The topic of exact, exponential-time algorithms for NP-hard problems has received a lot of attention, particularly with the focus of producing algorithms with stronger theoretical guarantees, e.g. upper bounds on the running time on the form O(c^n) for some c. Better methods of analysis may have an impact not only on these bounds, but on the nature of the algorithms as well. The most classic method of analysis of the running time of DPLL-style ("branching" or "backtracking") recursive algorithms consists of counting the number of variables that the algorithm removes at every step. Notable improvements include Kullmann's work on complexity measures, and Eppstein's work on solving multivariate recurrences through quasiconvex analysis. Still, one limitation that remains in Eppstein's framework is that it is difficult to introduce (non-trivial) restrictions on the applicability of a possible recursion. We introduce two new kinds of complexity measures, representing two ways to add such restrictions on applicability to the analysis. In the first measure, the execution of the algorithm is viewed as moving between a finite set of states (such as the presence or absence of certain structures or properties), where the current state decides which branchings are applicable, and each branch of a branching contains information about the resultant state. In the second measure, it is instead the relative sizes of the modelled attributes (such as the average degree or other concepts of density) that controls the applicability of branchings. We adapt both measures to Eppstein's framework, and use these tools to provide algorithms with stronger bounds for a number of problems. The problems we treat are satisfiability for sparse formulae, exact 3-satisfiability, 3-hitting set, and counting models for 2- and 3-satisfiability formulae, and in every case the bound we prove is stronger than previously known bounds.

Deciding the truth of a Quantified Boolean Formula (QBF) is a canonical PSPACE-complete problem. It provides a powerful framework for encoding problems that lie in PSPACE. These include many problems in automatic verification, and problems with discrete uncertainty or non-determinism. Two person adversarial games are another type of problem that are naturally encoded in QBF. It is standard practice to use Conjunctive Normal Form (CNF) when representing QBFs. Any propositional formula can be efficiently translated to CNF via the addition of new variables, and solvers can be implemented more efficiently due to the structural simplicity of CNF. However, the translation to CNF involves a loss of some structural information. This thesis shows that this structural information is important for efficient QBF solving, and shows how this structural information can be utilized to improve state-of-the-art QBF solving. First, a non-CNF circuit-based solver is presented. It makes use of information not present in CNF to improve its performance. We present techniques that allow it to exploit the duality between solutions and conflicts that is lost when working with CNF. This duality can also be utilized in the production of certificates, allowing both true and false formulas to have easy-to-verify certificates of the same form. Then, it is shown that most modern CNF-based solvers can benefit from the additional information derived from duality using only minor modifications. Furthermore, even partial duality information can be helpful. We show that for standard methods for conversion to CNF, some of the required information can be reconstructed from the CNF and greatly benefit the solver.

Boolean Satisfiability (SAT) is a fundamental NP-complete problem of determining whether there exists an assignment of variables which makes a Boolean formula evaluate to True. SAT is a convenient representation for many naturally occurring optimization and decisions problems such as planning and circuit verification. SAT is most commonly solved by a form of backtracking search which systematically explores the space of possible variable assignments. We show that the order in which variable polarities are assigned can have a significant impact on the performance of backtracking algorithms. We present several ways of transforming SAT instances into non-linear objective functions and describe three value-ordering methods based on iterative optimization techniques. We implement and test these heuristics in the widely-recognized MiniSAT framework. The first approach determines polarities by applying Newton's Method to a sparse system of non-linear objective functions whose roots correspond to the satisfying assignments of the propositional formula. The second approach determines polarities by minimizing an objective function corresponding to the number of clauses conflicting with each assignment. The third approach determines preferred polarities by performing stochastic gradient descent on objective functions sampled from a family of continuous potentials. The heuristics are evaluated on a set of standard benchmarks including random, crafted and industrial problems. We compare our results to five existing heuristics, and show that MiniSAT equipped with our heuristics often outperforms state-of-the-art SAT solvers.

Many problems in nature can be represented as some kind of a satisfiability problem. Several SAT solvers and SMT solvers have been developed in the last decade with the goal of checking the satisfiability of different SAT problems. An all-solution satisfiability modulo theories on top of the Z3 SMT solver is presented that uses the clause blocking algorithm to find all the solution sets of a SAT problem. Then, an incremental All-SMT solver has been presented based on the all-SMT solver which is able to find the satisfiable answers of an incremental SMT problem based on the solution set of the original problem.

