Enhancing Clause Learning by Symmetry in SAT Solvers

The satisfiability problem (SAT) is shown to be the first decision NP-complete problem. It is central in complexity theory. A CNF formula usually contains an interesting number of symmetries. That is, the formula remains invariant under some variable permutations. Such permutations are the symmetries of the formula, their elimination can lead to make a short proof for a satisfiability proof procedure. On other hand, many improvements had been done in SAT solving, Conflict-Driven Clause Learning (CDCL) SAT solvers are now able to solve great size and industrial SAT instances efficiently. The main theoretical key behind these modern solvers is, they use lazy data structures, a restart policy and perform clause learning at each fail end point in the search tree. Although symmetry and clause learning are shown to be powerful principles for SAT solving, but their combination, as far as we now, is not investigated. In this paper, we will show how symmetry can be used to improve clause learning in CDCL SAT solvers. We implemented the symmetry clause learning approach on the MiniSat solver and experimented it on several SAT instances. We compared both MiniSat with and without symmetry and the results obtained are very promising and show that clause learning by symmetry is profitable for CDCL SAT solvers.