Solving planning problems via translation to satisfiability (SAT) is one of the most successful approaches to automated planning. In this thesis we describe several ways of encoding a planning problem represented in the SAS+ formalism into SAT. We review and adapt existing encoding schemes as well as introduce new original encodings. We compare the encodings by calculating upper bounds on the size of the formulas they produce as well as by running extensive experiments on benchmark problems from the 2011 International Planning Competition (IPC). In the experimental section we also compare our encodings with the state-of-the-art encodings of the planner Madagascar. The experiments show, that our techniques can outperform these state-of-the-art encodings. In the presented thesis we also deal with a special case of post-planning optimization -- elimination of redundant actions. The elimination of all redundant actions is NP-complete. We review the existing polynomial heuristic approaches and propose our own heuristic approach which can eliminate a higher number and more costly redundant actions than the existing techniques. We also propose a SAT encoding for the problem of plan redundancy which together with MaxSAT solvers allows us to solve the problem of action elimination optimally. Experiments done with...

The Boolean satisfiability (SAT) problem is central to many applications involving the verification and optimization of digital systems. These combinatorial problems are typically solved by using a decision-based approach, however the lengthy compute time of SAT can make it prohibitively impractical for some applications. We discuss how the underlying physical characteristics of various technologies affect the practicality of SAT solvers. Power dissipation and other physical limitations are increasingly restricting the improvement in performance of conventional software on CPUs. We use heterogeneous computing to maximize the strengths of different underlying technologies as well as different computing architectures. In this thesis, we present a custom hardware architecture for accelerating the common computation within a SAT solver. Algorithms and data structures must be fundamentally redesigned in order to maximize the strengths of customized computing. Generalizable optimizations are proposed to maximize the throughput, minimize communication latencies, and aggressively compact the memory. We tightly integrate as well as jointly optimize the hardware accelerator and the software host. Our fully implemented system is significantly faster than pure software on real-life SAT problems. Due to our insights and optimizations, we are able to benchmark SAT in uncharted territory. / Thesis / Doctor of Philosophy (PhD)

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Many problems that arise in the real world are difficult to solve partly because they present computational challenges. Many of these challenging problems are optimization problems. In the real world we are generally interested not just in solutions but in the cost or benefit of these solutions according to different metrics. Hence, finding optimal solutions is often highly desirable and sometimes even necessary. The most effective computational approach for solving such problems is to first model them in a mathematical or logical language, and then solve them by applying a suitable algorithm. This thesis is concerned with developing practical algorithms to solve optimization problems modeled in a particular logical language, MAXSAT. MAXSAT is a generalization of the famous Satisfiability (SAT) problem, that associates finite costs with falsifying various desired conditions where these conditions are expressed as propositional clauses. Optimization problems expressed in MAXSAT typically have two interacting components: the logical relationships between the variables expressed by the clauses, and the optimization component involving minimizing the falsified clauses. The interaction between these components greatly contributes to the difficulty of solving MAXSAT. The main contribution of the thesis is a new hybrid approach, MaxHS, for solving MAXSAT. Our hybrid approach attempts to decouple these two components so that each can be solved with a different technology. In particular, we develop a hybrid solver that exploits two sophisticated technologies with divergent strengths: SAT for solving the logical component, and Integer Programming (IP) solvers for solving the optimization component. MaxHS automatically and incrementally splits the MAXSAT problem into two parts that are given to the SAT and IP solvers, which work together in a complementary way to find a MAXSAT solution. The thesis investigates several improvements to the MaxHS approach and provides empirical analysis of its behaviour in practise. The result is a new solver, MaxHS, that is shown to be the most robust existing solver for MAXSAT.

