Speeding up SAT solver by exploring CNF symmetries : Revisited

Boolean Satisfiability solvers have gone through dramatic improvements in their performances and scalability over the last few years by considering symmetries. It has been shown that by using graph symmetries and generating symmetry breaking predicates (SBPs) it is possible to break symmetries in Conjunctive Normal Form (CNF). The SBPs cut down the search space to the nonsymmetric regions of the space without affecting the satisfiability of the CNF formula. The symmetry breaking predicates are created by representing the formula as a graph, finding the graph symmetries and using some symmetry extraction mechanism (Crawford et al.). Here in this paper we take one non-trivial CNF and explore its symmetries. Finally, we generate the SBPs and adding it to CNF we show how it helps to prune the search tree, so that SAT solver would take short time. Here we present the pruning procedure of the search tree from scratch, starting from the CNF and its graph representation. As we explore the whole mechanism by a non-trivial example, it would be easily comprehendible. Also we have given a new idea of generating symmetry breaking predicates for breaking symmetry in CNF, not derived from Crawford’s conditions. At last we propose a backtrack SAT solver with inbuilt SBP generator. 1 Problem description Boolean satisfiability problem checks if a Boolean expression in CNF has some truth assignments of its variables for which the expression evaluates to true. CNF formulas are vastly used in automatic theorem proving and in Electronic Design Automation. As these formulas contain human-design artifacts and therefore often contain a great deal of symmetry, which causes satisfiability solvers to explore many redundant truth –assignments. Symmetry breaking predicates can be appended to the CNF formula to remove the symmetry while keeping satisfiability same. These Symmetry breaking predicates are created by expressing the formula as a graph, finding the graph symmetries, and then applying SBP construction algorithms on graph symmetries. The new CNF formula has the same satisfiability, but has pruned search space and therefore can be solved faster. The problem is described by using one trivial example below: For e.g. θ is a