theories. This result is turned into coherent characterizations of the different notions of equivalence by countermodels, as well as by a mixture of HT-models and countermodels (so-called equivalence interpretations). Moreover, we generalize the so-called notion of relativized hyperequivalence for programs to propositional theories, and apply the same methodology in order to obtain a semantic characterization which is amenable to infinite settings. This allows for a lifting of the results to first-order theories under a very general semantics given in terms of a quantified version of HT. We thus obtain a general framework for the study of various notions of equivalence for theories under answer-set semantics. Moreover, we prove an expedient property that allows for a simplified treatment of extended signatures, and provide further results for non-ground logic programs. In particular, uniform equivalence coincides under open and ordinary answer-set semantics, and for finite non-ground programs under these semantics, also the usual characterization of uniform equivalence in terms of maximal and total HT-models of the grounding is correct, even for infinite domains, when corresponding ground programs are infinite.

scholarly journals Possibilistic ASP Base Revision by Certain Input

2018 ◽  
Author(s):  
Laurent Garcia ◽  
Claire Lefèvre ◽  
Odile Papini ◽  
Igor Stéphan ◽  
Eric Würbel
Keyword(s):  
Answer Set Programming ◽  
Semantic Content ◽  
Logic Programs ◽  
Rule Based ◽  
Belief Base ◽  
Programming Framework ◽  
Base Revision ◽  
Possibilistic Logic ◽  
Answer Set Semantics ◽  
Answer Set

Belief base revision has been studied within the answer set programming framework. We go a step further by introducing uncertainty and studying belief base revision when beliefs are represented by possibilistic logic programs under possibilistic answer set semantics and revised by certain input. The paper proposes two approaches of rule-based revision operators and presents their semantic characterization in terms of possibilistic distribution. This semantic characterization allows for equivalently considering the evolution of syntactic logic programs and the evolution of their semantic content. It then studies the logical properties of the proposed operators and gives complexity results.

scholarly journals Better Paracoherent Answer Sets with Less Resources

2019 ◽  
Vol 19 (5-6) ◽  
pp. 757-772 ◽  
Author(s):  
GIOVANNI AMENDOLA ◽  
CARMINE DODARO ◽  
FRANCESCO RICCA
Keyword(s):  
Problem Solving ◽  
Equilibrium Model ◽  
State Of The Art ◽  
Answer Set Programming ◽  
Evaluation Technique ◽  
Answer Set Semantics ◽  
Computational Resources ◽  
Answer Sets ◽  
Modeling Feature ◽  
Answer Set

AbstractAnswer Set Programming (ASP) is a well-established formalism for logic programming. Problem solving in ASP requires to write an ASP program whose answers sets correspond to solutions. Albeit the non-existence of answer sets for some ASP programs can be considered as a modeling feature, it turns out to be a weakness in many other cases, and especially for query answering. Paracoherent answer set semantics extend the classical semantics of ASP to draw meaningful conclusions also from incoherent programs, with the result of increasing the range of applications of ASP. State of the art implementations of paracoherent ASP adopt the semi-equilibrium semantics, but cannot be lifted straightforwardly to compute efficiently the (better) split semi-equilibrium semantics that discards undesirable semi-equilibrium models. In this paper an efficient evaluation technique for computing a split semi-equilibrium model is presented. An experiment on hard benchmarks shows that better paracoherent answer sets can be computed consuming less computational resources than existing methods.

Answer Sets and the Language of Answer Set Programming

AI Magazine ◽  
2016 ◽  
Vol 37 (3) ◽  
pp. 7-12 ◽  
Author(s):  
Vladimir Lifschitz
Keyword(s):  
Answer Set Programming ◽  
Declarative Programming ◽  
Logic Programs ◽  
Special Issue ◽  
Programming Paradigm ◽  
Introductory Article ◽  
Answer Set Semantics ◽  
Answer Sets ◽  
Semantics Of Logic Programs ◽  
Answer Set

Answer set programming is a declarative programming paradigm based on the answer set semantics of logic programs. This introductory article provides the mathematical background for the discussion of answer set programming in other contributions to this special issue.

scholarly journals A Semantic Characterization ASP Base Revision

2019 ◽  
Vol 66 ◽  
pp. 989-1029
Author(s):  
Laurent Garcia ◽  
Claire Lefèvre ◽  
Igor Stéphan ◽  
Odile Papini ◽  
Éric Würbel
Keyword(s):  
Classical Logic ◽  
Answer Set Programming ◽  
Semantic Content ◽  
Logic Programs ◽  
Rule Based ◽  
Complexity Results ◽  
Base Revision ◽  
Answer Sets ◽  
Answer Set

The paper deals with base revision for Answer Set Programming (ASP). Base revision in classical logic is done by the removal of formulas. Exploiting the non-monotonicity of ASP allows one to propose other revision strategies, namely addition strategy or removal and/or addition strategy. These strategies allow one to define families of rule-based revision operators. The paper presents a semantic characterization of these families of revision operators in terms of answer sets. This semantic characterization allows for equivalently considering the evolution of syntactic logic programs and the evolution of their semantic content. It then studies the logical properties of the proposed operators and gives complexity results.  

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