Many problems that arise in the real world are difficult to solve partly because they present computational challenges. Many of these challenging problems are optimization problems. In the real world we are generally interested not just in solutions but in the cost or benefit of these solutions according to different metrics. Hence, finding optimal solutions is often highly desirable and sometimes even necessary. The most effective computational approach for solving such problems is to first model them in a mathematical or logical language, and then solve them by applying a suitable algorithm. This thesis is concerned with developing practical algorithms to solve optimization problems modeled in a particular logical language, MAXSAT. MAXSAT is a generalization of the famous Satisfiability (SAT) problem, that associates finite costs with falsifying various desired conditions where these conditions are expressed as propositional clauses. Optimization problems expressed in MAXSAT typically have two interacting components: the logical relationships between the variables expressed by the clauses, and the optimization component involving minimizing the falsified clauses. The interaction between these components greatly contributes to the difficulty of solving MAXSAT. The main contribution of the thesis is a new hybrid approach, MaxHS, for solving MAXSAT. Our hybrid approach attempts to decouple these two components so that each can be solved with a different technology. In particular, we develop a hybrid solver that exploits two sophisticated technologies with divergent strengths: SAT for solving the logical component, and Integer Programming (IP) solvers for solving the optimization component. MaxHS automatically and incrementally splits the MAXSAT problem into two parts that are given to the SAT and IP solvers, which work together in a complementary way to find a MAXSAT solution. The thesis investigates several improvements to the MaxHS approach and provides empirical analysis of its behaviour in practise. The result is a new solver, MaxHS, that is shown to be the most robust existing solver for MAXSAT.

Phasing genotypes from sequence data is an important step betweendata gathering and downstream analysis in population genetics,disease studies, and multiple other fields. This determination ofthe sequences of markers corresponding to the individualchromosomes can be done on data where the markers are in lowdensity across the chromosome, such as from single nucleotidepolymorphism (SNP) microarrays, or on data with a higher localdensity of markers like in next generation sequencing (NGS). Thesorted markers may then be used for many different analyses anddata processing such as linkage analysis, or inference of missinggenotypes in the process of imputation cnF2freq is a haplotype phasing program that uses an uncommonapproach allowing it to divide big groups of related individualsinto smaller ones. It sets an initial haplotype phase and theniteratively changes it using estimations from Hidden MarkovModels. If a marker is judged to have been placed in the wronghaplotype, a switch needs to be made so that it belongs to thecorrect phase. The objective of this project was to go fromallowing only one individual within a group to be switched in aniteration to allowing multiple switches that are dependent on eachother. The result of this project is a theoretical solution for allowingmultiple dependent switches in cnF2freq, and an implementedsolution using the max-SAT solver toulbar2.

Satisfiability (SAT) and satisfiability modulo theories (SMT) solvers are high-performance automated propositional and first-order theorem provers, used as underlying tools in many formal verification and artificial intelligence systems. Theoretic and engineering advancement of solver technologies improved the performance of modern solvers; however, the increased complexity of those solvers calls for formal verification of those tools themselves. This thesis discusses two methods to formally certify SAT/SMT solvers. The first method is generating proofs from solvers and certifying those proofs. Because new theories are constantly added to SMT solvers, a flexible framework to safely add new inference rules is necessary. The proposal is to use a meta-language called LFSC, which is based on Edinburgh Logical Framework. SAT/SMT logics have been encoded in LFSC, and the encoding can be easily and modularly extended for new logics. It is shown that an optimized LFSC checker can certify SMT proofs efficiently. The second method is using a verified programming language to implement a SAT solver and verify the code statically. Guru is a pure functional programming language with support for dependent types and theorem proving; Guru also allows for efficient code generation by means of resource typing. A modern SAT solver, called versat, has been implemented and verified to be correct in Guru. The performance of versat is shown to be comparable with that of the current proof checking technology.

