Efficient Algorithms for Description Problems over Finite Totally Ordered Domains

Given a finite set of vectors over a finite totally ordered domain, we study the problem of computing a constraint in conjunctive normal form such that the set of solutions for the produced constraint is identical to the original set. We develop an efficient polynomial-time algorithm for the general case, followed by specific polynomial-time algorithms producing Horn, dual Horn, and bijunctive formulas for sets of vectors closed under the operations of conjunction, disjunction, and median, respectively. Our results generalize the work of Dechter and Pearl on relational data, as well as the papers by Hebrard and Zanuttini. They complement the results of Hahnle et al. on multivalued logics and Jeavons et al. on the algebraic approach to constraints.

[1]  Tomás Feder,et al.  The Computational Structure of Monotone Monadic SNP and Constraint Satisfaction: A Study through Datalog and Group Theory , 1999, SIAM J. Comput..

[2]  Reiner Hähnle.  Short Conjunctive Normal Forms in Finitely Valued Logics , 1994, J. Log. Comput..

[3]  Bernhard Beckert,et al.  Transformations between signed and classical clause logic , 1999, Proceedings 1999 29th IEEE International Symposium on Multiple-Valued Logic (Cat. No.99CB36329).

[4]  Gernot Salzer,et al.  Efficient Algorithms for Constraint Description Problems over Finite Totally Ordered Domains: Extended Abstract , 2004, IJCAR.

[5]  Bernhard Beckert,et al.  The 2-SAT problem of regular signed CNF formulas , 2000, Proceedings 30th IEEE International Symposium on Multiple-Valued Logic (ISMVL 2000).

[6]  Reiner Hähnle.  Exploiting Data Dependencies in Many-Valued Logics , 1996, J. Appl. Non Class. Logics.

[7]  Bruno Zanuttini,et al.  A unified framework for structure identification , 2002, Inf. Process. Lett..

[8]  Sanjeev Khanna,et al.  Complexity classifications of Boolean constraint satisfaction problems , 2001, SIAM monographs on discrete mathematics and applications.

[9]  Rina Dechter,et al.  Structure Identification in Relational Data , 1992, Artif. Intell..

[10]  Pierre Marquis,et al.  Consistency restoration and explanations in dynamic CSPs Application to configuration , 2002, Artif. Intell..

[11]  Carlos Ansótegui,et al.  New logical and complexity results for Signed-SAT , 2003, 33rd International Symposium on Multiple-Valued Logic, 2003. Proceedings..

[12]  Peter Jeavons,et al.  On the Algebraic Structure of Combinatorial Problems , 1998, Theor. Comput. Sci..

[13]  Reiner Hähnle,et al.  Complexity of Many-Valued Logics. , 2001 .

[14]  Bruno Zanuttini,et al.  An efficient algorithm for Horn description , 2003, Inf. Process. Lett..

[15]  Marc Gyssens,et al.  Closure properties of constraints , 1997, JACM.

[16]  Christian G. Fermüller,et al.  Automated Deduction for Many-Valued Logics , 2001, Handbook of Automated Reasoning.

[17]  Martin C. Cooper,et al.  Characterising Tractable Constraints , 1994, Artif. Intell..

[18]  Reiner Hähnle,et al.  A modular reduction of regular logic to classical logic , 2001, Proceedings 31st IEEE International Symposium on Multiple-Valued Logic.

[19]  Andrei A. Bulatov,et al.  A dichotomy theorem for constraints on a three-element set , 2002, The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings..

[20]  Jaroslav Nesetril,et al.  Graphs and homomorphisms , 2004, Oxford lecture series in mathematics and its applications.

[21]  Neil V. Murray,et al.  Adapting Classical Inference Techniques to Multiple-Valued Logics Using Signed Formulas , 1994, Fundam. Informaticae.

[22]  Andrei A. Bulatov,et al.  Tractable conservative constraint satisfaction problems , 2003, 18th Annual IEEE Symposium of Logic in Computer Science, 2003. Proceedings..

[23]  Martin C. Cooper,et al.  Tractable Constraints on Ordered Domains , 1995, Artif. Intell..

[24]  Thomas J. Schaefer,et al.  The complexity of satisfiability problems , 1978, STOC.

[25]  Phokion G. Kolaitis,et al.  Conjunctive-Query Containment and Constraint Satisfaction , 2000, J. Comput. Syst. Sci..