Proceedings of SAT Competition 2013 : Solver and Benchmark Descriptions

Peer reviewe

Complete Randomized Cutting Plane Algorithms for Propositional Satisfiability

The propositional satisfiability problem (SAT) is a fundamental problem in computer science and combinatorial optimization. A considerable number of prior researchers have investigated SAT, and much is already known concerning limitations of known algorithms for SAT. In particular, some necessary conditions are known, such that any algorithm not meeting those conditions cannot be efficient. This paper reports a research to develop and test a new algorithm that meets the currently known necessary conditions. In chapter three, we give a new characterization of the convex integer hull of SAT, and two new algorithms for finding strong cutting planes. We also show the importance of choosing which vertex to cut, and present heuristics to find a vertex that allows a strong cutting plane. In chapter four, we describe an experiment to implement a SAT solving algorithm using the new algorithms and heuristics, and to examine their effectiveness on a set of problems. In chapter five, we describe the implementation of the algorithms, and present computational results. For an input SAT problem, the output of the implemented program provides either a witness to the satisfiability or a complete cutting plane proof of satisfiability. The description, implementation, and testing of these algorithms yields both empirical data to characterize the performance of the new algorithms, and additional insight to further advance the theory. We conclude from the computational study that cutting plane algorithms are efficient for the solution of a large class of SAT problems

Decision procedures for linear arithmetic

In this thesis, we present new decision procedures for linear arithmetic in the context of SMT solvers and theorem provers: 1) CutSat++, a calculus for linear integer arithmetic that combines techniques from SAT solving and quantifier elimination in order to be sound, terminating, and complete. 2) The largest cube test and the unit cube test, two sound (although incomplete) tests that find integer and mixed solutions in polynomial time. The tests are especially efficient on absolutely unbounded constraint systems, which are difficult to handle for many other decision procedures. 3) Techniques for the investigation of equalities implied by a constraint system. Moreover, we present several applications for these techniques. 4) The Double-Bounded reduction and the Mixed-Echelon-Hermite transformation, two transformations that reduce any constraint system in polynomial time to an equisatisfiable constraint system that is bounded. The transformations are beneficial because they turn branch-and-bound into a complete and efficient decision procedure for unbounded constraint systems. We have implemented the above decision procedures (except for Cut- Sat++) as part of our linear arithmetic theory solver SPASS-IQ and as part of our CDCL(LA) solver SPASS-SATT. We also present various benchmark evaluations that confirm the practical efficiency of our new decision procedures.In dieser Arbeit präsentieren wir neue Entscheidungsprozeduren für lineare Arithmetik im Kontext von SMT-Solvern und Theorembeweisern: 1) CutSat++, ein korrekter und vollständiger Kalkül für ganzzahlige lineare Arithmetik, der Techniken zur Entscheidung von Aussagenlogik mit Techniken aus der Quantorenelimination vereint. 2) Der Größte-Würfeltest und der Einheitswürfeltest, zwei korrekte (wenn auch unvollständige) Tests, die in polynomieller Zeit (gemischt-)ganzzahlige Lösungen finden. Die Tests sind besonders effizient auf vollständig unbegrenzten Systemen, welche für viele andere Entscheidungsprozeduren schwer sind. 3) Techniken zur Ermittlung von Gleichungen, die von einem linearen Ungleichungssystem impliziert werden. Des Weiteren präsentieren wir mehrere Anwendungsmöglichkeiten für diese Techniken. 4) Die Beidseitig-Begrenzte-Reduktion und die Gemischte-Echelon-Hermitesche- Transformation, die ein Ungleichungssystem in polynomieller Zeit auf ein erfüllbarkeitsäquivalentes System reduzieren, das begrenzt ist. Vereint verwandeln die Transformationen Branch-and-Bound in eine vollständige und effiziente Entscheidungsprozedur für unbeschränkte Ungleichungssysteme. Wir haben diese Techniken (ausgenommen CutSat++) in SPASS-IQ (unserem theory solver für lineare Arithmetik) und in SPASS-SATT (unserem CDCL(LA) solver) implementiert. Basierend darauf präsentieren wir Benchmark-Evaluationen, die die Effizienz unserer Entscheidungsprozeduren bestätigen

