The Complexity of Satisfiability for Sub-Boolean Fragments of ALC

The standard reasoning problem, concept satisfiability, in the basic description logic ALC is PSPACE-complete, and it is EXPTIME-complete in the presence of unrestricted axioms. Several fragments of ALC, notably logics in the FL, EL, and DL-Lite family, have an easier satisfiability problem; sometimes it is even tractable. All these fragments restrict the use of Boolean operators in one way or another. We look at systematic and more general restrictions of the Boolean operators and establish the complexity of the concept satisfiability problem in the presence of axioms. We separate tractable from intractable cases.Comment: 17 pages, accepted (in short version) to Description Logic Workshop 201

LTL Fragments are Hard for Standard Parameterisations

We classify the complexity of the LTL satisfiability and model checking problems for several standard parameterisations. The investigated parameters are temporal depth, number of propositional variables and formula treewidth, resp., pathwidth. We show that all operator fragments of LTL under the investigated parameterisations are intractable in the sense of parameterised complexity.Comment: TIME 2015 conference versio

A Team Based Variant of CTL

We introduce two variants of computation tree logic CTL based on team semantics: an asynchronous one and a synchronous one. For both variants we investigate the computational complexity of the satisfiability as well as the model checking problem. The satisfiability problem is shown to be EXPTIME-complete. Here it does not matter which of the two semantics are considered. For model checking we prove a PSPACE-completeness for the synchronous case, and show P-completeness for the asynchronous case. Furthermore we prove several interesting fundamental properties of both semantics.Comment: TIME 2015 conference version, modified title and motiviatio

The model checking fingerprints of CTL operators

The aim of this study is to understand the inherent expressive power of CTL operators. We investigate the complexity of model checking for all CTL fragments with one CTL operator and arbitrary Boolean operators. This gives us a fingerprint of each CTL operator. The comparison between the fingerprints yields a hierarchy of the operators that mirrors their strength with respect to model checking

Parametrised enumeration

In this thesis, we develop a framework of parametrised enumeration complexity. At first, we provide the reader with preliminary notions such as machine models and complexity classes besides proving them to be well-chosen. Then, we study the interplay and the landscape of these classes and present connections to classical enumeration classes. Afterwards, we translate the fundamental methods of kernelisation and self-reducibility into equivalent techniques in the setting of parametrised enumeration. Subsequently, we illustrate the introduced classes by investigating the parametrised enumeration complexity of Max-Ones-SAT and strong backdoor sets as well as sharpen the first result by presenting a dichotomy theorem for Max-Ones-SAT. After this, we extend the definitions of parametrised enumeration algorithms by allowing orders on the solution space. In this context, we study the relations ``order by size'' and ``lexicographic order'' for graph modification problems and observe a trade-off between enumeration delay and space requirements of enumeration algorithms. These results then yield an enumeration technique for generalised modification problems that is illustrated by applying this method to the problems closest string, weak and strong backdoor sets, and weighted satisfiability. Eventually, we consider the enumeration of satisfying teams of formulas of poor man's propositional dependence logic. There, we present an enumeration algorithm with FPT delay and exponential space which is one of the first enumeration complexity results of a problem in a team logic. Finally, we show how this algorithm can be modified such that only polynomial space is required, however, by increasing the delay to incremental FPT time.In diesem Werk begründen wir die Theorie der parametrisierten Enumeration, präsentieren die grundlegenden Definitionen und prüfen ihre Sinnhaftigkeit. Im nächsten Schritt, untersuchen wir das Zusammenspiel der eingeführten Komplexitätsklassen und zeigen Verbindungen zur klassischen Enumerationskomplexität auf. Anschließend übertragen wir die zwei fundamentalen Techniken der Kernelisierung und Selbstreduzierbarkeit in Entsprechungen in dem Gebiet der parametrisierten Enumeration. Schließlich untersuchen wir das Problem Max-Ones-SAT und das Problem der Aufzählung starker Backdoor-Mengen als typische Probleme in diesen Klassen. Die vorherigen Resultate zu Max-Ones-SAT werden anschließend in einem Dichotomie-Satz vervollständigt. Im nächsten Abschnitt erweitern wir die neuen Definitionen auf Ordnungen (auf dem Lösungsraum) und erforschen insbesondere die zwei Relationen \glqq Größenordnung\grqq\ und \glqq lexikographische Reihenfolge\grqq\ im Kontext von Graphen-Modifikationsproblemen. Hierbei scheint es, als müsste man zwischen Delay und Speicheranforderungen von Aufzählungsalgorithmen abwägen, wobei dies jedoch nicht abschließend gelöst werden kann. Aus den vorherigen Überlegungen wird schließlich ein generisches Enumerationsverfahren für allgemeine Modifikationsprobleme entwickelt und anhand der Probleme Closest String, schwacher und starker Backdoor-Mengen sowie gewichteter Erfüllbarkeit veranschaulicht. Im letzten Abschnitt betrachten wir die parametrisierte Enumerationskomplexität von Erfüllbarkeitsproblemen im Bereich der Poor Man's Propositional Dependence Logic und stellen einen Aufzählungsalgorithmus mit FPT Delay vor, der mit exponentiellem Platz arbeitet. Dies ist einer der ersten Aufzählungsalgorithmen im Bereich der Teamlogiken. Abschließend zeigen wir, wie dieser Algorithmus so modifiziert werden kann, dass nur polynomieller Speicherplatz benötigt wird, bezahlen jedoch diese Einsparung mit einem Anstieg des Delays auf inkrementelle FPT Zeit (IncFPT)

Counting Complexity for Reasoning in Abstract Argumentation

In this paper, we consider counting and projected model counting of extensions in abstract argumentation for various semantics. When asking for projected counts we are interested in counting the number of extensions of a given argumentation framework while multiple extensions that are identical when restricted to the projected arguments count as only one projected extension. We establish classical complexity results and parameterized complexity results when the problems are parameterized by treewidth of the undirected argumentation graph. To obtain upper bounds for counting projected extensions, we introduce novel algorithms that exploit small treewidth of the undirected argumentation graph of the input instance by dynamic programming (DP). Our algorithms run in time double or triple exponential in the treewidth depending on the considered semantics. Finally, we take the exponential time hypothesis (ETH) into account and establish lower bounds of bounded treewidth algorithms for counting extensions and projected extension.Comment: Extended version of a paper published at AAAI-1