Un mot de données (resp. un arbre de données) est un mot (resp. arbre) fini, dont chaque position est étiquetée avec une lettre d'un alphabet fini et une donnée d'un domaine infini. Dans cette thèse, nous étudions des automates et des logiques sur des mots et des arbres de données ayant des propriétés décidables: nous nous concentrons sur le problème du test du vide dans le cas des automates, et sur le problème de la satisfaisabilité dans le cas des logiques. Sur les mots de données, nous présentons une extension décidable du modèle d'automate alternant avec registre étudié par Demri et Lazic. En outre, nous montrons la décidabilité du problème de satisfaisabilité pour la logique du temps linéaire sur les mots de données LTL(X,F,U) (étudié par Demri et Lazic) étendue avec une quantification sur des données. Nous montrons aussi que la borne inférieure de non-récursivité primitive montré par Demri et Lazic pour LTL(X,F) est déjà valable pour LTL(F). Sur les arbres de données, nous considérons trois modèles décidables d'automates avec des caractéristiques différentes. Nous commençons par introduire l'automate avec donnée ``downward'' (automates DD). Son exécution consiste en une transduction ré-étiquetant la partie finie de l'étiquetage de l'arbre, et une vérification des propriétés des données de chaque sous-arbre de l'arbre résultant de la transduction. Ce modèle est clos par les opérations booléennes, mais les tests autorisés sur l'ordre des noeuds ayant le même père sont très limités. Son problème du vide est dans 2ExpTime. Au contraire, les deux autres modèles d'automates que nous introduisons ont un problème du vide avec une complexité non récursive primitive, et sont clos par intersection et union, mais par par complémentation. Ils ont tous les deux un contrôle alternant ainsi qu'un registre pour stocker et comparer les données. La classe des automates ATRA(guess,spread) généralise le modèle d'automate top-down ATRA de Jurdzinski et Lazic. Nous introduisons des extensions décidables similaires à celles que nous avons étudiées dans le cas de mots de données. Cette classe d'automates généralise la notion de langage rationnel d'arbre, ---contrairement aux automates DD. Enfin, nous considérons un modèle d'automate bottom-up avec un contrôle alternant et un registre (appelé BUDA). Bien que les BUDA soient bottom-up, ils peuvent tester des propriétés sur les données en navigant dans l'arbre dans les deux directions: vers le haut et vers le bas. Au contraire de ATRA(guess,spread), ce modèle d'automate ne peut pas tester de propriétés sur la séquence des noeuds ayant le même père (comme, par exemple, l'ordre dans lequel apparaissent leurs étiquettes). Ces trois modèles d'automates ont des liens avec la logique XPath---une logique conçue pour les documents XML, qui peuvent être vus comme des arbres de données. En utilisant les automates que nous avons mentionnés ci-dessus, nous montrons que la satisfaisabilité de trois fragments naturels de XPath sont décidables. Ces fragments sont: downward XPath, où la navigation ne peut se faire que via les axes child et descendant- forward XPath, où la navigation permet également les axes next sibling ainsi que sa clôture transitive, et vertical XPath, dont la navigation est limitée aux axes child, descendant, parent et ancestor. Alors que downward XPath est ExpTime-complet, les fragments forward et vertical de XPath ont une borne inférieure de non-récursivité primitive. / A data word (resp. a data tree) is a &#64257-nite word (resp. tree) whose every position carries a letter from a &#64257-nite alphabet and a datum form an in&#64257-nite domain. In this thesis we investigate automata and logics for data words and data trees with decidable reasoning problems: we focus on the emptiness problem in the case of automata, and the satis&#64257-ability problem in the case of logics. On data words, we present a decidable extension of the model of alternating register automata studied by Demri and Lazi´c. Further, we show the decidability of the satis&#64257-ability problem for the linear-time temporal logic on data words LTL_\downarrow (X, F, U) (studied by Demri and Lazi´c) with quanti&#64257-cation over data values. We also prove that the lower bounds of non-primitive recursiveness shown by Demri and Lazi´c for LTL&#8595- (X, F) carry over to LTL&#8595- (F). On data trees, we consider three decidable automata models with di&#64256-erent characteristics. We &#64257-rst introduce the Downward Data automaton (DD automata). Its execution consists in a transduction of the &#64257-nite labeling of the tree, and a veri&#64257-cation of data properties for every subtree of the transduced tree. This model is closed under boolean operations, but the tests it can make on the order of the siblings is very limited. Its emptiness problem is 2ExpTime. On the contrary, the other two automata models we introduce have an emptiness problem with a non-primitive recursive complexity, and are closed under intersection and union, but not complementation. They are both alternating automata with one register to store and compare data values. The automata class ATRA(guess, spread) extends the top-down automata ATRA of Jurdzinski and Lazic. We exhibit similar decidable extensions as the one showed in the case of data words. This class can test for any tree regular language—in contrast to DD automata. Finally, we consider a bottom-up alternating tree automaton with one register (called BUDA). Although the BUDA class is one-way, it has features that allow to test data properties by navigating the tree in both directions: upward and downward. In opposition to ATRA(guess, spread), this automaton cannot test for properties on the the sequence of siblings (like, for example, the order in which labels appear). All these three models have connections with the logic XPath—a logic conceived for xml documents, which can be seen as data trees. Through the aforementioned automata we show that the satis&#64257-ability of three natural fragments of XPath are decidable. These fragments are: downward XPath, where navigation can only be done by child and descendant axes- forward XPath, where navigation also contains the next sibling axis and its transitive closure- and vertical XPath, whose navigation consists in the child, descendant, parent and ancestor axes. Whereas downward XPath is ExpTime-complete, forward and vertical XPath have non-primitive recursive lower bounds.

