Meta-Kernelization with Structural Parameters

Meta-kernelization theorems are general results that provide polynomial kernels for large classes of parameterized problems. The known meta-kernelization theorems, in particular the results of Bodlaender et al. (FOCS'09) and of Fomin et al. (FOCS'10), apply to optimization problems parameterized by solution size. We present the first meta-kernelization theorems that use a structural parameters of the input and not the solution size. Let C be a graph class. We define the C-cover number of a graph to be a the smallest number of modules the vertex set can be partitioned into, such that each module induces a subgraph that belongs to the class C. We show that each graph problem that can be expressed in Monadic Second Order (MSO) logic has a polynomial kernel with a linear number of vertices when parameterized by the C-cover number for any fixed class C of bounded rank-width (or equivalently, of bounded clique-width, or bounded Boolean width). Many graph problems such as Independent Dominating Set, c-Coloring, and c-Domatic Number are covered by this meta-kernelization result. Our second result applies to MSO expressible optimization problems, such as Minimum Vertex Cover, Minimum Dominating Set, and Maximum Clique. We show that these problems admit a polynomial annotated kernel with a linear number of vertices

Towards Uniform Certification in QBF

We pioneer a new technique that allows us to prove a multitude of previously open simulations in QBF proof complexity. In particular, we show that extended QBF Frege p-simulates clausal proof systems such as IR-Calculus, IRM-Calculus, Long-Distance Q-Resolution, and Merge Resolution. These results are obtained by taking a technique of Beyersdorff et al. (JACM 2020) that turns strategy extraction into simulation and combining it with new local strategy extraction arguments. This approach leads to simulations that are carried out mainly in propositional logic, with minimal use of the QBF rules. Our proofs therefore provide a new, largely propositional interpretation of the simulated systems. We argue that these results strengthen the case for uniform certification in QBF solving, since many QBF proof systems now fall into place underneath extended QBF Frege

Towards Uniform Certification in QBF

We pioneer a new technique that allows us to prove a multitude of previously open simulations in QBF proof complexity. In particular, we show that extended QBF Frege p-simulates clausal proof systems such as IR-Calculus, IRM-Calculus, Long-Distance Q-Resolution, and Merge Resolution. These results are obtained by taking a technique of Beyersdorff et al. (JACM 2020) that turns strategy extraction into simulation and combining it with new local strategy extraction arguments. This approach leads to simulations that are carried out mainly in propositional logic, with minimal use of the QBF rules. Our proofs therefore provide a new, largely propositional interpretation of the simulated systems. We argue that these results strengthen the case for uniform certification in QBF solving, since many QBF proof systems now fall into place underneath extended QBF Frege

Three Modern Roles for Logic in AI

We consider three modern roles for logic in artificial intelligence, which are based on the theory of tractable Boolean circuits: (1) logic as a basis for computation, (2) logic for learning from a combination of data and knowledge, and (3) logic for reasoning about the behavior of machine learning systems.Comment: To be published in PODS 202

Quantified CDCL with Universal Resolution

Quantified Conflict-Driven Clause Learning (QCDCL) solvers for QBF generate Q-resolution proofs. Pivot variables in Q-resolution must be existentially quantified. Allowing resolution on universally quantified variables leads to a more powerful proof system called QU-resolution, but so far, QBF solvers have used QU-resolution only in very limited ways. We present a new version of QCDCL that generates proofs in QU-resolution by leveraging propositional unit propagation. We detail how conflict analysis must be adapted to handle universal variables assigned by propagation, and show that the procedure is still sound and terminating. We further describe how dependency learning can be incorporated in the algorithm to increase the flexibility of decision heuristics. Experiments with crafted instances and benchmarks from recent QBF evaluations demonstrate the viability of the resulting version of QCDCL