Doctor of Philosophy

dissertationFormal verification of hardware designs has become an essential component of the overall system design flow. The designs are generally modeled as finite state machines, on which property and equivalence checking problems are solved for verification. Reachability analysis forms the core of these techniques. However, increasing size and complexity of the circuits causes the state explosion problem. Abstraction is the key to tackling the scalability challenges. This dissertation presents new techniques for word-level abstraction with applications in sequential design verification. By bundling together k bit-level state-variables into one word-level constraint expression, the state-space is construed as solutions (variety) to a set of polynomial constraints (ideal), modeled over the finite (Galois) field of 2^k elements. Subsequently, techniques from algebraic geometry -- notably, Groebner basis theory and technology -- are researched to perform reachability analysis and verification of sequential circuits. This approach adds a "word-level dimension" to state-space abstraction and verification to make the process more efficient. While algebraic geometry provides powerful abstraction and reasoning capabilities, the algorithms exhibit high computational complexity. In the dissertation, we show that by analyzing the constraints, it is possible to obtain more insights about the polynomial ideals, which can be exploited to overcome the complexity. Using our algorithm design and implementations, we demonstrate how to perform reachability analysis of finite-state machines purely at the word level. Using this concept, we perform scalable verification of sequential arithmetic circuits. As contemporary approaches make use of resolution proofs and unsatisfiable cores for state-space abstraction, we introduce the algebraic geometry analog of unsatisfiable cores, and present algorithms to extract and refine unsatisfiable cores of polynomial ideals. Experiments are performed to demonstrate the efficacy of our approaches

Automatic Numerical Solving for Auto-active Verification of Floating-Point Programs

We present a new process for the verification of numerical programs with tight functional specifications that feature exact arithmetic including selected transcendental functions. The process, which simplifies, derives bounds, and safely eliminates floating-point operations from Verification Conditions (VCs) produced by Why3, is capable of automatically verifying such specifications and is implemented in our new open source tool named PropaFP. We evaluate PropaFP alongside the state-of-the-art in formal verification of floating-point programs where we find that the process is able to verify specifications that current tools are unable to verify. We also present novel branch-and-prune contractions based on linearisations of conjunctions that consist of nonlinear real inequalities with differentiable expressions. These linearisations and contractions are implemented in our new open source numerical prover named LPPaver. The contractions we have discovered are used to significantly improve the ‘pruning’ step of our branch-and-prune algorithm. We evaluate LPPaver alongside state-of-the-art automated solvers for problems involving nonlinear real arithmetic. LPPaver performs comparably and, in some cases, better than these solvers. Together, PropaFP and LPPaver yield the first fully automatically verified implementations of the sine and square root functions with tight functional specifications

Proceedings of the 21st Conference on Formal Methods in Computer-Aided Design – FMCAD 2021

The Conference on Formal Methods in Computer-Aided Design (FMCAD) is an annual conference on the theory and applications of formal methods in hardware and system verification. FMCAD provides a leading forum to researchers in academia and industry for presenting and discussing groundbreaking methods, technologies, theoretical results, and tools for reasoning formally about computing systems. FMCAD covers formal aspects of computer-aided system design including verification, specification, synthesis, and testing

The Performance Optimization of ASP Solving Based on Encoding Rewriting and Encoding Selection

Answer set programming (ASP) has long been used for modeling and solving hard search problems. These problems are modeled in ASP as encodings, a collection of rules that declaratively describe the logic of the problem without explicitly listing how to solve it. It is common that the same problem has several different but equivalent encodings in ASP. Experience shows that the performance of these ASP encodings may vary greatly from instance to instance when processed by current state-of-the-art ASP grounder/solver systems. In particular, it is rarely the case that one encoding outperforms all others. Moreover, running an ASP system on one encoding for a specific instance may “take forever,” while running it on another encoding for this instance may yield a solution in a fraction of a second. The selection of a ”good” encoding for each instance is crucial to the performance of ASP solving. In this dissertation, I propose methods to improve the performance of ASP solving that exploit these observations. First, I designed and implemented methods that, given an encoding for a problem, rewrite it in several ways into new different but equivalent encodings. Second, I designed and implemented a system that given a set of input encodings of a problem, a set of problem instances, and an ASP grounder/solver system, automatically generates equivalent encodings and builds for each selected encoding its performance model. The model predicts for any instance the execution time that the grounder/solver system takes to process the instance under the corresponding encoding. These performance models are then used to improve solving efficiency: whenever a new instance arrives, the system selects the encoding predicted to perform the best on the instance and invokes the grounder/solver. The system also supports a scheduled execution and an interleaved execution of encodings, which are complementary to machine learning techniques. Third, I implemented algorithms that generate hard structured instances for several combinatorial problems I selected for our experimental study of the efficacy of the methods I developed. Hard instances can serve as the benchmark for evaluating the hardness of specific problems and contribute as training data to the platform I created to help build encoding selection models. The process can also provide meaningful insights into finding hard instances of other combinatorial problems

Tools and Algorithms for the Construction and Analysis of Systems

This open access book constitutes the proceedings of the 28th International Conference on Tools and Algorithms for the Construction and Analysis of Systems, TACAS 2022, which was held during April 2-7, 2022, in Munich, Germany, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2022. The 46 full papers and 4 short papers presented in this volume were carefully reviewed and selected from 159 submissions. The proceedings also contain 16 tool papers of the affiliated competition SV-Comp and 1 paper consisting of the competition report. TACAS is a forum for researchers, developers, and users interested in rigorously based tools and algorithms for the construction and analysis of systems. The conference aims to bridge the gaps between different communities with this common interest and to support them in their quest to improve the utility, reliability, exibility, and efficiency of tools and algorithms for building computer-controlled systems