In recent years, Satisfiability Modulo Theories (SMT) solvers have emerged as powerful tools in many formal methods applications, including verification, automated theorem proving, planning and software synthesis. The expressive power of SMT allows problems from many disciplines to be handled in a single unified approach. While SMT solvers are highly effective at handling certain classes of problems due to highly tuned implementations of efficient ground decision procedures, their ability is often limited when reasoning about universally quantified first-order formulas. Since generally this class of problems is undecidable, most SMT solvers use heuristic techniques for answering unsatisfiable when quantified formulas are present. On the other hand, when the problem is satisfiable, solvers using these techniques will either run indefinitely, or give up after some predetermined amount of effort. In a majority of formal methods applications, it is critical that the SMT solver be able to determine when such a formula is satisfiable, especially when it can return some representation of a model for the formula. This dissertation introduces new techniques for finding models for SMT formulas containing quantified first-order formulas. We will focus our attention on finding finite models, that is, models whose domain elements can be represented as a finite set. We give a procedure that is both finite model complete and refutationally complete for a fragment of first-order logic that occurs commonly in practice.

Abstract Solving Quantified Boolean Formulas Horst Samulowitz Doctor of Philosophy Graduate Department of Computer Science University of Toronto 2008 Many real-world problems do not have a simple algorithmic solution and casting these problems as search problems is often not only the simplest way of casting them, but also the most efficient way of solving them. In this thesis we will present several techniques to advance search-based algorithms in the context of solving quantified boolean formulas (QBF). QBF enables complex realworld problems including planning, two-player games and verification to be captured in a compact and quite natural fashion. We will discuss techniques ranging from straight forward pre-processing methods utilizing strong rules of inference to more sophisticated online approaches such as dynamic partitioning. Furthermore, we will show that all of the presented techniques achieve an essential improvement of the search process when solving QBF. At the same time the displayed empirical results also reveal the orthogonality of the different techniques with respect to performance. Generally speaking each approach performs well on a particular subset of benchmarks, but performs poorly on others. Consequently, an adaptive employment of the available techniques that maximizes the performance would result in further improvements. We will demonstrate that such an adaptive approach can pay off in practice, by presenting the results of using machine learning methods to dynamically select the best variable ordering heuristics during search.