Sémantique fonctionnelle pour les QBF non-prénexes

We propose in this article a functional semantics for non-prenex Quantified Boolean Formulae. The proposed formalism is symetrical for validity and non-validity and allows to give different interpretations to the quantifiers. This formalism is also sufficiently flexible to allow the definition of a certificate for the search-based algorithms for non-prenex QBF

On quantified propositional logics and the exponential time hierarchy

We study quantified propositional logics from the complexity theoretic point of view. First we introduce alternating dependency quantified boolean formulae (ADQBF) which generalize both quantified and dependency quantified boolean formulae. We show that the truth evaluation for ADQBF is AEXPTIME(poly)-complete. We also identify fragments for which the problem is complete for the levels of the exponential hierarchy. Second we study propositional team-based logics. We show that DQBF formulae correspond naturally to quantified propositional dependence logic and present a general NEXPTIME upper bound for quantified propositional logic with a large class of generalized dependence atoms. Moreover we show AEXPTIME(poly)-completeness for extensions of propositional team logic with generalized dependence atoms.University of AucklandAcademy of Finlan

09461 Abstracts Collection -- Algorithms and Applications for Next Generation SAT Solvers

From 8th to 13th November 2009, the Dagstuhl Seminar 09461 "Algorithms and Applications for Next Generation SAT Solvers" was held in Schloss Dagstuhl--Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts, slides or full papers are provided, if available

Symbolic reactive synthesis

In this thesis, we develop symbolic algorithms for the synthesis of reactive systems. Synthesis, that is the task of deriving correct-by-construction implementations from formal specifications, has the potential to eliminate the need for the manual—and error-prone—programming task. The synthesis problem can be formulated as an infinite two-player game, where the system player has the objective to satisfy the specification against all possible actions of the environment player. The standard synthesis algorithms represent the underlying synthesis game explicitly and, thus, they scale poorly with respect to the size of the specification. We provide an algorithmic framework to solve the synthesis problem symbolically. In contrast to the standard approaches, we use a succinct representation of the synthesis game which leads to improved scalability in terms of the symbolically represented parameters. Our algorithm reduces the synthesis game to the satisfiability problem of quantified Boolean formulas (QBF) and dependency quantified Boolean formulas (DQBF). In the encodings, we use propositional quantification to succinctly represent different parts of the implementation, such as the state space and the transition function. We develop highly optimized satisfiability algorithms for QBF and DQBF. Based on a counterexample-guided abstraction refinement (CEGAR) loop, our algorithms avoid an exponential blow-up by using the structure of the underlying symbolic encodings. Further, we extend the solving algorithms to extract certificates in the form of Boolean functions, from which we construct implementations for the synthesis problem. Our empirical evaluation shows that our symbolic approach significantly outperforms previous explicit synthesis algorithms with respect to scalability and solution quality.In dieser Dissertation werden symbolische Algorithmen für die Synthese von reaktiven Systemen entwickelt. Synthese, d.h. die Aufgabe, aus formalen Spezifikationen korrekte Implementierungen abzuleiten, hat das Potenzial, die manuelle und fehleranfällige Programmierung überflüssig zu machen. Das Syntheseproblem kann als unendliches Zweispielerspiel verstanden werden, bei dem der Systemspieler das Ziel hat, die Spezifikation gegen alle möglichen Handlungen des Umgebungsspielers zu erfüllen. Die Standardsynthesealgorithmen stellen das zugrunde liegende Synthesespiel explizit dar und skalieren daher schlecht in Bezug auf die Größe der Spezifikation. Diese Arbeit präsentiert einen algorithmischen Ansatz, der das Syntheseproblem symbolisch löst. Im Gegensatz zu den Standardansätzen wird eine kompakte Darstellung des Synthesespiels verwendet, die zu einer verbesserten Skalierbarkeit der symbolisch dargestellten Parameter führt. Der Algorithmus reduziert das Synthesespiel auf das Erfüllbarkeitsproblem von quantifizierten booleschen Formeln (QBF) und abhängigkeitsquantifizierten booleschen Formeln (DQBF). In den Kodierungen verwenden wir propositionale Quantifizierung, um verschiedene Teile der Implementierung, wie den Zustandsraum und die Übergangsfunktion, kompakt darzustellen. Wir entwickeln hochoptimierte Erfüllbarkeitsalgorithmen für QBF und DQBF. Basierend auf einer gegenbeispielgeführten Abstraktionsverfeinerungsschleife (CEGAR) vermeiden diese Algorithmen ein exponentielles Blow-up, indem sie die Struktur der zugrunde liegenden symbolischen Kodierungen verwenden. Weiterhin werden die Lösungsalgorithmen um Zertifikate in Form von booleschen Funktionen erweitert, aus denen Implementierungen für das Syntheseproblem abgeleitet werden. Unsere empirische Auswertung zeigt, dass unser symbolischer Ansatz die bisherigen expliziten Synthesealgorithmen in Bezug auf Skalierbarkeit und Lösungsqualität deutlich übertrifft

Reasons for Hardness in QBF Proof Complexity

Quantified Boolean Formulas (QBF) extend the canonical NP-complete satisfiability problem by including Boolean quantifiers. Determining the truth of a QBF is PSPACE-complete; this is expected to be a harder problem than satisfiability, and hence QBF solving has much wider applications in practice. QBF proof complexity forms the theoretical basis for understanding QBF solving, as well as providing insights into more general complexity theory, but is less well understood than propositional proof complexity. We begin this thesis by looking at the reasons underlying QBF hardness, and in particular when the hardness is propositional in nature, rather than arising due to the quantifiers. We introduce relaxing QU-Res, a previous model for identifying such propositional hardness, and construct an example where relaxing QU-Res is unsuccessful in this regard. We then provide a new model for identifying such hardness which we prove captures this concept. Now equipped with a means of identifying ‘genuine’ QBF hardness, we prove a new lower bound technique for tree-like QBF proof systems. Lower bounds using this technique allows us to show a new separation between tree-like and dag-like systems. We give a characterisation of lower bounds for a large class of tree-like proof systems, in which such lower bounds play a prominent role. Further to the tree-like bound, we provide a new lower bound technique for QBF proof systems in general. This technique has some similarities to the above technique for tree-like systems, but requires some refinement to provide bounds for dag-like systems. We give applications of this new technique by proving lower bounds across several systems. The first such lower bounds are for a very simple family of QBFs. We then provide a construction to combine false QBFs to give formulas for which we can show lower bounds in this way, allowing the generation of the first random QBF proof complexity lower bounds