xii, 136 p. : ill. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. / Many algorithms are known to work well in practice on a variety of different problem instances. Reusing existing algorithms for problems besides the one that they were designed to solve is often quite valuable. This is accomplished by transforming an instance of the new problem into an input for the algorithm and transforming the output of the algorithm into the correct answer for the new problem. To capitalize on the efficiency of the algorithm, it is essential that these transformations are efficient. Clearly not all problems will have efficient transformations to a particular algorithm so there are limitations on the scope of an algorithm. There is no previous study of which I am aware on determining the capability of an algorithm in terms of the complexity of problems that it can be used to solve. Two examples of this concept will be presented in proving the exact capability of the most well known algorithms for solving Satisfiability (SAT) and for solving Quantified Boolean Formula (QBF). The most well known algorithm for solving SAT is called DPLL. It has been well studied and is continuously being optimized in an effort to develop faster SAT solvers. The amount of work being done on optimizing DPLL makes it a good candidate for solving other problems. The notion of algorithm capability proved useful in applying DPLL to two areas of AI: Planning and Nonmonotonic Reasoning. Planning is PSPACE Complete in general, but NP Complete when restricted to problems that have polynomial length plans. Trying to optimize the plan length or introducing preferences increases the complexity of the problem. Despite the fact that these problems are harder than SAT, they are with in the scope of what DPLL can handle. Most problems in nonmonotonic reasoning are also harder than SAT. Despite this fact, DPLL is a candidate solution for nonmonotonic logics. The complexity of nonmonotonic reasoning in general is beyond the scope of what DPLL can handle. By knowing the capability of DPLL, one can analyze subsets of nonmonotonic reasoning that it can be used to solve. For example, DPLL is capable of solving the problem of model checking in normal default logic. Again, this problem is harder than SAT, but can still be solved with a single call to a SAT solver. The idea of algorithm capability led to the fascinating discovery that SAT solvers can solve problems that are harder than SAT. / Advisers: Matthew Ginsberg, Christopher Wilson

The memory cells used in modern field programmable gate arrays (FPGAs) are highly susceptible to single event upsets (SEUs). The typical mitigation strategy in the industry is some form of hardware redundancy in the form of duplication with comparison (DWC) or triple modular redundancy (TMR). While this strategy is highly effective in masking out the effect of faults, it incurs a large hardware cost. In this thesis, we explore a different approach to hardware redundancy. The core idea of our approach is to exploit the conflict-driven clause learning (CDCL) mechanism in modern Boolean satisfiability (SAT) solvers to provide us with invariants which can be realized as hardware checkers. Furthermore, we develop the algorithms required to select a subset of these invariants to be included as part of a checker circuit. This checker circuit then augments the original FPGA design. We find which look-up table (LUT) memory cells are sensitive to bitflips, then we automatically generate a checker circuit consisting of hardware invariants targeted towards those faults. We aim to reach 100% coverage of sensitizable faults. After extensive experimentation, we conclude that this approach is not competitive with DWC with respect to hardware area. However, we demonstrate that many bitflips will have reduced a detection latency compared to DWC. / Thesis / Master of Applied Science (MASc)

No description available.

Temporal logics are a very popular family of logical languages, used to specify properties of abstracted systems. In the last few years, many extensions of temporal logics have been proposed, in order to address the need to express more than just abstract properties. In our work we study temporal logics extended by local constraints, which allow to express quantitative properties on data values from an arbitrary relational structure called the concrete domain. An example of concrete domain can be (Z, <, =), where the integers are considered as a relational structure over the binary order relation and the equality relation. Formulas of temporal logics with constraints are evaluated on data-words or data-trees, in which each node or position is labeled by a vector of data from the concrete domain. We call the constraints local because they can only compare values at a fixed distance inside such models. Several positive results regarding the satisfiability of LTL (linear temporal logic) with constraints over the integers have been established in the past years, while the corresponding results for branching time logics were only partial. In this work we prove that satisfiability of CTL* (computation tree logic) with constraints over the integers is decidable and also lift this result to ECTL*, a proper extension of CTL*. We also consider other classes of concrete domains, particularly ones that are \"tree-like\". We consider semi-linear orders, ordinal trees and trees of a fixed height, and prove decidability in this framework as well. At the same time we prove that our method cannot be applied in the case of the infinite binary tree or the infinitely branching infinite tree. We also look into extending the expressiveness of our logic adding non-local constraints, and find that this leads to undecidability of the satisfiability problem, even on very simple domains like (Z, <, =). We then find a way to restrict the power of the non-local constraints to regain decidability.

Boolean satisfiability (SAT) is the problem of determining whether there exists an assignment of the Boolean variables to the truth values such that a given Boolean formula evaluates to true. SAT was the first example of an NP-complete problem. Only two decades ago SAT was mainly considered as of a theoretical interest. Nowadays, the picture is very different. SAT solving becomes mature and is a successful approach for tackling a large number of applications, ranging from artificial intelligence to industrial hardware design and verification. SAT solving consists of encodings and solvers. In order to benefit from the tremendous advances in the development of solvers, one must first encode the original problems into SAT instances. These encodings should not only be easily generated, but should also be efficiently processed by SAT solvers. Furthermore, an increasing number of practical applications in computer science can be expressed as constraint satisfaction problems (CSPs). However, encoding a CSP to SAT is currently regarded as more of an art than a science, and choosing an appropriate encoding is considered as important as choosing an algorithm. Moreover, it is much easier and more efficient to benefit from highly optimized state-of-the-art SAT solvers than to develop specialized tools from scratch. Hence, finding appropriate SAT encodings of CSPs is one of the most fascinating challenges for solving problems by SAT. This thesis studies SAT encodings of CSPs and aims at: 1) conducting a comprehensively profound study of SAT encodings of CSPs by separately investigating encodings of CSP domains and constraints; 2) proposing new SAT encodings of CSP domains; 3) proposing new SAT encoding of the at-most-one constraint, which is essential for encoding CSP variables; 4) introducing the redundant encoding and the hybrid encoding that aim to benefit from both two efficient and common SAT encodings (i.e., the sparse and order encodings) by using the channeling constraint (a term used in Constraint Programming) for SAT; and 5) revealing interesting guidelines on how to choose an appropriate SAT encoding in the way that one can exploit the availability of many efficient SAT solvers to solve CSPs efficiently and effectively. Experiments show that the proposed encodings and guidelines improve the state-of-the-art SAT encodings of CSPs.

