Reducing the size of resolution proofs in linear time
Abstract
DPLL-based SAT solvers progress by implicitly applying bi-
nary resolution. The resolution proofs that they generate are used, afterthe SAT solver’s run has terminated, for various purposes. Most notable uses in formal verification are: extracting an unsatisfiable core, extracting
an interpolant, and detecting clauses that can be reused in an incremental satisfiability setting (the latter uses the proof only implicitly, during the run of the SAT solver). Making the resolution proof smaller can benefit all of these goals: it can lead to smaller cores, smaller interpolants, and smaller clauses that are propagated to the next SAT instance in an incremental setting. We suggest two methods that are linear in the size
of the proof for doing so. Our first technique, called Recycle-Units ,
uses each learned constant (unit clause) (x) for simplifying resolution
steps in which x was the pivot, prior to when it was learned. Our second
technique, called Recycle-Pivots, simplifies proofs in which there are
several nodes in the resolution graph, one of which dominates the others,
that correspond to the same pivot. Our experiments with industrial in-
stances show that these simplifications reduce the core by ≈ 5% and the
proof by ≈ 13%. It reduces the core less than competing methods such
as run-till-fix, but whereas our algorithms are linear in the size of the
proof, the latter and other competing techniques are all exponential as
they are based on SAT runs. If we consider the size of the proof (the
resolution graph) as being polynomial in the number of variables (it is
not necessarily the case in general), this gives our method an exponen-
tial time reduction comparing to existing tools for small core extraction.
Our experiments show that this result is evident in practice more so for
the second method: rarely it takes more than a few seconds, even when
competing tools time out, and hence it can be used as a cheap proof
post-processing procedure.
Research Areas
