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The Boolean conjunctive normal form (CNF) satisability problem, called SAT for short, gets as input a CNF formula and has to decide whether this formula admits a satisfying truth assignment. As is well known, the remarkable result by S. Cook in 1971 established SAT as the first and genuine complete problem for the complexity class NP. In this thesis we consider SAT for a subclass of CNF, the so called Mixed Horn formula class (MHF). A formula F 2 MHF consists of a 2-CNF part P and a Horn part H. We propose that MHF has a central relevance in CNF because many prominent NP-complete problems, e.g. Feedback Vertex Set, Vertex Cover, Dominating Set and Hitting Set, can easily be encoded as MHF. Furthermore, we show that SAT remains NP-complete for some interesting subclasses of MHF. We also provide algorithms for some of these subclasses solving SAT in a better running time than O(2^0.5284n) which is the best bound for MHF so far. In addition, we investigate the computational complexity of some prominent variants of SAT, namely not-all-equal SAT (NAE-SAT) and exact SAT (XSAT) restricted to the class of linear CNF formulas.

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Erscheinungsdatum: 28.06.2015, Medium: Taschenbuch, Einband: Kartoniert / Broschiert, Titel: Computational Complexity of SAT, XSAT and NAE-SAT, Titelzusatz: For linear and mixed Horn CNF formulas, Autor: Schmidt, Tatjana, Verlag: Südwestdeutscher Verlag für Hochschulschriften AG Co. KG, Sprache: Englisch, Rubrik: Naturwissenschaften // Technik allg., Seiten: 172, Informationen: Paperback, Gewicht: 272 gr, Verkäufer: averdo

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High Quality Content by WIKIPEDIA articles! The Valiant Vazirani theorem is an important result in computational complexity theory. It was proven by Leslie Valiant and Vijay Vazirani in their paper titled "NP is as easy as detecting unique solutions" published in 1986. The theorem states that if there is a polynomial time algorithm for UNIQUE-SAT, then NP=RP. The theorem implies that even if the number of satisfying assignments is very small, SAT (which is an NP-complete problem) still remains a hard problem. UNIQUE-SAT is a promise problem that decides whether a given Boolean formula is unsatisfiable or has exactly one satisfying assignment. In the first case a UNIQUE-SAT algorithm would reject, and in the second it would accept the formula. If the formula has more than one satisfying assignment then the behavior of the UNIQUE-SAT algorithm does not matter.

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The verification of systems to guarantee their correct behavior is discussed in this book. The mainly applied algorithmic method is the model checking technique combined with algorithms for solving the satisfiability problem (short: SAT). SAT-based verification of discrete systems has become one of the most effective technique within the last 10 years, such that industrial as well as academic applications heavily rely on it. The book covers the whole range of a SAT-based tool application. We propose extensions and concepts that concentrate on the core of a SAT-solver. However, these proposals are then transferred to novel verification models. Moreover, we describe approaches that incorporate the structure of the problem to exploit knowledge gained during the verification process on the level of the SAT-solver. The main focus of the book is on the verification of incomplete system designs, which occur for example in the early phase of a design. We describe various SAT-based modeling concepts that vary regarding their expressiveness and computational resources. The proposed methods are evaluated experimentally to guarantee their applicability in practice.

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High Quality Content by WIKIPEDIA articles! In computational complexity theory, a branch of computer science, Schaefer's theorem states necessary and sufficient conditions under which a finite set S of Boolean relations yields polynomial-time or NP-complete problems when the relations of S are used to constrain some of the propositional variables. More precisely, Schaefer defines a decision problem which he calls the Generalized Satisfiability problem for S (denoted SAT(S)). The problem is to determine whether the given formula is satisfiable, in other words if the variables can be assigned values such that they satisfy all the constraints. Special cases of SAT(S) include the variants of Boolean satisfiability problem and the problem can also be viewed as a constraint satisfaction problem over the Boolean domain.

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This thesis deals with a general modeling framework for large-scale biological systems which is, on the one hand, applied to various practical instances, and on the other hand, strictly formalized and mathematically analyzed with respect to its complexity and structure. For the biological application initially an overview of existing analytic methods for biological systems is presented, and the proposed modeling framework is classified in this context. The framework is based on logical implication formulas. It allows for the verification of a biological model, the prediction of its response to prescribed stimuli, as well as the identification of possible intervention strategies for diseases or failure modes. This basic model is afterwards extended into two directions: First, timing information of reactions in the biological unit are incorporated. This generalization additionally enables to detect possible unknown timing information or inconsistencies that arise due to modeling errors. Besides this, it provides a method to consistently integrate the logical models of related biological units into one model. Second, the purely binary basic framework is enhanced by including a fine discretization of a biological component's activity level. This permits to express different effects depending on different levels of activity of one component, and therefore the predictions of the model become more sophisticated. On the mathematical side the logical framework and its extensions are derived and formalized. The basic model evolves to a special type of satisfiability problem (SAT) whose complexity is classified to be generally hard but mathematically easy subclasses are identified. The correspondence between SAT and integer programming is exploited and the underlying polyhedra are analyzed. Interestingly, the SAT problem allows for a wider class of polynomially solvable problems than its integer programming equivalent. Nevertheless, the computational results provided proof that the integer programming approach is computationally feasible. The basic SAT problem can additionally be translated into a bipartite digraph for which algorithms are adapted, and their practical use is discussed. Furthermore, for a special class of biological units a duality framework based on linear programming duality is derived, which completes the theory of such biological units. The dynamic extension of the basic framework yields a related SAT problem that contains the original one as a special case, and is thus hard to solve as well. The focus for this extension is on the analysis of maximally feasible and minimally infeasible solutions of the extended SAT problem. Therefore, it is necessary to optimize over the set of solutions of the SAT problem which suggests to employ the equivalent integer programming approach. To enumerate all maximally feasible and minimally infeasible solutions the Joint Generation algorithm is utilized. To this end, a monotone reformulation of the extended SAT problem is derived that preserves the maximally feasible and minimally infeasible solutions, and at the same time significantly reduces the size of the description. In certain very restrictive cases the resulting integer optimization problems are even computationally tractable. Finally, the minimally infeasible solutions are completely characterized by means of graph structures in the original digraph, and an alternative method for computing all minimally infeasible solutions via polyhedral projection is obtained. The discrete extension of the logical framework leads to a generalization of the SAT problem, the so called interval satisfiability problem. In this setting the variables are integer valued and associated intervals provide the set of values for which the expression becomes TRUE. To computationally determine feasible solutions, this problem is transformed to a system of polynomials which can be checked for feasibility by means of Hilbert's Nullstellensatz. Moreover, the general interval satisfiability problem is analyzed with respect to complexity and satisfiability. Concerning the computational complexity, it is shown to be generally hard even if assuming certain restrictions for the formulas. Concerning the satisfiability behavior the well known threshold phenomenon of classical random SAT, which has been observed for interval satisfiability, is examined and lower bounds on specific thresholds are identified.

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