Two proof procedures for a cardinality based language in propositional calculus

  • Belaid Benhamou
  • Lakhdar Sais
  • Pierre Siegel
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 775)


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 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 words

theorem proving propositional calculus symmetry and cardinality 


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Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  • Belaid Benhamou
    • 1
  • Lakhdar Sais
    • 1
  • Pierre Siegel
    • 1
  1. 1.L.I.U.P. Case AterUFR-MIM - Université de ProvenceMarseille cedex 3France

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