Submitted by Marcos Roberto Gomes (mrobertosg@gmail.com) on 2017-06-22T15:28:21Z No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Dissertacao_Rodrigo_Farias_Araujo.pdf: 2432590 bytes, checksum: a0accf6a453257550a0ea9f75b50b687 (MD5) / Approved for entry into archive by Divisão de Documentação/BC Biblioteca Central (ddbc@ufam.edu.br) on 2017-06-23T14:38:14Z (GMT) No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Dissertacao_Rodrigo_Farias_Araujo.pdf: 2432590 bytes, checksum: a0accf6a453257550a0ea9f75b50b687 (MD5) / Approved for entry into archive by Divisão de Documentação/BC Biblioteca Central (ddbc@ufam.edu.br) on 2017-06-23T14:44:39Z (GMT) No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Dissertacao_Rodrigo_Farias_Araujo.pdf: 2432590 bytes, checksum: a0accf6a453257550a0ea9f75b50b687 (MD5) / Made available in DSpace on 2017-06-23T14:44:39Z (GMT). No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Dissertacao_Rodrigo_Farias_Araujo.pdf: 2432590 bytes, checksum: a0accf6a453257550a0ea9f75b50b687 (MD5) Previous issue date: 2017-03-30 / This work presents a new method of optimization applied to different classes of problems, such as non-convex and convex. The methodology consists in the use the counterexample generated from the model checking technique based on Boolean satisfiability theory (SAT) and satisfiability modulo theory (SMT), to guide the optimization process. Three algorithms of optimization are developed: Generic Algorithm, applied to any class of optimization problem, it will be used in the optimization of non-convex functions, Simplified Algorithm, used in the optimization of functions in which there is some previous knowledge, e. g., semi-defined or defined positive functions and Fast Algorithm, used to optimize convex functions. In addition, convergence proofs are provided for the respective algorithms. The algorithms are implemented using two model verifiers, CBMC which uses the SAT-based MiniSAT solver as back-end, and the ESBMC, which supports SMT-based solvers, such as Z3, Boolector and MathSAT. For perfomance evaluation, the algorithms are applied to a set of thirty functions taken from the literature and used to test optimization algorithms, they are also compared with traditional optimization algorithms usually used in solving non-convex optimization problems, such as genetic algorithm, particle swarm, pattern search, simulated annealing and nonlinear programming. Through the analysis of the results it can be concluded that the developed algorithms are suitable the classes of functions for which they were developed and have a higher rate of success in the search for the optimal value in comparison with the other algorithms. Finally, the developed methodology is applied to solve optimization problems in the context of the two-dimensional path planning for autonomous mobile robots. / Este trabalho apresenta um novo método de otimização aplicado a diferentes classes de problemas, como não-convexos e convexos. A metodologia consiste na utilização do contraexemplo gerado a partir da técnica de verificação de modelos, baseada na teoria de satisfatibilidade booleana (SAT) ou na teoria do módulo de satisfatibilidade (SMT), para guiar o processo de otimização. São desenvolvidos três algoritmos de otimização, são eles: Algoritmo Genérico, aplicado a qualquer classe de problema de otimização, neste será utilizado na otimização de funções não-convexas, Algoritmo Simplificado, empregado na otimização de funções nas quais tem-se algum conhecimento prévio, por exemplo, funções semi-definidas ou definidas positivas e Algoritmo Rápido, utilizado para otimização de funções convexas. Adicionalmente, são fornecidas as provas de convergência para os respectivos algoritmos. Os algoritmos são implementados utilizando dois verificadores de modelos, o CBMC que utiliza como back-end o solucionador MiniSAT baseado em SAT, e o ESBMC, que tem suporte aos solucionadores baseados em SMT, como: Z3, Boolector e MathSAT. Para avaliação de desempenho, os algoritmos são aplicados a um conjunto de trinta funções retiradas da literatura e utilizadas para teste de algoritmos de otimização, os mesmos também são comparados com algoritmos de otimização tradicionais usualmente empregados na resolução de problemas de otimização não-convexa, como: algoritmo genético, enxame de partícula, busca de padrões, recozimento simulado e programação não-linear. Através da análise dos resultados pode-se concluir que os algoritmos desenvolvidos são adequados as classes de funções para os quais foram desenvolvidos e possuem maior taxa de acerto na busca pelo valor ótimo em comparação com os outros algoritmos. Finalmente a metodologia desenvolvida é aplicada para resolver problemas de otimização no contexto de planejamento de caminhos bidimensionais para robô móveis autônomos.