Parallelization of SAT on Reconfigurable Hardware

Quoique très difficile à résoudre, le problème de satisfiabilité Booléenne (SAT) est fréquemment utilisé lors de la modélisation d’applications industrielles. À cet effet, les deux dernières décennies ont vu une progression fulgurante des outils conçus pour trouver des solutions à ce problème NP-complet. Deux grandes avenues générales ont été explorées afin de produire ces outils, notamment l’approche logicielle et matérielle. Afin de raffiner et améliorer ces solveurs, de nombreuses techniques et heuristiques ont été proposées par la communauté de recherche. Le but final de ces outils a été de résoudre des problèmes de taille industrielle, ce qui a été plus ou moins accompli par les solveurs de nature logicielle. Initialement, le but de l’utilisation du matériel reconfigurable a été de produire des solveurs pouvant trouver des solutions plus rapidement que leurs homologues logiciels. Cependant, le niveau de sophistication de ces derniers a augmenté de telle manière qu’ils restent le meilleur choix pour résoudre SAT. Toutefois, les solveurs modernes logiciels n’arrivent toujours pas a trouver des solutions de manière efficace à certaines instances SAT. Le but principal de ce mémoire est d’explorer la résolution du problème SAT dans le contexte du matériel reconfigurable en vue de caractériser les ingrédients nécessaires d’un solveur SAT efficace qui puise sa puissance de calcul dans le parallélisme conféré par une plateforme FPGA. Le prototype parallèle implémenté dans ce travail est capable de se mesurer, en termes de vitesse d’exécution à d’autres solveurs (matériels et logiciels), et ce sans utiliser aucune heuristique. Nous montrons donc que notre approche matérielle présente une option prometteuse vers la résolution d’instances industrielles larges qui sont difficilement abordées par une approche logicielle.Though very difficult to solve, the Boolean satisfiability problem (SAT) is extensively used to model various real-world applications and problems. Over the past two decades, researchers have tried to provide tools that are used, to a certain degree, to find solutions to the Boolean satisfiability problem. The nature of these tools is broadly divided in software and reconfigurable hardware solvers. In addition, the main algorithms used to solve this problem have also been complemented with heuristics of various levels of sophistication to help overcome some of the NP-hardness of the problem. The end goal of these tools has been to provide solutions to industrial-sized problems of enormous size. Initially, reconfigurable hardware tools provided a promising avenue to accelerating SAT solving over traditional software based solutions. However, the level of sophistication of software solvers overcame their hardware counterparts, which remained limited to smaller problem instances. Even so, modern state-of-the-art software solvers still fail unpredictably on some instances. The main focus of this thesis is to explore solving SAT on reconfigurable hardware in order to gain an understanding of what would be essential ingredients to add (and discard) to a very efficient hardware SAT solver that obtains its processing power from the raw parallelism of an FPGA platform. The parallel prototype solver that was implemented in this work has been found to be comparable with other hardware and software solvers in terms of execution speed even though no heuristics or other helping techniques were implemented. We thus show that our approach provides a very promising avenue to solving large, industrial SAT instances that might be difficult to handle by software solvers

Solving graph coloring and SAT problems using field programmable gate arrays.

