Two proof procedures for a cardinality based language in propositional calculus
In this paper we use the cardinality to increase the expressiveness efficiency of propositional calculus and improve the efficiency of resolution methods. Hence to express propositional problems and logical constraints we introduce the pair formulas (ρ, ℒ) which mean that “at least ρ literals among those of a list ℒ are true”. This makes a generalization of propositional clauses which express ”At least one literal is true among those of the clause”. We propose a cardinality resolution proof system for which we prove both completenesss and decidability. A linear proof for Pigeon-hole problem is given in this system showing the advantage of cardinality.
On other hand we provide an enumerative method (DPC) which is Davis and Putnam procedure adapted with Cardinality. Good results are obtained on many known problems such as Pigeon-hole problem, Queenes and some other instances derived from mathematical theorems (Ramsey, Schur's lemma) when this method is augmented with the principle of symmetry.
Key wordstheorem proving propositional calculus symmetry and cardinality
Unable to display preview. Download preview PDF.
- 1.A. S. M. Aguirre. How to use symmetries in boolean constraints solving. PhD thesis, GIA-Luminy (Marseille), 1992.Google Scholar
- 2.B. Benhamou and L. Sais. Cardinality formulas in propositional calculus. Technical Report 1, Université de provence, 1992.Google Scholar
- 3.B. Benhamou and L. Sais. Theoretical study of symmetries in propositional calculus and application. Eleventh International Conference on Automated Deduction, Saratoga Springs,NY, USA, 1992.Google Scholar
- 4.A. Colmerauer. An introduction to prolog III. CACM, 4(28):412–418, 1990.Google Scholar
- 5.C. C. Cook, W. Cook and Gy. Cook, on the complexity of cutting-planes proofs, working paper, Cornell university, ithaca, ny. 1985.Google Scholar
- 6.M. Davis and H. Putnam. A computing procedure for quatification theory. JACM, (7):201–215, 1960.Google Scholar
- 7.M. Dincbas, P. V. Hentenryck, H. Simonis, A. Aggoun, T. Grof, and F.Berthier. The constraint logic programing language CHIP. In the International Conference on Fifth Generation Computer Systems, Tokyo, Japon, December 1988.Google Scholar
- 8.P. V. Hentenryck and Y. Deville. The cardinality operator: A new logical connective for constraint logic programming. Technical report, CS Departement, Brown University, Technical Report, October, 1990.Google Scholar
- 9.J. N. Hooker. Generalized resolution and cutting planes. Approches to Intelligent Decision Suport, a volume in Annals of Operations Researchs series.Google Scholar
- 10.J. N. Hooker. A quantitive approach to logical inference. Decision Suport Systems, (4):45–69, 1988.Google Scholar
- 11.J. Jaffar and J. L. Lassez. Constraint logic programing. POPL-87,Munich, FRG, January 1988.Google Scholar
- 12.R. Kowalski and D. Kuehner. Linear resolution with selection function. Artificial Intelligence, (2):227–260, 1971.Google Scholar
- 13.B. Krishnamurty. Short proofs for tricky formulas. Acta informatica, (22):253–275, 1985.Google Scholar
- 14.L. Oxusoff and A. Rauzy. L'évaluation sémantique en calcul propositionnel. PhD thesis, GIA-Luminy (Marseille), 1989.Google Scholar