Although satisfiability problems (SAT) are NP-complete, state-of-the-art SAT solvers are able to solve large practical instances. The notion of backdoors has been introduced to capture structural properties of instances. Backdoors are a set of variables for which there exists some value assignment that leads to a polynomial-time solvable sub-problem. I address in this thesis the problem of finding all minimal backdoors, which is essential for studying value and variable ordering mistakes. I discuss our definition of sub-solvers and propose algorithms for finding backdoors. I implement our proposed algorithms by modifying a state-of-the-art SAT solver, Minisat. I analyze experimental results comparing our proposed algorithms to previous algorithms applied to random 3SAT, structured, and real-world instances. Our proposed algorithms improve over previous algorithms for finding backdoors in two ways. First, our algorithms often find smaller backdoors. Second, our algorithms often find a much larger number of backdoors.

Although satisfiability problems (SAT) are NP-complete, state-of-the-art SAT solvers are able to solve large practical instances. The notion of backdoors has been introduced to capture structural properties of instances. Backdoors are a set of variables for which there exists some value assignment that leads to a polynomial-time solvable sub-problem. I address in this thesis the problem of finding all minimal backdoors, which is essential for studying value and variable ordering mistakes. I discuss our definition of sub-solvers and propose algorithms for finding backdoors. I implement our proposed algorithms by modifying a state-of-the-art SAT solver, Minisat. I analyze experimental results comparing our proposed algorithms to previous algorithms applied to random 3SAT, structured, and real-world instances. Our proposed algorithms improve over previous algorithms for finding backdoors in two ways. First, our algorithms often find smaller backdoors. Second, our algorithms often find a much larger number of backdoors.

This paper presents a novel MEMS based DNA computer for solving SAT problems. No time-consuming sample preparation procedures and delicate sample applying equipment were required for the computing process. Moreover, experimental results show the bound DNA sequences can sustain the chemical solutions during computing processes such that the proposed method shall be useful in dealing with large scale problems. An algorithm based on a modified sticker model accompanied with a state-of-the-art MEMS-based microarray experiment is demonstrated to solve SAT problem which has long served as a benchmark in DNA computing. Unlike conventional DNA computing algorithms need an initial data pool to cover all correct and incorrect answers and further execute a series of separation procedures to destroy the unwanted ones, we built solutions in parts to satisfy one clause in one step, and eventually solve the entire Boolean formula through steps. Accordingly this algorithm greatly reduces the formation of unnecessary candidate solutions and shall be very practical as problem size grows. In this study, a novel MEMS-based technology including utilizing blank mask as the microarray substrate to prevent the self-fluorescent effect, a twin-mask back-side exposure process to improve the computing speed and a low-temperature backing process to prevent DNA damage during computing procedure. In addition, the minimal time requirement for DNA hybridization was also evaluated experimentally. The paper reports a novel computing method for solving SAT problem utilizing a state-of-art MEMS-based microarray. The advantage of this method is as the problem size scales up, it only needs to linearly increase the variety of sequences standing for variables and augment the array size. Therefore, while solving a complicated SAT problem, the numbers of DNA sample and the time for the computing process can be dramatically reduced with this approach.

Thesis (M. Ed.)--Kutztown University of Pennsylvania, 2001. / Source: Masters Abstracts International, Volume: 45-06, page: 2797. Typescript. Abstract precedes thesis as preliminary leaves. Includes bibliographical references (leaves 41-43).

Thesis (Ph. D.)--University of Chicago, Dept. of Education, June 2003. / Includes bibliographical references. Also available on the Internet.

