Towards Understanding and Harnessing the Potential of Clause Learning

Efficient implementations of DPLL with the addition of clause learning are the fastest complete Boolean satisfiability solvers and can handle many significant real-world problems, such as verification, planning and design. Despite its importance, little is known of the ultimate strengths and limitations of the technique. This paper presents the first precise characterization of clause learning as a proof system (CL), and begins the task of understanding its power by relating it to the well-studied resolution proof system. In particular, we show that with a new learning scheme, CL can provide exponentially shorter proofs than many proper refinements of general resolution (RES) satisfying a natural property. These include regular and Davis-Putnam resolution, which are already known to be much stronger than ordinary DPLL. We also show that a slight variant of CL with unlimited restarts is as powerful as RES itself. Translating these analytical results to practice, however, presents a challenge because of the nondeterministic nature of clause learning algorithms. We propose a novel way of exploiting the underlying problem structure, in the form of a high level problem description such as a graph or PDDL specification, to guide clause learning algorithms toward faster solutions. We show that this leads to exponential speed-ups on grid and randomized pebbling problems, as well as substantial improvements on certain ordering formulas

Grammatical Evolution with Restarts for Fast Fractal Generation

In a previous work, the authors proposed a Grammatical Evolution algorithm to automatically generate Lindenmayer Systems which represent fractal curves with a pre-determined fractal dimension. This paper gives strong statistical evidence that the probability distributions of the execution time of that algorithm exhibits a heavy tail with an hyperbolic probability decay for long executions, which explains the erratic performance of different executions of the algorithm. Three different restart strategies have been incorporated in the algorithm to mitigate the problems associated to heavy tail distributions: the first assumes full knowledge of the execution time probability distribution, the second and third assume no knowledge. These strategies exploit the fact that the probability of finding a solution in short executions is non-negligible and yield a severe reduction, both in the expected execution time (up to one order of magnitude) and in its variance, which is reduced from an infinite to a finite value.Comment: 26 pages, 13 figures, Extended version of the paper presented at ANNIE'0

Approaches to grid-based SAT solving

In this work we develop techniques for using distributed computing resources to efficiently solve instances of the propositional satisfiability problem (SAT). The computing resources considered in this work are assumed to be geographically distributed and connected by a non-dedicated network. Such systems are typically referred to as computational grid environments. The time a modern SAT solver consumes while solving an instance varies according to a random distribution. Unlike many other methods for distributed SAT solving, this work identifies the random distribution as a valuable resource for solving-time reduction. The methods which use randomness in the run times of a search algorithm, such as the ones discussed in this work, are examples of multi-search. The main contribution of this work is in developing and analyzing the multi-search approach in SAT solving and showing its efficiency with several experiments. For the purpose of the analysis, the work introduces a grid simulation model which captures several of the properties of a grid environment which are not observed in more traditional parallel computing systems. The work develops two algorithmic frameworks for multi-search in SAT. The first, SDSAT, is based on using properties of the distribution of the solving time so that the expected time required to solve an instance is reduced. Based on the analysis of SDSAT, the work proposes an algorithm for efficiently using large number of computing resources simultaneously to solve collections of SAT instances. The analysis of SDSAT also motivates the second algorithmic framework, CL-SDSAT. The framework is used to efficiently solve many industrial SAT instances by carefully combining information learned in the distributed SAT solvers. All methods described in the work are directly applicable in a wide range of grid environments and can be used together with virtually unmodified state-of-the-art SAT solvers. The methods are experimentally verified using standard benchmark SAT instances in a production-level grid environment. The experiments show that using the relatively simple methods developed in the work, SAT instances which cannot be solved efficiently in sequential settings can be now solved in a grid environment

Understanding and Improving SAT Solvers via Proof Complexity and Reinforcement Learning

