Augmenting problem variables in a quantified Boolean formula with definition variables enables a compact representation in clausal form. Generally these definition variables are placed in the innermost quantifier level. To restore some structural information, we introduce a preprocessing technique that moves definition variables to the quantifier level closest to the variables that define them. We express the movement in the QRAT proof system to allow verification by independent proof checkers. We evaluated definition variable movement on the QBFEVAL’20 competition benchmarks. Movement significantly improved performance for the competition’s top solvers. Combining variable movement with the preprocessor Bloqqer improves solver performance compared to using Bloqqer alone.

The propagation redundant (PR) proof system generalizes the resolution and resolution asymmetric tautology proof systems used by conflict-driven clause learning (CDCL) solvers. PR allows short proofs of unsatisfiability for some problems that are difficult for CDCL solvers. Previous attempts to automate PR clause learning used hand-crafted heuristics that work well on some highly-structured problems. For example, the solver SaDiCaL incorporates PR clause learning into the CDCL loop, but it cannot compete with modern CDCL solvers due to its fragile heuristics. We present PReLearn, a preprocessing technique that learns short PR clauses. Adding these clauses to a formula reduces the search space that the solver must explore. By performing PR clause learning as a preprocessing stage, PR clauses can be found efficiently without sacrificing the robustness of modern CDCL solvers. On a large portion of SAT competition benchmarks we found that preprocessing with PReLearn improves solver performance. In addition, there were several satisfiable and unsatisfiable formulas that could only be solved after preprocessing with PReLearn. PReLearn supports proof logging, giving a high level of confidence in the results.

Modern SAT solvers produce proofs of unsatisfiability to justify the correctness of their results. These proofs, which are usually represented in the well-known DRAT format, can often become huge, requiring multiple gigabytes of disk storage. We present a technique for semantic proof compression that selects a subset of important clauses from a proof and stores them as a so-called proof skeleton. This proof skeleton can later be used to efficiently reconstruct a full proof by exploiting parallelism. We implemented our approach on top of the award-winning SAT solver CaDiCaL and the proof checker DRAT-trim. In an experimental evaluation, we demonstrate that we can compress proofs into skeletons that are 100 to 5, 000 times smaller than the original proofs. For almost all problems, proof reconstruction using a skeleton improves the solving time on a single core, and is around five times faster when using 24 cores.