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

2010 ◽  
Vol 11 (6) ◽  
pp. 881-904 ◽  
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.

scholarly journals Semantics for Possibilistic Disjunctive Programs

2011 ◽  
Vol 13 (1) ◽  
pp. 33-70 ◽  
Author(s):  
JUAN CARLOS NIEVES ◽  
MAURICIO OSORIO ◽  
ULISES CORTÉS
Keyword(s):  
Logic Programming ◽  
Knowledge Base ◽  
Logic Program ◽  
Programming Approach ◽  
Logic Programs ◽  
Possibilistic Logic ◽  
Point Operator ◽  
Disjunctive Logic Programs ◽  
Answer Sets ◽  
Disjunctive Programs

AbstractIn this paper, a possibilistic disjunctive logic programming approach for modeling uncertain, incomplete, and inconsistent information is defined. This approach introduces the use of possibilistic disjunctive clauses, which are able to capture incomplete information and states of a knowledge base at the same time. By considering a possibilistic logic program as a possibilistic logic theory, a construction of a possibilistic logic programming semantic based on answer sets and the proof theory of possibilistic logic is defined. It shows that this possibilistic semantics for disjunctive logic programs can be characterized by a fixed-point operator. It is also shown that the suggested possibilistic semantics can be computed by a resolution algorithm and the consideration of optimal refutations from a possibilistic logic theory. In order to manage inconsistent possibilistic logic programs, a preference criterion between inconsistent possibilistic models is defined. In addition, the approach of cuts for restoring consistency of an inconsistent possibilistic knowledge base is adopted. The approach is illustrated in a medical scenario.

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 Beyond NP: Quantifying over Answer Sets

2019 ◽  
Vol 19 (5-6) ◽  
pp. 705-721
Author(s):  
GIOVANNI AMENDOLA ◽  
FRANCESCO RICCA ◽  
MIROSLAW TRUSZCZYNSKI
Keyword(s):  
Answer Set Programming ◽  
The Other ◽  
Polynomial Hierarchy ◽  
Programming Paradigm ◽  
Quantified Boolean Formulas ◽  
Boolean Formulas ◽  
The One ◽  
Image Position ◽  
Answer Sets ◽  
Computational Properties

AbstractAnswer Set Programming (ASP) is a logic programming paradigm featuring a purely declarative language with comparatively high modeling capabilities. Indeed, ASP can model problems in NP in a compact and elegant way. However, modeling problems beyond NP with ASP is known to be complicated, on the one hand, and limited to problems in $\[\Sigma _2^P\]$ on the other. Inspired by the way Quantified Boolean Formulas extend SAT formulas to model problems beyond NP, we propose an extension of ASP that introduces quantifiers over stable models of programs. We name the new language ASP with Quantifiers (ASP(Q)). In the paper we identify computational properties of ASP(Q); we highlight its modeling capabilities by reporting natural encodings of several complex problems with applications in artificial intelligence and number theory; and we compare ASP(Q) with related languages. Arguably, ASP(Q) allows one to model problems in the Polynomial Hierarchy in a direct way, providing an elegant expansion of ASP beyond the class NP.

An abductive framework for computing knowledge base updates

2003 ◽  
Vol 3 (6) ◽  
pp. 671-715 ◽  
Author(s):  
CHIAKI SAKAMA ◽  
KATSUMI INOUE
Keyword(s):  
Computational Complexity ◽  
Knowledge Base ◽  
Knowledge Bases ◽  
Logic Programs ◽  
Types Of Knowledge ◽  
Special Cases ◽  
Different Types ◽  
View Updates ◽  
Answer Sets ◽  
Disjunctive Programs

This paper introduces an abductive framework for updating knowledge bases represented by extended disjunctive programs. We first provide a simple transformation from abductive programs to update programs which are logic programs specifying changes on abductive hypotheses. Then, extended abduction, which was introduced by the same authors as a generalization of traditional abduction, is computed by the answer sets of update programs. Next, different types of updates, view updates and theory updates are characterized by abductive programs and computed by update programs. The task of consistency restoration is also realized as special cases of these updates. Each update problem is comparatively assessed from the computational complexity viewpoint. The result of this paper provides a uniform framework for different types of knowledge base updates, and each update is computed using existing procedures of logic programming.

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 Inconsistency Proofs for ASP: The ASP - DRUPE Format

2019 ◽  
Vol 19 (5-6) ◽  
pp. 891-907
Author(s):  
MARIO ALVIANO ◽  
CARMINE DODARO ◽  
JOHANNES K. FICHTE ◽  
MARKUS HECHER ◽  
TOBIAS PHILIPP ◽  
...  
Keyword(s):  
Logic Program ◽  
Answer Set Programming ◽  
Boolean Satisfiability ◽  
Logic Programs ◽  
Disjunctive Logic Programs ◽  
Normal Logic ◽  
Consistency Problem ◽  
Unit Propagation ◽  
Answer Sets ◽  
Answer Set

AbstractAnswer Set Programming (ASP) solvers are highly-tuned and complex procedures that implicitly solve the consistency problem, i.e., deciding whether a logic program admits an answer set. Verifying whether a claimed answer set is formally a correct answer set of the program can be decided in polynomial time for (normal) programs. However, it is far from immediate to verify whether a program that is claimed to be inconsistent, indeed does not admit any answer sets. In this paper, we address this problem and develop the new proof format ASP-DRUPE for propositional, disjunctive logic programs, including weight and choice rules. ASP-DRUPE is based on the Reverse Unit Propagation (RUP) format designed for Boolean satisfiability. We establish correctness of ASP-DRUPE and discuss how to integrate it into modern ASP solvers. Later, we provide an implementation of ASP-DRUPE into the wasp solver for normal logic programs.

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 On elementary loops of logic programs

2011 ◽  
Vol 11 (6) ◽  
pp. 953-988 ◽  
Author(s):  
MARTIN GEBSER ◽  
JOOHYUNG LEE ◽  
YULIYA LIERLER
Keyword(s):  
Nonmonotonic Reasoning ◽  
The Body ◽  
The Other ◽  
Stable Model ◽  
Logic Programs ◽  
Graph Theoretic ◽  
Theoretic Characterization ◽  
Definition Of ◽  
Disjunctive Programs

AbstractUsing the notion of an elementary loop, Gebser and Schaub (2005. Proceedings of the Eighth International Conference on Logic Programming and Nonmonotonic Reasoning (LPNMR'05), 53–65) refined the theorem on loop formulas attributable to Lin and Zhao (2004) by considering loop formulas of elementary loops only. In this paper, we reformulate the definition of an elementary loop, extend it to disjunctive programs, and study several properties of elementary loops, including how maximal elementary loops are related to minimal unfounded sets. The results provide useful insights into the stable model semantics in terms of elementary loops. For a nondisjunctive program, using a graph-theoretic characterization of an elementary loop, we show that the problem of recognizing an elementary loop is tractable. On the other hand, we also show that the corresponding problem is coNP-complete for a disjunctive program. Based on the notion of an elementary loop, we present the class of Head-Elementary-loop-Free (HEF) programs, which strictly generalizes the class of Head-Cycle-Free (HCF) programs attributable to Ben-Eliyahu and Dechter (1994. Annals of Mathematics and Artificial Intelligence 12, 53–87). Like an HCF program, an HEF program can be turned into an equivalent nondisjunctive program in polynomial time by shifting head atoms into the body.

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