Despite the fact that the Boolean satisfiability (SAT) problem is NP-complete and believed to be intractable, SAT solvers are routinely used by practitioners to solve hard problems in wide variety of fields such as software engineering, formal methods, security, and AI. This gap between theory and practice has motivated an entire line of research whose primary goals are twofold: first, to develop a better theoretical framework aimed at accurately capturing solver behavior and thus prove tighter complexity bounds; and second, to further experimentally verify the soundness of the theory thus developed via rigorous empirical analysis and design theory-inspired techniques to improve solver performance. This interplay between theory and practice is at the heart of the work presented here. More precisely, this thesis contains a collection of results which attempt to resolve the above-described discrepancy between theory and practice. The first two sets of results are centered around the restart problem. Restarts are classes of methods which aim at erasing part of the progress a solver may have made at run time, in order to help solvers escape from the ``bad parts'' of the search space. We provide a detailed theoretical analysis of the power of restarts used in modern Conflict-Driven Clause Learning (CDCL) SAT solvers, where we prove a series of equivalence and separation results for various configurations of solvers with and without restarts. From the intuition developed via this theoretical analysis, we design and implement a machine learning based reset policy, where resets are variants of restarts that erase activity scores in addition to the parts of the solver state erased by restarts. We perform extensive experimental work to show that our reset policy outperforms both baseline and state-of-the-art solvers over a class of cryptographic instances derived from bitcoin mining problems. In a different direction, we propose the concept of hierarchical community structure (HCS) for Boolean formulas. We first theoretically show that formulas with ``good'' HCS parameter values have short CDCL proofs. Then we construct an Empirical Hardness Model using the HCS parameters. These HCS parameters exhibit a robust correlation with solver run time, leading to the development of a classifier capable of accurately distinguishing between easily solvable industrial instances and challenging random/crafted scenarios. We also present scaling studies of formulas with HCS structures to further support of theoretical analysis. In the latter part of the thesis, the focus shifts to satisfaction-driven clause-learning (SDCL) solvers, known to be being exponentially more powerful than CDCL solvers. Despite the theoretical strength of SDCL, it remains a challenge to automate and determinize such solvers. To address this, we again leverage machine learning techniques to strategically decide when to invoke an SDCL subroutine, with the goal of minimizing the associated overhead. The resulting SDCL solver, enhanced with MaxSAT techniques and conflict analysis, outperforms existing solvers on combinatorial benchmarks, particularly demonstrating superior efficacy on Mutilated Chess Board (MCB) problems

Identifying sources of global contention in constraint satisfaction search

Much work has been done on learning from failure in search to boost solving of combinatorial problems, such as clause-learning and clause-weighting in boolean satisfiability (SAT), nogood and explanation-based learning, and constraint weighting in constraint satisfaction problems (CSPs). Many of the top solvers in SAT use clause learning to good effect. A similar approach (nogood learning) has not had as large an impact in CSPs. Constraint weighting is a less fine-grained approach where the information learnt gives an approximation as to which variables may be the sources of greatest contention. In this work we present two methods for learning from search using restarts, in order to identify these critical variables prior to solving. Both methods are based on the conflict-directed heuristic (weighted-degree heuristic) introduced by Boussemart et al. and are aimed at producing a better-informed version of the heuristic by gathering information through restarting and probing of the search space prior to solving, while minimizing the overhead of these restarts. We further examine the impact of different sampling strategies and different measurements of contention, and assess different restarting strategies for the heuristic. Finally, two applications for constraint weighting are considered in detail: dynamic constraint satisfaction problems and unary resource scheduling problems

Random perfect lattices and the sphere packing problem

Motivated by the search for best lattice sphere packings in Euclidean spaces of large dimensions we study randomly generated perfect lattices in moderately large dimensions (up to d=19 included). Perfect lattices are relevant in the solution of the problem of lattice sphere packing, because the best lattice packing is a perfect lattice and because they can be generated easily by an algorithm. Their number however grows super-exponentially with the dimension so to get an idea of their properties we propose to study a randomized version of the algorithm and to define a random ensemble with an effective temperature in a way reminiscent of a Monte-Carlo simulation. We therefore study the distribution of packing fractions and kissing numbers of these ensembles and show how as the temperature is decreased the best know packers are easily recovered. We find that, even at infinite temperature, the typical perfect lattices are considerably denser than known families (like A_d and D_d) and we propose two hypotheses between which we cannot distinguish in this paper: one in which they improve Minkowsky's bound phi\sim 2^{-(0.84+-0.06) d}, and a competitor, in which their packing fraction decreases super-exponentially, namely phi\sim d^{-a d} but with a very small coefficient a=0.06+-0.04. We also find properties of the random walk which are suggestive of a glassy system already for moderately small dimensions. We also analyze local structure of network of perfect lattices conjecturing that this is a scale-free network in all dimensions with constant scaling exponent 2.6+-0.1.Comment: 19 pages, 22 figure