FLP answer set semantics without circular justifications for general logic programs

The answer set semantics presented by Faber et al. [27] has been widely used to define so called FLP answer sets for different types of logic programs. However, it was recently observed that when being extended from normal to more general classes of logic programs, this approach may produce answer sets with circular justifications that are caused by self-supporting loops. The main reason for this behavior is that the FLP answer set semantics is not fully constructive by a bottom up construction of answer sets. In this paper, we overcome this problem by enhancing the FLP answer set semantics with a level mapping formalism such that every answer set I can be built by fixpoint iteration of a one-step provability operator (more precisely, an extended van Emden-Kowalski operator for the FLP reduct fΠI). This is inspired by the fact that under the standard answer set semantics, each answer set I of a normal logic program Π is obtainable by fixpoint iteration of the standard van Emden-Kowalski one-step provability operator for the Gelfond-Lifschitz reduct ΠI, which induces a level mapping. The enhanced FLP answer sets, which we call well-justified FLP answer sets, are thanks to the level mapping free of circular justifications. As a general framework, the well-justified FLP answer set semantics applies to logic programs with first-order formulas, logic programs with aggregates, description logic programs, hex-programs etc., provided that the rule satisfaction is properly extended to such general logic programs. We study in depth the computational complexity of FLP and well-justified FLP answer sets for general classes of logic programs. Our results show that the level mapping does not increase the worst-case complexity of FLP answer sets. Furthermore, we describe an implementation of the well-justified FLP answer set semantics, and report about an experimental evaluation, which indicates a potential for performance improvements by the level mapping in practice.

[1]  Vladimir Lifschitz,et al.  Thirteen Definitions of a Stable Model , 2010, Fields of Logic and Computation.

[2]  M. Fitting.  First-order logic and automated theorem proving (2nd ed.) , 1996 .

[3]  Phan Minh Dung,et al.  On the Acceptability of Arguments and its Fundamental Role in Nonmonotonic Reasoning, Logic Programming and n-Person Games , 1995, Artif. Intell..

[4]  Joohyung Lee,et al.  Integrating Rules and Ontologies in the First-Order Stable Model Semantics (Preliminary Report) , 2011, LPNMR.

[5]  Hans Tompits,et al.  Combining answer set programming with description logics for the Semantic Web , 2004, Artif. Intell..

[6]  David Pearce,et al.  Equilibrium logic , 2006, Annals of Mathematics and Artificial Intelligence.

[7]  Victor W. Marek,et al.  Logic Programs with Abstract Constraint Atoms , 2004, AAAI.

[8]  Paolo Mancarella,et al.  Computing ideal sceptical argumentation , 2007, Artif. Intell..

[9]  Thomas Eiter,et al.  Pushing Efficient Evaluation of HEX Programs by Modular Decomposition , 2011, LPNMR.

[10]  Thomas Eiter,et al.  Nonmonotonic Multi-Context Systems: A Flexible Approach for Integrating Heterogeneous Knowledge Sources , 2011, Logic Programming, Knowledge Representation, and Nonmonotonic Reasoning.

[11]  Ian Horrocks,et al.  The Even More Irresistible SROIQ , 2006, KR.

[12]  Michael Gelfond,et al.  Classical negation in logic programs and disjunctive databases , 1991, New Generation Computing.

[13]  Thomas Lukasiewicz.  A Novel Combination of Answer Set Programming with Description Logics for the Semantic Web , 2007, ESWC.

[14]  Melvin Fitting,et al.  First-Order Logic and Automated Theorem Proving , 1990, Graduate Texts in Computer Science.

[15]  Thomas Eiter,et al.  Exploiting Unfounded Sets for HEX-Program Evaluation , 2012, JELIA.

[16]  Fausto Giunchiglia,et al.  Local Models Semantics, or Contextual Reasoning = Locality + Compatibility , 1998, KR.

[17]  Thomas Eiter,et al.  Equilibria in Heterogeneous Nonmonotonic Multi-Context Systems , 2007, AAAI.

[18]  Antonius Weinzierl,et al.  Finding explanations of inconsistency in multi-context systems , 2010, Artif. Intell..

[19]  Diego Calvanese,et al.  Tractable Reasoning and Efficient Query Answering in Description Logics: The DL-Lite Family , 2007, Journal of Automated Reasoning.

[20]  Enrico Pontelli,et al.  A Constructive semantic characterization of aggregates in answer set programming , 2007, Theory Pract. Log. Program..

[21]  Joohyung Lee,et al.  First-Order Extension of the FLP Stable Model Semantics via Modified Circumscription , 2011, IJCAI.

[22]  Yuri Gurevich,et al.  The Classical Decision Problem , 1997, Perspectives in Mathematical Logic.

[23]  Yevgeny Kazakov,et al.  SRIQ and SROIQ are Harder than SHOIQ , 2008, Description Logics.

[24]  Chitta Baral.  Knowledge Representation, Reasoning and Declarative Problem Solving: Query answering and answer set computing systems , 2003 .

[25]  Georg Gottlob,et al.  On the computational cost of disjunctive logic programming: Propositional case , 1995, Annals of Mathematics and Artificial Intelligence.

[26]  Thomas Eiter,et al.  Conflict-driven ASP solving with external sources , 2012, Theory and Practice of Logic Programming.

[27]  Nils J. Nilsson,et al.  Artificial Intelligence , 1974, IFIP Congress.

[28]  Peter F. Patel-Schneider,et al.  Reducing OWL entailment to description logic satisfiability , 2004, Journal of Web Semantics.