No estudo da complexidade de problemas computacionais destacam-se duas classes conhecidas como P e NP. A questao P=NP e um dos maiores problemas nao resolvidos em Ciencia da Compu- tacao teorica e Matematica contemporanea. O problema SAT foi o primeiro problema reconhecido como NP-completo e consiste em verificar se uma determinada formula da logica proposicional clas- sica e ou nao satisfazivel. As implementacoes de algoritmos para resolver problemas SAT sao conhe- cidas como resolvedores SAT (SAT Solvers). Existem diversas aplicacoes em Ciencia da Computacao que podem ser realizadas com SAT Solvers e com outros resolvedores de problemas NP-completos que podem ser reduzidos ao SAT como por exemplo problemas de coloracao de grafos, problemas de agendamento e problemas de planejamento. Dentre os mais eficientes algoritmos para resolvedores de SAT estao Sato, Grasp, Chaff, MiniSat e Berkmin. O Algoritmo Chaff e baseado no Algoritmo DPLL o qual existe a mais de 40 anos e e a estrategia mais utilizada para os Sat Solvers. Essa dissertacao apresenta um estudo aprofundado do comportamento do zChaff (uma implementacao muito eficiente do Chaff) para saber o que esperar de suas execucoes em geral . / In the study of computational complexity stand out two classes known as P and NP. The question P = NP is one of the greatest unsolved problems in theoretical computer science and contemporary mathematics. The SAT problem was first problem recognized as NP-complete and consists to check whether a certain formula of classical propositional logic is satisfiable or not. The implementations of algorithms to solve SAT problems are known as SAT solvers. There are several applications in computer science that can be performed with SAT solvers and other solvers NP- complete problems can be reduced to SAT problems such as graph coloring, scheduling problems and planning problems. Among the most efficient algorithms for SAT solvers are Sato, Grasp, Chaf, MiniSat and Berkmin. The Chaff algorithm is based on the DPLL algorithm which there is more than 40 years and is the most used strategy for Sat Solvers. This dissertation presents a detailed study of the behavior of zChaff (a very efficient implementation of the Chaff) to know what to expect from their performance in general.

This thesis is concerned with design and implementation of a complete SAT solver accelerated on GPU. The achitecture of modern graphics cards is described as well as the CUDA platform and a list of common algorithms used for solving the boolean satisfiability problem (the SAT problem). The presented solution is based on the 3-SAT DC alogirthm, which belongs to the family of well-known DPLL based algorithms. This work describes problems encountered during the design and implementation. The resulting application was then analyzed and optimized. The presented solver cannot compete with state of the art solvers, but proves that it can be up to 21x faster than an equivalent sequential version. Unfortunately, the current implementation can only handle formulas of a limited size. Suggestions on further improvements are given in final sections.

This thesis presents a new approach to improve the efficiency of defect screening during manufacturing test of digital integrated circuits through the use of multiple observations during test generation. To address the limitations of test sets generated based on the single stuck-at fault model, we combine the advantages of multiple-detect and detection at all observable outputs in order to generate test sets that can improve surrogate detection. Imposing additional constraints, such as multiple observations, on the test generation process motivates the development of a new constrained automatic test pattern generation (ATPG) work flow that leverages the recent advancements in the Boolean satisfiability (SAT) problem. Building this ATPG work flow brings its own technical challenges and solutions described in detail in this thesis. To assess the effectiveness of the test sets generated by the proposed ATPG work flow, we evaluate them using coverage metrics for fault models that are not targeted explicitly during test generation. / Thesis / Master of Applied Science (MASc)

Due to the widespread demands for efficient Propositional Satisfiability (SAT) solvers and its derivatives in Electronic Design Automation applications, methods to boost the performance of the SAT solver are highly desired. This dissertation aims to enhance the performance of SAT and related SAT solving problems. A hybrid solution to boost SAT solver performance is proposed as an initial attack in this dissertation, via an integration of local and DPLL-based search approaches. Next, a different hybrid strategy is attempted that takes advantage of the conflicts in the SAT search, which plays a critical role in modern SAT solvers. Usually a learned conflict-induced clause is added back to the clause database. Although conflict-induced clauses help to block a portion of the search space, they can also become a burden due to the added cost in memory consumption and Boolean Constraint Propagation (BCP). We thus propose a novel double-layer conflict-driven learning to store only those "primary" conflict clauses back into the clause database while keeping the other clauses as pseudo Boolean constraints. With this approach our experiments demonstrate that the approach can improve both in performance and memory consumption. This work opens the door on how to assess the usefulness of conflict induced clauses. Besides the aforementioned works about enhancing SAT solver performance and reducing memory cost, this dissertation also proposed a contributing work on the extended SAT problem solving. The current SAT solvers can provide an assignment for a satisfiable propositional formula. However, the capability for a SAT solver to return an "optimal" solution for a given objective function is severely lacking. MIN-ONE SAT is an optimization problem which requires the satisfying assignment with the minimal number of Ones, and it can be easily extended to minimize an arbitrary linear objective function. While some research has been conducted on MIN-ONE SAT, the existing algorithms do not scale very well on large formulas. This dissertation presents a novel approximation algorithm (RelaxSAT) for MIN-ONE SAT. RelaxSAT generates a set of constraints from the objective function to guide the search. The constraints are gradually relaxed to eliminate the conflicts with the original Boolean SAT formula until a solution is found. The experiments demonstrate that RelaxSAT is able to handle very large instances which cannot be solved by existing MIN-ONE algorithms; furthermore, RelaxSAT is able to obtain a very tight bound on the solution with one to two orders of magnitude speedup. Based on the proposed powerful MIN-ONE SAT algorithm, we built a MAX-SAT solver which achieved more than one order of magnitude speed up compared with the-state-of-art MAX-SAT solver. We also discuss a promising application of this MAX-SAT solver in formal verification. / Ph. D.

