On the Non-degeneracy of Unsatisfiability Proof Graphs Produced by SAT Solvers

Rohan Fossé, Laurent Simon

Abstract: Despite the important effort in developing fast and powerful SAT solvers, many aspects of their behaviors remains largely unexplained. We analyze the properties of learnt clauses derived by a typical Conflict Driven Clause Learning algorithm (CDCL) and study how they are linked to their ancestors, in the dependency graph generated by the resolution steps during conflict analysis and clauses minimizations. We show that all these graphs share a common structure: they are non k-degenerated with surprising large values, which mean they contain a very dense subgraph, the K-Core. We unveil the existence of large K-Cores, even on parallelized SAT solvers with clauses exchanges. We show that the analysis of the K-Core allows a good prediction of which literals will occur in future learnt clauses, until the very end of the computation. Moreover, we show that the analysis of the graph allows to identify a set of learnt clauses that will be necessary for deriving the final contradiction. At last, we demonstrate that the analysis of the dependency graph is possible with a reasonable cost in any CDCL.