USING A THEOREM PROVER FOR REASONING ON CONSTRAINT PROBLEMS

The efficiency of systems for constraint programming (CP) is currently highly affected by the actual formulation of the input problem. To this end, several choices have to be made by modelers in order to write efficient specifications and handle instances of realistic size, and this, of course, represents a major obstacle to reach full declarativeness. Several structural properties of problem specifications have been investigated in order to provide techniques that reformulate a constraint program into one which is more efficiently evaluable by the solver at hand. In this paper we consider two such properties, symmetries and functional dependencies among variables, and show that, by characterizing problem specifications as logical formulae, the task of deciding whether such properties hold, and consequently that of performing the relevant reformulations, can be practically mechanized by means of automated theorem proving (ATP) technology. In particular, we report the results on using ATP technology for checking the existence of symmetries, checking whether a given constraint is symmetry-breaking, and checking the existence of functional dependencies in a specification. The output of the reasoning phase is a transformed constraint program, consisting in a reformulated specification and, possibly, a search strategy. We show our techniques on problems such as graph coloring, Sailco inventory, and protein folding.

[1]  William McCune,et al.  MACE 2.0 Reference Manual and Guide , 2001, ArXiv.

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

[3]  C. Yangt,et al.  Constraint Networks: A Survey , 2004 .

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

[5]  Toby Walsh,et al.  Breaking Row and Column Symmetries in Matrix Models , 2002, CP.

[6]  Jean-François Puget,et al.  On the Satisfiability of Symmetrical Constrained Satisfaction Problems , 1993, ISMIS.

[7]  Toni Mancini,et al.  Detecting and breaking symmetries on specications , 2003 .

[8]  Toni Mancini,et al.  Exploiting functional dependencies in declarative problem specifications , 2004, Artif. Intell..

[9]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[10]  Enrique Castillo,et al.  Building and Solving Mathematical Programming Models in Engineering and Science , 2001 .

[11]  Gerald Pfeifer,et al.  The KR System dlv: Progress Report, Comparisons and Benchmarks , 1998, KR.

[12]  Enrique Francisco Castillo Ron,et al.  Building and solving mathematical programming models in engineering and science , 2002 .

[13]  Chu Min Li,et al.  Integrating Equivalency Reasoning into Davis-Putnam Procedure , 2000, AAAI/IAAI.

[14]  Carme Torras,et al.  Solving Strategies for Highly Symmetric CSPs , 1999, IJCAI.

[15]  David Kelley.  A theory of abstraction. , 1984 .

[16]  Mihalis Yannakakis,et al.  On the Complexity of Protein Folding , 1998, J. Comput. Biol..

[17]  Brian W. Kernighan,et al.  AMPL: A Modeling Language for Mathematical Programming , 1993 .

[18]  James M. Crawford,et al.  Symmetry-Breaking Predicates for Search Problems , 1996, KR.

[19]  Toni Mancini,et al.  Detecting and Breaking Symmetries by Reasoning on Problem Specifications , 2005, SARA.

[20]  William McCune,et al.  OTTER 3.0 Reference Manual and Guide , 1994 .

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

[22]  Enrico Giunchiglia,et al.  Applying the Davis-Putnam Procedure to Non-clausal Formulas , 1999, AI*IA.

[23]  Bradford D. Allen.  Building and solving mathematical programming models inengineering and science by Enrique Castillo, Antonio J. Conejo,Pablo Pedregal, Ricardo Garcia, and Natalia Alguacil , 2002 .

[24]  Toni Mancini,et al.  Automated reformulation of specifications by safe delay of constraints , 2004, Artif. Intell..

[25]  K. Dill,et al.  A lattice statistical mechanics model of the conformational and sequence spaces of proteins , 1989 .

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

[27]  Gwynedd Pickett.  Vampire , 2006, Canadian Medical Association Journal.

[28]  Phokion G. Kolaitis.  Constraint Satisfaction, Databases, and Logic , 2003, IJCAI.

[29]  Toby Walsh,et al.  Using Auxiliary Variables and Implied Constraints to Model Non-Binary Problems , 2000, AAAI/IAAI.

[30]  Stuart C. Shapiro,et al.  Encyclopedia of artificial intelligence, vols. 1 and 2 (2nd ed.) , 1992 .

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

[32]  William McCune,et al.  OTTER 3.3 Reference Manual , 2003, ArXiv.

[33]  Wolfgang Bibel,et al.  Constraint Satisfaction from a Deductive Viewpoint , 1988, Artif. Intell..

[34]  Paul Walton Purdom,et al.  Backtrack Searching in the Presence of Symmetry , 1988, Nord. J. Comput..

[35]  Toby Walsh,et al.  CGRASS: A System for Transforming Constraint Satisfaction Problems , 2002, International Workshop on Constraint Solving and Constraint Logic Programming.