The combinatorics of minimal unsatisfiability: connecting to graph theory

Minimally Unsatisfiable CNFs (MUs) are unsatisfiable CNFs where removing any clause destroys unsatisfiability. MUs are the building blocks of unsatisfia-bility, and our understanding of them can be very helpful in answering various algorithmic and structural questions relating to unsatisfiability. In this thesis we study MUs from a combinatorial point of view, with the aim of extending the understanding of the structure of MUs. We show that some important classes of MUs are very closely related to known classes of digraphs, and using arguments from logic and graph theory we characterise these MUs.Two main concepts in this thesis are isomorphism of CNFs and the implica-tion digraph of 2-CNFs (at most two literals per disjunction). Isomorphism of CNFs involves renaming the variables, and flipping the literals. The implication digraph of a 2-CNF F has both arcs (¬a → b) and (¬b → a) for every binary clause (a ∨ b) in F .In the first part we introduce a novel connection between MUs and Minimal Strong Digraphs (MSDs), strongly connected digraphs, where removing any arc destroys the strong connectedness. We introduce the new class DFM of special MUs, which are in close correspondence to MSDs. The known relation between 2-CNFs and implication digraphs is used, but in a simpler and more direct way, namely that we have a canonical choice of one of the two arcs. As an application of this new framework we provide short and intuitive new proofs for two im-portant but isolated characterisations for nonsingular MUs (every literal occurs at least twice), both with ingenious but complicated proofs: Characterising 2-MUs (minimally unsatisfiable 2-CNFs), and characterising MUs with deficiency 2 (two more clauses than variables).In the second part, we provide a fundamental addition to the study of 2-CNFs which have efficient algorithms for many interesting problems, namely that we provide a full classification of 2-MUs and a polytime isomorphism de-cision of this class. We show that implication digraphs of 2-MUs are “Weak Double Cycles” (WDCs), big cycles of small cycles (with possible overlaps). Combining logical and graph-theoretical methods, we prove that WDCs have at most one skew-symmetry (a self-inverse fixed-point free anti-symmetry, re-versing the direction of arcs). It follows that the isomorphisms between 2-MUs are exactly the isomorphisms between their implication digraphs (since digraphs with given skew-symmetry are the same as 2-CNFs). This reduces the classifi-cation of 2-MUs to the classification of a nice class of digraphs.Finally in the outlook we discuss further applications, including an alter-native framework for enumerating some special Minimally Unsatisfiable Sub-clause-sets (MUSs)

Proof Theory of Finite-valued Logics

The proof theory of many-valued systems has not been investigated to an extent comparable to the work done on axiomatizatbility of many-valued logics. Proof theory requires appropriate formalisms, such as sequent calculus, natural deduction, and tableaux for classical (and intuitionistic) logic. One particular method for systematically obtaining calculi for all finite-valued logics was invented independently by several researchers, with slight variations in design and presentation. The main aim of this report is to develop the proof theory of finite-valued first order logics in a general way, and to present some of the more important results in this area. In Systems covered are the resolution calculus, sequent calculus, tableaux, and natural deduction. This report is actually a template, from which all results can be specialized to particular logics

Counting Problems on Quantum Graphs: Parameterized and Exact Complexity Classifications

Quantum graphs, as defined by Lovász in the late 60s, are formal linear combinations of simple graphs with finite support. They allow for the complexity analysis of the problem of computing finite linear combinations of homomorphism counts, the latter of which constitute the foundation of the structural hardness theory for parameterized counting problems: The framework of parameterized counting complexity was introduced by Flum and Grohe, and McCartin in 2002 and forms a hybrid between the classical field of computational counting as founded by Valiant in the late 70s and the paradigm of parameterized complexity theory due to Downey and Fellows which originated in the early 90s. The problem of computing homomorphism numbers of quantum graphs subsumes general motif counting problems and the complexity theoretic implications have only turned out recently in a breakthrough regarding the parameterized subgraph counting problem by Curticapean, Dell and Marx in 2017. We study the problems of counting partially injective and edge-injective homomorphisms, counting induced subgraphs, as well as counting answers to existential first-order queries. We establish novel combinatorial, algebraic and even topological properties of quantum graphs that allow us to provide exhaustive parameterized and exact complexity classifications, including necessary, sufficient and mostly explicit tractability criteria, for all of the previous problems.Diese Arbeit befasst sich mit der Komplexit atsanalyse von mathematischen Problemen die als Linearkombinationen von Graphhomomorphismenzahlen darstellbar sind. Dazu wird sich sogenannter Quantengraphen bedient, bei denen es sich um formale Linearkombinationen von Graphen handelt und welche von Lov asz Ende der 60er eingef uhrt wurden. Die Bestimmung der Komplexit at solcher Probleme erfolgt unter dem von Flum, Grohe und McCartin im Jahre 2002 vorgestellten Paradigma der parametrisierten Z ahlkomplexit atstheorie, die als Hybrid der von Valiant Ende der 70er begr undeten klassischen Z ahlkomplexit atstheorie und der von Downey und Fellows Anfang der 90er eingef uhrten parametrisierten Analyse zu verstehen ist. Die Berechnung von Homomorphismenzahlen zwischen Quantengraphen und Graphen subsumiert im weitesten Sinne all jene Probleme, die das Z ahlen von kleinen Mustern in gro en Strukturen erfordern. Aufbauend auf dem daraus resultierenden Durchbruch von Curticapean, Dell und Marx, das Subgraphz ahlproblem betre end, behandelt diese Arbeit die Analyse der Probleme des Z ahlens von partiell- und kanteninjektiven Homomorphismen, induzierten Subgraphen, und Tre ern von relationalen Datenbankabfragen die sich als existentielle Formeln ausdr ucken lassen. Insbesondere werden dabei neue kombinatorische, algebraische und topologische Eigenschaften von Quantengraphen etabliert, die hinreichende, notwendige und meist explizite Kriterien f ur die Existenz e zienter Algorithmen liefern

Efficient local search for Pseudo Boolean Optimization

Algorithms and the Foundations of Software technolog

Pseudo-contractions as Gentle Repairs

Updating a knowledge base to remove an unwanted consequence is a challenging task. Some of the original sentences must be either deleted or weakened in such a way that the sentence to be removed is no longer entailed by the resulting set. On the other hand, it is desirable that the existing knowledge be preserved as much as possible, minimising the loss of information. Several approaches to this problem can be found in the literature. In particular, when the knowledge is represented by an ontology, two different families of frameworks have been developed in the literature in the past decades with numerous ideas in common but with little interaction between the communities: applications of AGM-like Belief Change and justification-based Ontology Repair. In this paper, we investigate the relationship between pseudo-contraction operations and gentle repairs. Both aim to avoid the complete deletion of sentences when replacing them with weaker versions is enough to prevent the entailment of the unwanted formula. We show the correspondence between concepts on both sides and investigate under which conditions they are equivalent. Furthermore, we propose a unified notation for the two approaches, which might contribute to the integration of the two areas

Computer Aided Verification

This open access two-volume set LNCS 11561 and 11562 constitutes the refereed proceedings of the 31st International Conference on Computer Aided Verification, CAV 2019, held in New York City, USA, in July 2019. The 52 full papers presented together with 13 tool papers and 2 case studies, were carefully reviewed and selected from 258 submissions. The papers were organized in the following topical sections: Part I: automata and timed systems; security and hyperproperties; synthesis; model checking; cyber-physical systems and machine learning; probabilistic systems, runtime techniques; dynamical, hybrid, and reactive systems; Part II: logics, decision procedures; and solvers; numerical programs; verification; distributed systems and networks; verification and invariants; and concurrency