[29]  Ilkka Niemelä,et al.  Logic programs with stable model semantics as a constraint programming paradigm , 1999, Annals of Mathematics and Artificial Intelligence.

[30]  Diego Calvanese,et al.  The Description Logic Handbook: Theory, Implementation, and Applications , 2003, Description Logic Handbook.

[31]  Hans Tompits,et al.  A Uniform Integration of Higher-Order Reasoning and External Evaluations in Answer-Set Programming , 2005, IJCAI.

[32]  Kewen Wang,et al.  Extending Logic Programs with Description Logic Expressions for the Semantic Web , 2011, SEMWEB.

[33]  Ian Horrocks,et al.  From SHIQ and RDF to OWL: the making of a Web Ontology Language , 2003, J. Web Semant..

[34]  François Fages,et al.  Consistency of Clark's completion and existence of stable models , 1992, Methods Log. Comput. Sci..

[35]  Miroslaw Truszczynski.  Reducts of propositional theories, satisfiability relations, and generalizations of semantics of logic programs , 2010, Artif. Intell..

[36]  Jia-Huai You,et al.  A Default Approach to Semantics of Logic Programs with Constraint Atoms , 2009, LPNMR.

[37]  Thomas Eiter,et al.  Distributed Nonmonotonic Multi-Context Systems , 2010, KR.

[38]  Boris Motik,et al.  Reconciling description logics and rules , 2010, JACM.

[39]  Thomas Eiter,et al.  Modular Nonmonotonic Logic Programming Revisited , 2009, ICLP.

[40]  L. Hemachandra.  The strong exponential hierarchy collapses , 1987, STOC 1987.

[41]  Maurice Bruynooghe,et al.  Well-founded and stable semantics of logic programs with aggregates , 2007, Theory Pract. Log. Program..

[42]  Vladimir Lifschitz,et al.  Answer set programming and plan generation , 2002, Artif. Intell..

[43]  Jia-Huai You,et al.  A Generalized Gelfond-Lifschitz Transformation for Logic Programs with Abstract Constraints , 2007, AAAI.

[44]  Vladimir Lifschitz,et al.  Mathematical Foundations of Answer Set Programming , 2005, We Will Show Them!.

[45]  Boris Motik,et al.  OWL 2: The next step for OWL , 2008, J. Web Semant..

[46]  Trevor J. M. Bench-Capon,et al.  Argumentation in artificial intelligence , 2007, Artif. Intell..

[47]  Maurice Bruynooghe,et al.  Ultimate Well-Founded and Stable Semantics for Logic Programs with Aggregates , 2001, ICLP.

[48]  Miroslaw Truszczynski,et al.  Answer set programming at a glance , 2011, Commun. ACM.

[49]  Robert A. Kowalski,et al.  The Semantics of Predicate Logic as a Programming Language , 1976, JACM.

[50]  Fausto Giunchiglia,et al.  What is Local Models Semantics , 2002 .

[51]  Wolfgang Faber,et al.  Recursive Aggregates in Disjunctive Logic Programs: Semantics and Complexity , 2004, JELIA.

[52]  Paolo Ferraris,et al.  Answer Sets for Propositional Theories , 2005, LPNMR.

[53]  Arthur B. Markman,et al.  Knowledge Representation , 1998 .

[54]  Yi-Dong Shen,et al.  Well-Supported Semantics for Description Logic Programs , 2011, IJCAI.

[55]  Wolfgang Faber,et al.  Semantics and complexity of recursive aggregates in answer set programming , 2011, Artif. Intell..

[56]  Georg Gottlob,et al.  Complexity and expressive power of logic programming , 1997, Proceedings of Computational Complexity. Twelfth Annual IEEE Conference.

[57]  Stephan Tobies,et al.  Complexity results and practical algorithms for logics in knowledge representation , 2001, ArXiv.

[58]  Miroslaw Truszczynski,et al.  Reducts of propositional theories, satisfiability relations, and generalizations of semantics of logic programs , 2009, Artif. Intell..

[59]  Joohyung Lee,et al.  Stable models and circumscription , 2011, Artif. Intell..

[60]  Albert R. Meyer,et al.  Word problems requiring exponential time(Preliminary Report) , 1973, STOC.

[61]  Jos de Bruijn,et al.  A semantical framework for hybrid knowledge bases , 2010, Knowledge and Information Systems.

[62]  Kewen Wang,et al.  FLP Semantics Without Circular Justifications for General Logic Programs , 2012, AAAI.

[63]  David Pearce,et al.  A New Logical Characterisation of Stable Models and Answer Sets , 1996, NMELP.

[64]  Jos de Bruijn,et al.  Embedding Approaches to Combining Rules and Ontologies into Autoepistemic Logic , 2008, KR.

[65]  Miroslaw Truszczynski,et al.  Logic programs with abstract constraint atoms: The role of computations , 2007, Artif. Intell..

[66]  Victor W. Marek,et al.  Stable models and an alternative logic programming paradigm , 1998, The Logic Programming Paradigm.

[67]  Yevgeny Kazakov,et al.  RIQ and SROIQ Are Harder than SHOIQ , 2008, KR.

[68]  Enrico Pontelli,et al.  Answer Sets for Logic Programs with Arbitrary Abstract Constraint Atoms , 2006, AAAI.

[69]  Chitta Baral,et al.  Knowledge Representation, Reasoning and Declarative Problem Solving , 2003 .

[70]  Franz Baader,et al.  Pushing the EL Envelope , 2005, IJCAI.