No abstract

We introduce proof systems for propositional logic that admit short proofs of hard formulas as well as the succinct expression of most techniques used by modern SAT solvers. Our proof systems allow the derivation of clauses that are not necessarily implied, but which are redundant in the sense that their addition preserves satisfiability. To guarantee that these added clauses are redundant, we consider various efficiently decidable redundancy criteria which we obtain by first characterizing clause redundancy in terms of a semantic implication relationship and then restricting this relationship so that it becomes decidable in polynomial time. As the restricted implication relation is based on unit propagation-a core technique of SAT solvers-it allows efficient proof checking too. The resulting proof systems are surprisingly strong, even without the introduction of new variables-a key feature of short proofs presented in the proof-complexity literature. We demonstrate the strength of our proof systems on the famous pigeon hole formulas by providing short clausal proofs without new variables.

No abstract

No abstract

Blocked clauses provide the basis for powerful reasoning techniques used in SAT, QBF, and DQBF solving. Their definition, which relies on a simple syntactic criterion, guarantees that they are both redundant and easy to find. In this paper, we lift the notion of blocked clauses to first-order logic. We introduce two types of blocked clauses, one for first-order logic with equality and the other for first-order logic without equality, and prove their redundancy. In addition, we give a polynomial algorithm for checking whether a clause is blocked. Based on our new notions of blocking, we implemented a novel first-order preprocessing tool. Our experiments showed that many first-order problems in the TPTP library contain a large number of blocked clauses whose elimination can improve the performance of modern theorem provers, especially on satisfiable problem instances.