CDCL SAT Solvers Research Articles

SAT Modulo Symmetries for Graph Generation and Enumeration

We propose a novel SAT-based approach to graph generation. Our approach utilizes the interaction between a CDCL SAT solver and a special symmetry propagator where the SAT solver runs on an encoding of the desired graph property. The symmetry propagator checks partially generated graphs for minimality with respect to a lexicographic ordering during the solving process. This approach has several advantages over a static symmetry breaking: (i) symmetries are detected early in the generation process, (ii) symmetry breaking is seamlessly integrated into the CDCL procedure, and (iii) the propagator performs a complete symmetry breaking without causing a prohibitively large initial encoding. We instantiate our approach by generating extremal graphs with certain restrictions in terms of forbidden subgraphs and diameter. In particular, we could confirm the Murty-Simon Conjecture (1979) on diameter-2-critical graphs for graphs up to 19 vertices and prove the exact number of Ramsey graphs \(\mathcal{R}(3,5,n)\) and \(\mathcal{R}(4,4,n)\) .

The state-of-the-art Boolean Satisfiability based cryptanalysis

Over the past two decades, the effectiveness of Conflict-Driven Boolean Satisfiability (CDCL SAT) solutions in a range of applications such as verification, monitoring, safety and the AI has improved dramatically in relation to industrial problems. The availability of powerful general-purpose search tools such as the SAT solver have led many scientists to propose satellite methods for cryptanalysis such as havoc collision techniques and symmetric encryption schemes. One feature of all of the above suggested SAT cryptanalysis is that it is a black box in that the problem with encrypted cryptanalysis is encoded as an instance with SAT and a CDCL SAT solver is called upon to solve that instance. A limitation is that the encoding produced may be too large to solve efficiently by any modern solver. Perhaps more importantly, the solver is in no way trained or tailored for solving the particular instance. Finally, very little work has been done in the field of SAT-based cryptanalysis to exploit parallelism.

Toward Verifying Nonlinear Integer Arithmetic

We eliminate a key roadblock to efficient verification of nonlinear integer arithmetic using CDCL SAT solvers, by showing how to construct short resolution proofs for many properties of the most widely used multiplier circuits. Such short proofs were conjectured not to exist. More precisely, we give n O (1) size regular resolution proofs for arbitrary degree 2 identities on array, diagonal, and Booth multipliers and n O (log n ) size proofs for these identities on Wallace tree multipliers.

Enhancing Static Symmetry Breaking with Dynamic Symmetry Handling in CDCL SAT Solvers

Symmetry exploitation techniques in SAT can be classified into two main categories: static symmetry breaking and dynamic symmetry handling. It is currently recognized that the most effective approach is the static one, which proceeds by preprocessing the formula to be solved in order to add symmetry-breaking predicates (SBPs) that guarantee the preservation of equisatisfiability but not equivalence. Due to the large size of the CNF symmetry groups, only a subset of the SBPs is generated resulting in a partial symmetry breaking. It is at this level that dynamic symmetry handling can intervene to help the solver avoid as much as possible, the exploration of isomorphic symmetrical subspaces for symmetries not (entirely) broken by the static approach. However, the use of both techniques within the same solver could compromise the result. We propose in this paper in addition to an optimization of the dynamic scheme SLS, an approach that allows the two methods of symmetry exploitation to coexist within the same solver in order to increase its efficiency while preserving its correctness. This is to the best of our knowledge, the first time such an integration is envisaged within the framework of SAT solving. Experimental results conducted on hard combinatorial instances drawn from SAT competitions show that symmetrical learning can indeed improve static symmetry breaking.

Reusing the Assignment Trail in CDCL Solvers

We present the solver RestartSAT which includes a novel technique to reduce the cost to perform a restart in CDCL SAT solvers. This technique, called ReusedTrail, exploits the observation that CDCL solvers often reassign the same variables to the same truth values after a restart. It computes a partial restart level for which it is guaranteed that all variables below this level will be reassigned after a full restart. RestartSAT, an extended version of MiniSAT, incorporates ReusedTrail – which can be implemented easily in almost any CDCL solver. On average, it saves over a third of the decisions and propagations necessary to solve a problem using a Luby restart policy with unit run 1. Experimental results show that RestartSAT solves over a dozen more application instances than the default MiniSAT.