Generalized Conflict-Clause Strengthening for Satisfiability Solvers

The dominant propositional satisfiability solvers of the past decade use a technique often called conflict-driven clause learning (cdcl), although nomenclature varies. The first half of the decade concentrated on deriving the best clause from the conflict graph that the technique constructs, also with much emphasis on speed. In the second half of the decade efforts have emerged to exploit other information that is derived by the technique as a by-product of generating the conflict graph and learning a conflict clause. The main thrust has been to strengthen the conflict clause by eliminating some of its literals, a process often called conflict-clause minimization, but more accurately described as conflictclause width reduction, or strengthening. This paper first introduces implication sequences as a general framework to represent all the information derived by the CDCL technique, some of which is not represented in the conflict graph. Then the paper analyzes the structure of this information. The first main result is that any conflict clause that is a logical consequence of an implication sequence may be derived by a particularly simple form of resolution, known as linear input regular. A key observation needed for this result is that the set of clauses in any implication sequence is Horn-renamable. The second main result is that, given an implication sequence, and a clause C derived (learned) from it, it is NP -hard to find a minimum-cardinality subset of C that is also derivable. This is in sharp contrast to the known fact that such a minimum subset can be found quickly if the derivation is restricted to using only clauses in the conflict graph.