“Somebody says, ‘You know, you people always say that space is continuous. How do you know when you get to a small enough dimension that there really are enough points in between, that it isn’t just a lot of dots separated by little distances? ’ Or they say, ‘You know those quantum mechanical amplitudes you told me about, they’re so complicated and absurd, what makes you think those are right? Maybe they aren’t right. ’ Such remarks are obvious and are perfectly clear to anybody who is working on this problem. It does not do any good to point this out.” —Richard Feynman [1, p.161]

Abstract. We study the runtime distributions of backtrack procedures for propositional satisfiability and constraint satisfaction. Such procedures often exhibit a large variability in performance. Our study reveals some intriguing properties of such distributions: They are often characterized by very long tails or “heavy tails”. We will show that these distributions are best characterized by a general class of distributions that can have infinite moments (i.e., an infinite mean, variance, etc.). Such nonstandard distributions have recently been observed in areas as diverse as economics, statistical physics, and geophysics. They are closely related to fractal phenomena, whose study was introduced by Mandelbrot. We also show how random restarts can effectively eliminate heavy-tailed behavior. Furthermore, for harder problem instances, we observe long tails on the left-hand side of the distribution, which is indicative of a non-negligible fraction of relatively short, successful runs. A rapid restart strategy eliminates heavy-tailed behavior and takes advantage of short runs, significantly reducing expected solution time. We demonstrate speedups of up to two orders of magnitude on SAT and CSP encodings of hard problems in planning, scheduling, and circuit synthesis. Key words: satisfiability, constraint satisfaction, heavy tails, backtracking 1.

Recent work by Kautz et al. provides tantalizing evidence that large, classical planning problems may be efficiently solved by translating them into propositional satisfiability problems, using stochastic search techniques, and translating the resulting truth assignments back into plans for the original problems. We explore the space of such transformations, providing a simple framework that generates eight major encodings (generated by selecting one of four action representations and one of two frame axioms) and a number of subsidiary ones. We describe a fully-implemented compiler that can generate each of these encodings, and we test the compiler on a suite of STRIPS planning problems in order to determine which encodings have the best properties.

The past several years have seen much progress in the area of propositional reasoning and satisfiability testing. There is a growing consensus by researchers on the key technical challenges that need to be addressed in order to maintain this momentum. This paper outlines concrete technical challenges in the core areas of systematic search, stochastic search, problem encodings, and criteria for evaluating progress in this area. 1

Compilation to boolean satisfiability has become a powerful paradigm for solving AI problems. However, domains...

Boolean satisfiability (SAT) is NP-complete. No known algorithm for SAT is of polynomial time complexity. Yet, many of the SAT instances generated as a means of solving real-world electronic design automation problems are simple enough, structurally, that modern solvers can decide them efficiently. Consequently, SAT solvers are widely used in industry for logic verification. The most robust solver algorithms are poorly understood and only vaguely described in the literature of the field. We refine these algorithms, and present them clearly. We introduce several new techniques for Boolean constraint propagation that substantially improve solver efficiency. We explain why literal count decision strategies succeed, and on that basis, we introduce a new decision strategy that outperforms the state of the art. The culmination of this work is the most powerful SAT solver publically available.

In this paper we provide algorithms for reasoning with partitions of related logical axioms in propositional and first-order logic (FOL). We also provide a greedy algorithm that automatically decomposes a set of logical axioms into partitions. Our motivation is two-fold. First, we are concerned with how to reason e#ectively with multiple knowledge bases that have overlap in content. Second, we are concerned with improving the e#ciency of reasoning over a set of logical axioms by partitioning the set with respect to some detectable structure, and reasoning over individual partitions. Many of the reasoning procedures we present are based on the idea of passing messages between partitions. We present algorithms for reasoning using forward message-passing and using backward message-passing with partitions of logical axioms. Associated with each partition is a reasoning procedure. We characterize a class of reasoning procedures that ensures completeness and soundness of our message-passing ...

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.

The backtracking based Davis Putnam (DPLL) procedure remains the dominant method for deciding the satisfiability of a CNF formula. In recent years there has been much work on improving the basic procedure by adding features like improved heuristics and data structures, intelligent backtracking, clause learning, etc. Reasoning with binary clauses in DPLL has been a much discussed possibility for achieving improved performance, but to date solvers based on this idea have not been competitive with the best unit propagation based DPLL solvers. In this paper we experiment with a DPLL solver called 2CLS+EQ that makes more extensive use of binary clause reasoning than has been tried before. The results are very encouraging---2CLS+EQ is competitive with the very best DPLL solvers. The techniques it uses also open up a number of other possibilities for increasing our ability to solve SAT problems.

Research in algorithms for Boolean satisfiability and their efficient implementations [26, 8] has recently outpaced benchmarking efforts. Most of the classic DIMACS benchmarks from the early 1990s [12] can be solved in seconds on commodity PCs. More recent benchmarks take longer to solve primarily because of their large size, but are still solved in minutes [28]. However, small and difficult SAT instances must exist because Boolean satisfiability is NP-complete. Our work articulates a number of SAT instances that are unusually difficult for their size, including satisfiable instances from global routing and detailed routing for FPGAs [22]. Using an efficient implementation to solve the graph automorphism problem [21, 23, 25], we show that in structured SAT instances difficulty is sometimes associated with large numbers of symmetries. We propose a new, improved construction of symmetry-breaking clauses [11] and apply them to empirically demonstrate very significant speed-ups over current state of the art in Boolean satisfiability. Our techniques are formulated as pre-processing and can be applied to an arbitrary SAT solver without modifying its source code. We also show that considerations of symmetry may lead to more efficient reductions to SAT in the routing domain and potentially other applications.