Diese Arbeit befasst sich mit Worst-Case-Algorithmen für das Erfüllbarkeitsproblem boolescher Ausdrücke in konjunktiver Normalform. Im Wesentlichen betrachten wir Laufzeitschranken drei verschiedener Algorithmen, zwei für 3-SAT und einen für Unique-k-SAT. Wir entwickeln einen randomisierten Algorithmus, der eine Lösung eines erfüllbaren 3-KNF-Ausdrucks G mit n Variablen mit einer erwarteten Laufzeit von O(1.32793^n) findet. Der Algorithmus basiert auf der Analyse sogenannter Strings, welche Sequenzen von Klauseln der Länge drei sind. Dabei dürfen einerseits nicht aufeinanderfolgende Klauseln keine Variablen und andererseits aufeinanderfolgende Klauseln ein oder zwei Variablen gemeinsam haben. Gibt es wenige Strings, so treffen wir wahrscheinlich bereits während der String-Suche auf eine Lösung von G. 1999 entwarf Schöning einen Algorithmus mit einer Schranke von O(1.3334^n) für 3-SAT. Viele Strings erlauben es, die Laufzeit dieses Algorithmusses zu verbessern. Weiterhin werden wir den PPSZ-Algorithmus für Unique-k-SAT derandomisieren. Der 1998 von Paturi, Pudlak, Saks und Zane vorgestellte PPSZ-Algorithmus hat die besondere Eigenschaft, dass die Lösung eines eindeutig erfüllbaren 3-KNF-Ausdrucks in höchstens O(1.3071^n) erwarteter Laufzeit gefunden wird. Die derandomisierte Variante des Algorithmusses für Unique-k-SAT hat im Wesentlichen die gleiche Laufzeitschranke. Wir erreichen damit die momentan beste deterministische Worst-Case-Schranke für Unique-k-SAT. Zur Derandomisierung wenden wir die "Methode der kleinen Zufallsräume" an. Schließlich verbessern wir die Schranke für den Algorithmus von Iwama und Tamaki. 2003 kombinierten Iwama und Tamaki den PPSZ-Algorithmus mit Schönigs Algorithmus und konnten eine Schranke von O(1.3238^n) bewiesen. Um seinen Beitrag zum kombinierten Algorithmus zu steigern, justieren wir die Schranke des PPSZ-Algorithmusses. Damit erhalten wir die momentan beste randomisierte Worst-Case-Schranke für das 3-SAT-Problem von O(1.32216^n). / This work deals with worst-case algorithms for the satisfiability problem regarding boolean formulas in conjunctive normal form. The main part of this work consists of the analysis of the running time of three different algorithms, two for 3-SAT and one for Unique-k-SAT. We establish a randomized algorithm that finds a satisfying assignment for a satisfiable 3-CNF formula G on n variables in O(1.32793^n) expected running time. The algorithm is based on the analysis of so-called strings, which are sequences of clauses of size three, whereby non-succeeding clauses do not share a variable, and succeeding clauses share one or two variables. If there are not many strings, it is likely that we already encounter a solution of G while searching for strings. In 1999, Schöning proved a bound of O(1.3334^n) for 3-SAT. If there are many strings, we use them to improve the running time of Schöning''s algorithm. Furthermore, we derandomize the PPSZ algorithm for Unique-k-SAT. The PPSZ algorithm presented by Paturi, Pudlak, Saks, and Zane in 1998 has the feature that the solution of a uniquely satisfiable 3-CNF formula can be found in expected running time at most O(1.3071^n). In general, we will obtain a derandomized version of the algorithm for Unique-k-SAT that has essentially the same bound as the randomized version. This settles the currently best known deterministic worst-case bound for the Unique-k-SAT problem. We apply the `Method of Small Sample Spaces'' in order to derandomize the algorithm. Finally, we improve the bound for the algorithm of Iwama and Tamaki to get the currently best known randomized worst-case bound for the 3-SAT problem of O(1.32216^n). In 2003 Iwama and Tamaki combined Schöning''s and the PPSZ algorithm to yield an O(1.3238^n) bound. We tweak the bound for the PPSZ algorithm to get a slightly better contribution to the combined algorithm.