Chu-Keung Chung.Thesis (M.Phil.)--Chinese University of Hong Kong, 1999.Includes bibliographical references (leaves 88-92).Abstracts in English and Chinese.Abstract --- p.iAcknowledgments --- p.iiiChapter 1 --- Introduction --- p.1Chapter 1.1 --- Motivation and Aims --- p.1Chapter 1.2 --- Contributions --- p.3Chapter 1.3 --- Structure of the Thesis --- p.4Chapter 2 --- Literature Review --- p.6Chapter 2.1 --- Introduction --- p.6Chapter 2.2 --- Complete Algorithms --- p.7Chapter 2.2.1 --- Parallel Checking --- p.7Chapter 2.2.2 --- Mom's --- p.8Chapter 2.2.3 --- Davis-Putnam --- p.9Chapter 2.2.4 --- Nonchronological Backtracking --- p.9Chapter 2.2.5 --- Iterative Logic Array (ILA) --- p.10Chapter 2.3 --- Incomplete Algorithms --- p.11Chapter 2.3.1 --- GENET --- p.11Chapter 2.3.2 --- GSAT --- p.12Chapter 2.4 --- Summary --- p.13Chapter 3 --- Algorithms --- p.14Chapter 3.1 --- Introduction --- p.14Chapter 3.2 --- Tree Search Techniques --- p.14Chapter 3.2.1 --- Depth First Search --- p.15Chapter 3.2.2 --- Forward Checking --- p.16Chapter 3.2.3 --- Davis-Putnam --- p.17Chapter 3.2.4 --- GRASP --- p.19Chapter 3.3 --- Incomplete Algorithms --- p.20Chapter 3.3.1 --- GENET --- p.20Chapter 3.3.2 --- GSAT Algorithm --- p.22Chapter 3.4 --- Summary --- p.23Chapter 4 --- Field Programmable Gate Arrays --- p.24Chapter 4.1 --- Introduction --- p.24Chapter 4.2 --- FPGA --- p.24Chapter 4.2.1 --- Xilinx 4000 series FPGAs --- p.26Chapter 4.2.2 --- Bitstream --- p.31Chapter 4.3 --- Giga Operations Reconfigurable Computing Platform --- p.32Chapter 4.4 --- Annapolis Wildforce PCI board --- p.33Chapter 4.5 --- Summary --- p.35Chapter 5 --- Implementation --- p.36Chapter 5.1 --- Parallel Graph Coloring Machine --- p.36Chapter 5.1.1 --- System Architecture --- p.38Chapter 5.1.2 --- Evaluator --- p.39Chapter 5.1.3 --- Finite State Machine (FSM) --- p.42Chapter 5.1.4 --- Memory --- p.43Chapter 5.1.5 --- Hardware Resources --- p.43Chapter 5.2 --- Serial Graph Coloring Machine --- p.44Chapter 5.2.1 --- System Architecture --- p.44Chapter 5.2.2 --- Input Memory --- p.46Chapter 5.2.3 --- Solution Store --- p.46Chapter 5.2.4 --- Constraint Memory --- p.47Chapter 5.2.5 --- Evaluator --- p.48Chapter 5.2.6 --- Input Mapper --- p.49Chapter 5.2.7 --- Output Memory --- p.49Chapter 5.2.8 --- Backtrack Checker --- p.50Chapter 5.2.9 --- Word Generator --- p.51Chapter 5.2.10 --- State Machine --- p.51Chapter 5.2.11 --- Hardware Resources --- p.54Chapter 5.3 --- Serial Boolean Satisfiability Solver --- p.56Chapter 5.3.1 --- System Architecture --- p.58Chapter 5.3.2 --- Solutions --- p.59Chapter 5.3.3 --- Solution Generator --- p.59Chapter 5.3.4 --- Evaluator --- p.60Chapter 5.3.5 --- AND/OR --- p.62Chapter 5.3.6 --- State Machine --- p.62Chapter 5.3.7 --- Hardware Resources --- p.64Chapter 5.4 --- GSAT Solver --- p.65Chapter 5.4.1 --- System Architecture --- p.65Chapter 5.4.2 --- Variable Memory --- p.65Chapter 5.4.3 --- Flip-Bit Vector --- p.66Chapter 5.4.4 --- Clause Evaluator --- p.67Chapter 5.4.5 --- Adder --- p.70Chapter 5.4.6 --- Random Bit Generator --- p.71Chapter 5.4.7 --- Comparator --- p.71Chapter 5.4.8 --- Sum Register --- p.71Chapter 5.5 --- Summary --- p.71Chapter 6 --- Results --- p.73Chapter 6.1 --- Introduction --- p.73Chapter 6.2 --- Parallel Graph Coloring Machine --- p.73Chapter 6.3 --- Serial Graph Coloring Machine --- p.74Chapter 6.4 --- Serial SAT Solver --- p.74Chapter 6.5 --- GSAT Solver --- p.75Chapter 6.6 --- Summary --- p.76Chapter 7 --- Conclusion --- p.77Chapter 7.1 --- Future Work --- p.78Chapter A --- Software Implementation of Graph Coloring in CHIP --- p.79Chapter B --- Density Improvements Using Xilinx RAM --- p.81Chapter C --- Bit stream Configuration --- p.83Bibliography --- p.88Publications --- p.9

