Strong Extension-Free Proof Systems

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.

Simulating Strong Practical Proof Systems with Extended Resolution

“…. , I n is either (1) a deletion d, C where C is an arbitrary clause, or (2) an addition a, C where C is a RUP in F i−1 , or (3) an addition a, C, ω where C is propagation-redundant with respect to F i−1 and ω; we define propagation-redundancy in the following, it is based on the notion of precluded assignments: With this we can now define propagation-redundancy [16]: Definition 6 Let F be a formula, C a clause, α the assignment precluded by C, and ω an assignment that satisfies C. Then, C is propagation redundant (PR) with respect to F and ω if…”

“…If C is a RAT on p in a formula F, then C is propagation-redundant with respect to F and some witness ω [16]. Hence, DPR can be seen as a generalization of DRAT.…”

“…Moreover, due to its close relationship to unit propagationwhich is a core part of modern SAT solvers-the correctness of DRAT proofs can be checked efficiently, leading to the development of formally verified DRAT proof checkers. Spinning the idea of propagation-based proof systems even further, the proof system DPR [13,16] has been introduced as a generalization of DRAT. DPR allows for short proofs without the need for new variables, thus making it a strong candidate for practical SAT solving.…”

cake_lpr: Verified Propagation Redundancy Checking in CakeML

“…A recently proposed proof system, called Propagation Redundancy (PR) [21], generalizes RAT. There exist short PR proofs without new variables for many problems that are hard for resolution, such as pigeonhole formulas, Tseitin problems, and mutilated chessboard problems [19].…”

“…There exist short PR proofs without new variables for many problems that are hard for resolution, such as pigeonhole formulas, Tseitin problems, and mutilated chessboard problems [19]. Due to the absence of new variables it is easier to find PR proofs automatically [20], and it is considered unlikely that there exist short RAT proofs for these problems that do not introduce new variables nor reuse eliminated variables [21]. Such PR proofs can be checked directly [21], or they can first be transformed into DRAT proofs or even Extended Resolution proofs by introducing new variables [18,25].…”