Struktur in #SAT und QBF

Abweichender Titel laut Übersetzung der Verfasserin/des VerfassersZsfassung in dt. SpracheIn der Komplexitätstheorie geht man davon aus, dass für zahlreiche zentrale Probleme keine effizienten Algorithmen existieren. Einige dieser Probleme lassen sich in der Praxis dennoch lösen, was üblicherweise damit begründet wird, dass praxisrelevante Instanzen "Struktur" aufweisen, die von Lösungsverfahren ausgenutzt werden kann. Die vorliegende Arbeit untersucht konkrete Ausprägungen dieses Strukturbegriffs für zwei äußerst schwierige Probleme: das Erfüllbarkeitsproblem quantifizierter boolescher Formeln (QSAT) und das Abzählproblem von Modellen aussagenlogischer Formeln (#SAT). QSAT. Die Alternierung von Existenz- und Allquantoren im Präfix quantifizierter boolescher Formeln erzeugt Abhängigkeiten unter Variablen, die von Lösungsverfahren für QSAT berücksichtigt werden müssen. Gängige Verfahren gehen davon aus, dass alle prinzipiell möglichen Abhängigkeiten tatsächlich bestehen. Oft ist jedoch nur ein Bruchteil dieser Abhängigkeiten triftig, während die übrigen, "falschen" Abhängigkeiten lediglich zu unnötigen Einschränkungen führen. Wir untersuchen Dependency Schemes als Mittel zur Identifikation solcher falscher Abhängigkeiten, mit folgenden Resultaten. * Wir zeigen, dass das Resolution-Path Dependency Scheme in Polynomialzeit berechnet werden kann. Unter den derzeit bekannten Dependency Schemes erkennt das Resolution-Path Dependency Scheme eine maximale Menge falscher Abhängigkeiten. * Wir definieren notwendige und hinreichende Bedingungen für den Einsatz von Dependency Schemes in suchbasierten Algorithmen für QSAT und zeigen, dass diese Bedingungen von den in DepQBF implementierten Dependency Schemes sowie einer Variante des Resolution-Path Dependency Schemes erfüllt werden. * Dependency Schemes waren ursprünglich zum Verschieben von Quantoren im Präfix quantifizierter boolescher Formeln gedacht. Wir zeigen, dass gängige Dependency Schemes eine allgemeinere Operation zur Manipulation des Präfixes erlauben, und demonstrieren, wie diese Operation zur Minimierung der Alternierungstiefe von Formeln verwendet werden kann. #SAT. Das Zählen von Modellen aussagenlogischer Formeln ist nicht nur im Allgemeinen schwer, sondern selbst für Formelklassen, für die das zugehörige Entscheidungsproblem in Polynomialzeit gelöst werden kann, beispielsweise für Horn- oder 2CNF-Formeln. Wir untersuchen den Effekt von strukturellen (über Graphenparameter definierten) Einschränkungen auf die Komplexität von #SAT und bestimmen neue Formelklassen, die das Zählen von Modellen in Polynomialzeit erlauben. * Das Kontrahieren von Modulen in Graphen ist eine gängige Technik zur Vereinfachung kombinatorischer Optimierungsprobleme. Wir definieren die modulare Baumweite eines Graphen als seine Baumweite nach dem Kontrahieren von Modulen und zeigen, dass #SAT für Formeln, deren Inzidenzgraphen beschränkte modulare Baumweite haben, in Polynomialzeit gelöst werden kann. * Die symmetrische Cliquenweite ist ein Parameter, der sowohl Baumweite als auch modulare Baumweite verallgemeinert. Wir zeigen, dass #SAT für Formelklassen, deren Inzidenzgraphen beschränkte symmetrische Cliquenweite aufweisen, in Polynomialzeit lösbar ist.Computational problems that are intractable in general can often be efficiently resolved in practice due to latent structure in real-world instances. This thesis considers structural properties that can be used in the design of more efficient algorithms for two highly intractable problems: the satisfiability problem of quantified Boolean formulas (QSAT) and propositional model counting (#SAT). QSAT. The nesting of existential and universal quantifiers in quantified Boolean formulas (QBFs) generates dependencies among variables that have to be respected by QSAT solvers. In standard decision algorithms, it is assumed that all possible variable dependencies exist. But often, only a fraction of these dependencies is realized, while the remaining, "spurious" dependencies lead to unnecessary restrictions that inhibit solver performance. We study dependency schemes as a means to identifying spurious dependencies and establish the following results. * Among dependency schemes considered in the literature, the resolution-path dependency scheme identifies a maximal set of spurious dependencies. We prove that the resolution-path dependency scheme can be computed in polynomial time. * We state sufficient conditions for the sound deployment of dependency schemes in search-based QSAT solvers and prove that these conditions are met by several dependency schemes, including those implemented in the solver DepQBF and a variant of the resolution-path dependency scheme. * We show that known dependency schemes support a reordering operation that is more powerful than quantifier shifting, and present an application to the reduction of quantifier alternations of a QBF. #SAT. The model counting problem (#SAT) asks for the number of satisfying assignments of a propositional formula in conjunctive normal form. This problem is hard even for classes that admit satisfiability testing in polynomial time, such as Horn or 2CNF formulas. We prove the following results on the complexity of #SAT with respect to structural parameters based on graph width measures, identifying new classes of formulas amenable to efficient model counting. * Contraction of modules in a graph is a commonly used preprocessing step in combinatorial optimization. We define the modular treewidth of a graph as its treewidth after contraction of modules, and prove that #SAT is polynomial-time tractable for classes of formulas with incidence graphs of bounded modular treewidth. * Symmetric clique-width is a graph parameter that generalizes treewidth as well as modular treewidth. We show that #SAT is polynomial-time tractable for classes of formulas with incidence graphs of bounded symmetric clique-width.14

Pedant: A Certifying DQBF Solver

Pedant is a solver for Dependency Quantified Boolean Formulas (DQBF) that combines propositional definition extraction with Counterexample-Guided Inductive Synthesis (CEGIS) to compute a model of a given formula. Pedant 2 improves upon an earlier version in several ways. We extend the notion of dependencies by allowing existential variables to depend on other existential variables. This leads to more and smaller definitions, as well as more concise repairs for counterexamples. Additionally, we reduce counterexamples by determining minimal separators in a conflict graph, and use decision tree learning to obtain default functions for undetermined variables. An experimental evaluation on standard benchmarks showed a significant increase in the number of solved instances compared to the previous version of our solver