Lean clause-sets: generalizations of minimally unsatisfiable clause-sets

  • Authors:

  • Oliver Kullmann

  • Affiliations:

  • Computer Science Department, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, UK

  • Venue:
  • Discrete Applied Mathematics - The renesse issue on satisfiability
  • Year:
  • 2003

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Abstract

We study the problem of (efficiently) deleting such clauses from conjunctive normal forms (clause-sets) which cannot contribute to any proof of unsatisfiability. For that purpose we introduce the notion of an autarky system A, which detects deletion of superfluous clauses from a clause-set F and yields a canonical normal form NA(F) ⊆ F. Clause-sets where no clauses can be deleted are called A-lean, a natural weakening of minimally unsatisfiable clause-sets opening the possibility for combinatorial approaches and including also satisfiable instances. Three special examples for autarky systems are considered: general autarkies, linear autarkies (based on linear programming) and matching autarkies (based on matching theory). We give new characterizations of ("absolutely") lean clause-sets in terms of qualitative matrix analysis, while matching lean clause-sets are characterized in terms of deficiency (the difference between the number of clauses and the number of variables), by having a cyclic associated transversal matroid, and also in terms of fully indecomposable matrices. Finally we discuss how to obtain polynomial time satisfiability decision for clause-sets with bounded deficiency, and we make a few steps towards a general theory of autarky systems.