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

We study the problem of (eeciently) deleting such clauses from conjunctive normal forms (clause-sets) which can not contribute to any proof of unsatissability. For that purpose we introduce the notion of an autarky system, associated with a canonical normal form for every clause-set by deleting superruous clauses. Clause-sets where no clauses can be deleted are called lean, a natural generalization of minimally unsatissable clause-sets, opening the possibility for combi-natorial approaches (and including also satissable 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 lean and linearly lean clause-sets by \universal linear programming problems," while matching lean clause-sets are characterized in terms of \dee-ciency," the diierence between the number of clauses and the number of variables, and also by having a cyclic associated transversal ma-troid. Finally we discuss the applications to minimally unsatissable clause-sets.