Optimizing propositional calculus formulas with regard to questions of deducibility

Abstract

We consider propositional calculus formulas α and are interested in the complexity of deciding α ¦= γ for a clause γ. We investigate the problem whether efficiency of an algorithm deciding α ¦= γ can be improved by learning from queries γ′ having been answered by the algorithm before. So we are looking for an optimized formula opt(α), which is obtained from α by adding suitable consequences. We restrict ourselves to the following aspects of this optimization problem: first, we query for (k + 1)-clauses allowing the addition of at most k-clauses. We prove that this problem is coNP-complete. Second, we analyze various resolution strategies. We establish a lower bound for the number of clauses which must be added to obtain an optimized formula. Finally, we investigate optimization aspects of SLD resolution.