Encoding Redundancy for Satisfaction-Driven Clause Learning

Abstract

Satisfaction-Driven Clause Learning (SDCL) is a recent SAT solving paradigm that aggressively trims the search space of possible truth assignments. To determine if the SAT solver is currently exploring a dispensable part of the search space, SDCL uses the so-called positive reduct of a formula: The positive reduct is an easily solvable propositional formula that is satisfiable if the current assignment of the solver can be safely pruned from the search space. In this paper, we present two novel variants of the positive reduct that allow for even more aggressive pruning. Using one of these variants allows SDCL to solve harder problems, in particular the well-known Tseitin formulas and mutilated chessboard problems. For the first time, we are able to generate and automatically check clausal proofs for large instances of these problems.

cake_lpr: Verified Propagation Redundancy Checking in CakeML

“…The shrinking was shown to be correct [21], but has not been implemented so far. We observed that the witnesses in the PR proofs produced by SaDiCaL [20] can be substantially compressed using this method. Fig.…”

“…This section compares the verified CakeML LPR proof checker against other verified checkers on two benchmark suites and a RAT microbenchmark. The first suite is a collection of problems with PR proofs generated by the satisfactiondriven clause learning (SDCL) solver SaDiCaL [20], while the second suite consists of unsatisfiable problems from the SAT Race 2019 competition. 6 The RAT microbenchmark consists of proofs for large mutilated chessboards generated by a BDD-based SAT solver [5].…”

“…The SaDiCaL solver produces PR proofs for hard SAT problems in its benchmark suite [20] and it is experimentally much faster than a plain DRAT-based CDCL solver on those problems [20,Section 7]. The PR proofs are directly checked by cake_lpr after conversion into LPR format with DPR-trim.…”

“…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].…”

Simulating Strong Practical Proof Systems with Extended Resolution

“…DPR allows for short proofs without the need for new variables, thus making it a strong candidate for practical SAT solving. In fact, the solver SaDiCaL [15], which implements the DPR-based satisfaction-driven clause learning (SDCL) paradigm [14], can automatically find short proofs of the pigeon-hole principle, Tseitin formulas over expander graphs [30], and mutilated chessboard problems [25]. All these problems are infamous in the proof-complexity literature for being extremely hard [1,8,9,31], thus causing usual conflict-driven clause learning (CDCL) [24,26] solvers some serious trouble.…”