Understanding and Improving a Modern SAT Solver

Propositional satisfiability (SAT) is an NP-complete problem, holding a central place in computer science and engineering. SAT has numerous applications in formal verification, artificial intelligence and other areas. Modern SAT solvers, using an enhanced version of the backtrack search DavisLogemann-Loveland (DLL) algorithm, are able to successfully cope with instances comprising millions of variables. This work is an attempt to shed new light on the functionality of a modern SAT solver. We also propose a number of enhancements that are practically useful, especially in the formal verification domain. First, we propose a new framework for presenting and analyzing a modern DLL-based SAT solver. Our approach exploits the inherent relation between backtracking and resolution. We show how to derive the algorithm of a modern SAT solver from DLL step-by-step. We analyze the inference power of Boolean Constraint Propagation, Non-Chronological Backtracking and 1UIP-based Conflict-Directed Backjumping. This work is intended to serve as a basis for future work on the inference power of a modern SAT solver and on practical SAT solver design. Second, we define new criteria for measuring the practical impact of different schemes for Conflict-Driven Learning: backward pruning – the number of resolution nodes skipped during backtracking; and forward pruning – the potential impact on reusing conflict clauses in the subsequent search. We show that the minimized 1UIP scheme for Conflict-Driven Learning is better than other known schemes in terms of both backward and forward pruning. This explains its empirical advantage over other schemes. We propose a practically efficient enhancement to the minimized 1UIP scheme, called Local Conflict Clause recording. Third, we propose a new decision heuristic for SAT, called the ClauseBased Heuristic. This heuristic is designed to increase the likelihood that interrelated variables will be chosen in proximity. It maintains a clause list containing both the initial and conflict clauses. The next decision literal is picked from the first unsatisfied clause. We propose various methods for initially organizing the clause list and for moving clauses within it. Our approach results in a significant performance boost over existing heuristics on real-world hard industrial benchmarks. Lastly, we present an algorithm for minimal unsatisfiable core extraction that is able to find a minimal unsatisfiable core for large real-world formulas. Benchmark families, arising in formal verification of hardware, are of particular interest to us. A modern SAT solver is able to produce a resolution refutation of a given unsatisfiable formula, whose sources are the input clauses and whose sink is the empty clause. Our method consists of removing the input clauses connected to the empty clause one by one from the resolution refutation, preserving the validity of the refutation by adding other clauses and resolution relations until no more input clauses can be removed. In the end, all the input clauses, connected to the empty clause, comprise the minimal unsatisfiable core.