Solving difficult instances of boolean satisfiability in the presence of symmetry

Abstract

Research in algorithms for Boolean satisfiability (SAT) and their implementations [45,41,10] has recently outpaced benchmarking efforts. Most of the classic DIMACS benchmarks [21] can now be solved in seconds on commodity PCs. More recent benchmarks [54] take longer to solve due of their large size, but are still solved in minutes. Yet, small and difficult SAT instances must exist if P NP. To this end, our work articulates SAT instances that are unusually difficult for their size, including satisfiable instances derived from Very Large Scale Integration (VLSI) routing problems. With an efficient implementation to solve the graph automorphism problem [39,50,51], we show that in structured SAT instances difficulty may be associated with large numbers of symmetries.We point out that a previously published symmetry-detection mechanism [18] based on a reduction to the graph automorphism problem often produces many spurious symmetries. Our work contributes two new reductions to graph automorphism, which detect all correct symmetries detected previously [18] as well as phase-shift symmetries not detected earlier. The correctness of our reductions is rigorously proven, and they are evaluated empirically.We also formulate an improved construction of symmetry-breaking clauses in terms of permutation cycles and propose to use only generators of symmetries in this process. These ideas are implemented in a fully automated flow that first detects symmetries in a given SAT instance, pre-processes it by adding symmetry-breaking clauses and then calls a state-of-the-art backtrack SAT solver. Significant speed-ups are shown on many benchmarks versus direct application of the solver.In an attempt to further improve the practicality of our approach, we propose a scheme for fast "opportunistic" symmetry detection and also show that considerations of symmetry may lead to more efficient reductions to SAT in the VLSI routing domain.

Symmetries in Modal Logics

“…Since then, many articles discuss how to detect and exploit syntactic symmetries in SAT solving [7,5,8,9,10,11,12,13,14]. Symmetries have been also extensively investigated and successfully exploited in other domains besides SAT like Constraint Satisfaction Problem [15,16], Integer Programming [17,18], Planning [19,20], Model Checking [21,22,23,24], Quantified Boolean Formulas (QBF) [25,26,27], and Satisfiability Modulo Theories (SMT) [28,29,30].…”

“…Both constructions are based on the MIN3C construction for propositional CNF formulas [11]. Unlike MIN3C, in our graphs we use two types of edges, coloring is more complex since we have to deal with different modalities, and we fix the representation of binary clauses and Boolean consistency by using a vertex and two edges to model binary clauses and an edge between the positive literal and negative literal vertices to model Boolean consistency.…”

Resolution cannot polynomially simulate compressed-BFS

“…In fact the XOR-Chain series of benchmarks, on which Cassatt peforms quite well, contains the smallest unsatisfiable benchmark that was unsolved in the SAT02 competition [28]. The two other series where Cassatt does well, the Bart series and Homer series, represent FPGA switch-box problems as described in [9]. clause 7 is opened.…”

“…Within a SAT instance of routing, for example, there can be many embedded PHP m n instances to enforce the capacities of routing channels [9]. The situation is similar on other structured problems such as planning and scheduling [10].…”

Faster symmetry discovery using sparsity of symmetries

“…One popular means of discovering the symmetries of a system is to first convert the system into a graph, and employ a general-purpose graph symmetry tool to uncover the symmetries [3]. These symmetries can then be reflected back into the original system domain.…”