Model Expansion as a Framework for Modelling and Solving Search Problems

We propose a framework for modelling and solving search problems using logic, and describe a project whose goal is to produce practically effective, general purpose tools for representing and solving search problems based on this framework. The mathematical foundation lies in the areas of finite model theory and descriptive complexity, which provide us with many classical results, as well as powerful techniques, not available to many other approaches with similar goals. We describe the mathematical foundations; explain an extension to classical logic with inductive definitions that we consider central; give a summary of complexity and expressiveness properties; describe an approach to implementing solvers based on grounding; present grounding algorithms based on an extension of the relational algebra; describe an implementation of our framework which includes use of inductive definitions, sorts and order; and give experimental results comparing the performance of our implementation with ASP solvers and another solver based on the same framework.

[1]  Moshe Y. Vardi.  The complexity of relational query languages (Extended Abstract) , 1982, STOC '82.

[2]  Pascal Van Hentenryck.  The OPL optimization programming language , 1999 .

[3]  Pierre Flener,et al.  Incremental Algorithms for Local Search from Existential Second-Order Logic , 2005, CP.

[4]  Eugenia Ternovska,et al.  A Logic for Non-Monotone Inductive Definitions , 2005, ArXiv.

[5]  Kenneth A. Ross,et al.  The well-founded semantics for general logic programs , 1991, JACM.

[6]  David G. Mitchell,et al.  A Framework for Representing and Solving NP Search Problems , 2005, AAAI.

[7]  Bart Selman,et al.  Planning as Satisfiability , 1992, ECAI.

[8]  Maurice Bruynooghe,et al.  Satisfiability Checking for PC(ID) , 2005, LPAR.

[9]  Wolfgang Faber,et al.  The DLV system for knowledge representation and reasoning , 2002, TOCL.

[10]  Miroslaw Truszczynski,et al.  Predicate-calculus-based logics for modeling and solving search problems , 2006, TOCL.

[11]  Erich Grädel,et al.  Capturing Complexity Classes by Fragments of Second-Order Logic , 1991, Theor. Comput. Sci..

[12]  Phokion G. Kolaitis,et al.  Conjunctive-query containment and constraint satisfaction , 1998, PODS.

[13]  Miklós Ajtai,et al.  ∑11-Formulae on finite structures , 1983, Ann. Pure Appl. Log..

[14]  Toni Mancini,et al.  Declarative constraint modelling and specification-level reasoning , 2003 .

[15]  Pierre Flener,et al.  Introducing ESRA, a Relational Language for Modelling Combinatorial Problems , 2003, CP.

[16]  Larry J. Stockmeyer,et al.  The Polynomial-Time Hierarchy , 1976, Theor. Comput. Sci..

[17]  Neil Immerman,et al.  Descriptive Complexity , 1999, Graduate Texts in Computer Science.

[18]  Jörg Flum,et al.  Query evaluation via tree-decompositions , 2001, JACM.

[19]  Johan Wittocx,et al.  The IDP framework for declarative problem solving , 2006 .

[20]  Ronald Fagin.  Generalized first-order spectra, and polynomial. time recognizable sets , 1974 .

[21]  Eugenia Ternovska,et al.  Inductive situation calculus , 2004, Artif. Intell..

[22]  Larry Joseph Stockmeyer,et al.  The complexity of decision problems in automata theory and logic , 1974 .

[23]  Georg Gottlob,et al.  Robbers, marshals, and guards: game theoretic and logical characterizations of hypertree width , 2001, PODS '01.

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

[25]  J.A. Medina,et al.  A syntactic characterization of NP-completeness , 1994, Proceedings Ninth Annual IEEE Symposium on Logic in Computer Science.

[26]  Marco Cadoli,et al.  Compiling Problem Specifications into SAT , 2001, ESOP.

[27]  Eugenia Ternovska,et al.  Grounding for Model Expansion in k-Guarded Formulas with Inductive Definitions , 2007, IJCAI.

[28]  Warwick Harvey,et al.  Essence: A constraint language for specifying combinatorial problems , 2007, Constraints.

[29]  Anuj Dawar,et al.  Fixed point logics , 2002, Bull. Symb. Log..

[30]  Armin Biere,et al.  Symbolic Model Checking without BDDs , 1999, TACAS.

[31]  Leonid Libkin,et al.  Elements of Finite Model Theory , 2004, Texts in Theoretical Computer Science.

[32]  Marc Denecker,et al.  Extending Classical Logic with Inductive Definitions , 2000, Computational Logic.

[33]  Constraint Programming with Unrestricted Quantification , 2005 .

[34]  Stephen A. Cook,et al.  The complexity of theorem-proving procedures , 1971, STOC.

[35]  Complexity of Expanding a Finite Structure and Related Tasks , 2006 .

[36]  Johan van Benthem,et al.  Modal Languages and Bounded Fragments of Predicate Logic , 1998, J. Philos. Log..

[37]  Victor W. Marek,et al.  On the Expressibility of Stable Logic Programming , 2001, LPNMR.

[38]  Ronald Fagin,et al.  Finite-Model Theory - A Personal Perspective , 1990, Theor. Comput. Sci..

[39]  Stephen A. Cook,et al.  A hierarchy for nondeterministic time complexity , 1972, J. Comput. Syst. Sci..

[40]  Eugenia Ternovska,et al.  A Logic of Non-monotone Inductive Definitions and Its Modularity Properties , 2004, LPNMR.

[41]  Neil D. Jones,et al.  Turing machines and the spectra of first-order formulas , 1974, Journal of Symbolic Logic.

[42]  Michael Sipser,et al.  Parity, circuits, and the polynomial-time hierarchy , 1981, 22nd Annual Symposium on Foundations of Computer Science (sfcs 1981).

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

[44]  Eugenia Ternovska,et al.  Reducing Inductive Definitions to Propositional Satisfiability , 2005, ICLP.

[45]  Y. Gurevich.  On Finite Model Theory , 1990 .

[46]  Luigi Palopoli,et al.  NP-SPEC: an executable specification language for solving all problems in NP , 1999, Comput. Lang..

[47]  Victor W. Marek,et al.  The Logic Programming Paradigm: A 25-Year Perspective , 2011 .