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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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This book is a concise, self-contained, up-to-date introduction to extremal combinatorics for nonspecialists. There is a strong emphasis on theorems with particularly elegant and informative proofs, they may be called gems of the theory. The author presents a wide spectrum of the most powerful combinatorial tools together with impressive applications in computer science: methods of extremal set theory, the linear algebra method, the probabilistic method, and fragments of Ramsey theory. No special knowledge in combinatorics or computer science is assumed - the text is self-contained and the proofs can be enjoyed by undergraduate students in mathematics and computer science. Over 300 exercises of varying difficulty, and hints to their solution, complete the text.This second edition has been extended with substantial new material, and has been revised and updated throughout. It offers three new chapters on expander graphs and eigenvalues, the polynomial method and error-correcting codes. Most of the remaining chapters also include new material, such as the Kruskal-Katona theorem on shadows, the Lovász-Stein theorem on coverings, large cliques in dense graphs without induced 4-cycles, a new lower bounds argument for monotone formulas, Dvir's solution of the finite field Kakeya conjecture, Moser's algorithmic version of the Lovász Local Lemma, Schöning's algorithm for 3-SAT, the Szemerédi-Trotter theorem on the number of point-line incidences, surprising applications of expander graphs in extremal number theory, and some other new results.

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This book constitutes the refereed proceedings of the Third International Symposium on NASA Formal Methods, NFM 2011, held in Pasadena, CA, USA, in April 2011. The 26 revised full papers presented together with 12 tool papers, 3 invited talks, and 2 invited tutorials were carefully reviewed and selected from 141 submissions. The topics covered by NFM 2011 included but were not limited to: theorem proving, logic model checking, automated testing and simulation, model-based engineering, real-time and stochastic systems, SAT and SMT solvers, symbolic execution, abstraction and abstraction refinement, compositional verifcation techniques, static and dynamic analysis techniques, fault protection, cyber security, specification formalisms, requirements analysis, and applications of formal techniques.

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This book constitutes the refereed proceedings of the 25th International Symposium on Model Checking Software, SPIN 2018, held in Malaga, Spain, in June 2018.The 14 papers presented, 1 short paper, and 1 demo-tool paper, were carefully reviewed and selected from 28 submissions. Topics covered include formal verification techniques for automated analysis of software, formal analysis for modeling languages, such as UML/state charts, formal specification languages, temporal logic, design-by-contract, model checking, automated theorem proving, including SAT and SMT, verifying compilers, abstraction and symbolic execution techniques, and much more.

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This book constitutes the refereed proceedings of the 7th International Symposium on NASA Formal Methods, NFM 2015, held in Pasadena, CA, USA, in April 2015.The 24 revised regular papers presented together with 9 short papers were carefully reviewed and selected from 108 submissions. The topics include model checking, theorem proving, SAT and SMT solving, symbolic execution, static analysis, runtime verification, systematic testing, program refinement, compositional verification, security and intrusion detection, modeling and specification formalisms, model-based development, model-based testing, requirement engineering, formal approaches to fault tolerance, and applications of formal methods.

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This book constitutes the refereed proceedings of the 23rd International Conference on Automated Deduction, CADE-23, held in Wroclaw, Poland, in July/August 2011.The 28 revised full papers and 7 system descriptions presented were carefully reviewed and selected from 80 submissions. Furthermore, four invited lectures by distinguished experts in the area were included. Among the topics addressed are systems and tools for automated reasoning, rewriting logics, security protocol verification, unification, theorem proving, clause elimination, SAT, satifiability, interactive theorem proving, theory reasoning, static analysis, decision procedures, etc.

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A decision procedure is an algorithm that, given a decision problem, terminates with a correct yes/no answer. Here, the authors focus on theories that are expressive enough to model real problems, but are still decidable. Specifically, the book concentrates on decision procedures for first-order theories that are commonly used in automated verification and reasoning, theorem-proving, compiler optimization and operations research. The techniques described in the book draw from fields such as graph theory and logic, and are routinely used in industry. The authors introduce the basic terminology of satisfiability modulo theories and then, in separate chapters, study decision procedures for each of the following theories: propositional logic, equalities and uninterpreted functions, linear arithmetic, bit vectors, arrays, pointer logic, and quantified formulas. They also study the problem of deciding combined theories and dedicate a chapter to modern techniques based on an interplay between a SAT solver and a decision procedure for the investigated theory.This textbook has been used to teach undergraduate and graduate courses at ETH Zurich, at the Technion, Haifa, and at the University of Oxford. Each chapter includes a detailed bibliography and exercises. Lecturers' slides and a C++ library for rapid prototyping of decision procedures are available from the authors' website.

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This book constitutes the proceedings of the 11th International Symposium on Frontiers of Combining Systems, FroCoS 2017, held in Brasília, Bazil, in September 2017.The 17 papers presented in this volume were carefully reviewed and selected from 26 submissions. They were organized in topical sections named: description and temporal logics, decision procedures, decidability and verification, SAT, SMT and automated theorem proving, term rewriting, and properties and combinations of logics.The paper 'Subtropical Satisfiability' is open access under a CC BY 4.0 license via link.springer.com.

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