Arquitecturas reconfiguráveis para problemas de optimização combinatória

Os problemas combinatórios têm uma gama extremamente ampla de aplicações numa variedade de áreas de engenharia, incluindo teste de circuitos electrónicos, reconhecimento de padrões, síntese lógica, etc. Muitos dos problemas de interesse pertencem às classes NP-hard e NP-complete, o que implica que os algoritmos relevantes têm no pior caso complexidade exponencial. Este facto impede a solução de muitos problemas práticos com a ajuda de computadores convencionais. As implementações em circuitos integrados específicos também não são viáveis, em particular por causa da própria heterogeneidade dos problemas combinatórios. Uma solução alternativa consiste no uso de dispositivos reconfiguráveis que podem ser personalizados para um algoritmo específico e reutilizados para outros algoritmos via uma simples reprogramação da sua estrutura interna. As implementações baseadas em hardware reconfigurável permitem optimizar a execução dos algoritmos relevantes com a ajuda de técnicas tais como processamento paralelo, unidades funcionais personalizadas, etc. Tais implementações possibilitam conter o efeito de crescimento exponencial do tempo de computação, permitindo deste modo a solução de problemas combinatórios complexos. Recentemente foram desenvolvidos vários sistemas reconfiguráveis destinados a resolver problemas combinatórios. Estes são principalmente baseados na ideia de hardware específico para a instância, em que para cada instância do problema é gerado um circuito particular. Nesta tese exploramos duas abordagens alternativas. A primeira é orientada para o domínio e permite processar uma variedade de problemas da área da computação combinatória. Para tal é projectado e implementado um processador combinatório reconfigurável e são desenvolvidos métodos e ferramentas que asseguram a sua reconfiguração dinâmica parcial. A segunda abordagem é orientada para a aplicação e é destinada a resolver um problema combinatório específico. Em particular, é proposta uma arquitectura inovadora para a solução do problema de satisfação booleana com a ajuda de uma combinação de software e de hardware reconfigurável. A técnica adoptada elimina a compilação de hardware específica à instância e permite processar problemas que excedem os recursos lógicos disponíveis. São também exploradas as possibilidades de implementação em hardware reconfigurável de estratégias evolutivas para o caso do problema do caixeiro viajante. Esta tese estende o domínio de aplicação da computação reconfigurável ao demonstrar que esta é capaz de acelerar algoritmos com fluxos de controlo complexos.Combinatorial problems have an extremely wide range of practical applications in a variety of engineering areas, including the testing of electronic circuits, pattern recognition, logic synthesis, etc. Many of the problems of interest belong to the classes NP-hard and NP-complete, which implies that the relevant algorithms have an exponential worst-case complexity. This fact precludes the solution of many practical problems with conventional computers. ASIC-based implementations are also not viable, in particular because of the inherent heterogeneity of combinatorial problems. Reconfigurable devices offer an alternative solution, which can be customized to the requirements of a specific algorithm and reutilized for other algorithms via a simple reprogramming of their internal structure. Implementations based on reconfigurable hardware permit the execution of the relevant algorithms to be optimized with the aid of such techniques as parallel processing, personalized functional units, etc. Such implementations allow the effect of exponential growth in the computation time to be delayed, thus enabling more complex problem instances to be solved. Recently, a few reconfigurable engines for combinatorial problems have been developed. They are mainly based on the idea of instance-specific hardware, which assumes that a particular circuit is generated for each problem instance. In this thesis we explore two alternative approaches. The first, domain-specific, approach enables a variety of problems in the area of combinatorial computation to be addressed. For this purpose, a reconfigurable combinatorial processor has been designed and implemented and a number of methods and tools that support its partial dynamic reconfiguration have been developed. The second, application-specific, approach is oriented towards solving individual combinatorial problems. In particular, a novel architecture is proposed for solving the Boolean satisfiability problem with the aid of software and reconfigurable hardware. The adopted technique avoids instance-specific hardware compilation and permits problems that exceed the available logic resources to be solved. The possibility of implementing evolutionary strategies for the traveling salesman problem in reconfigurable hardware is also explored. This thesis extends the application domain of reconfigurable computing by demonstrating that it is effective in accelerating algorithms with complex control flows