Ontology-based data access (OBDA) augments classical query answering in databases by including domain knowledge provided by an ontology. An ontology captures the terminology of an application domain and describes domain knowledge in a machine-processable way. Formal ontology languages additionally provide semantics to these specifications. Systems for OBDA thus may apply logical reasoning to answer queries; they use the ontological knowledge to infer new information, which is only implicitly given in the data. Moreover, they usually employ the open-world assumption, which means that knowledge not stated explicitly in the data or inferred is neither assumed to be true nor false. Classical OBDA regards the knowledge however only w.r.t. a single moment, which means that information about time is not used for reasoning and hence lost; in particular, the queries generally cannot express temporal aspects. We investigate temporal query languages that allow to access temporal data through classical ontologies. In particular, we study the computational complexity of temporal query answering regarding ontologies written in lightweight description logics, which are known to allow for efficient reasoning in the atemporal setting and are successfully applied in practice. Furthermore, we present a so-called rewritability result for ontology-based temporal query answering, which suggests ways for implementation. Our results may thus guide the choice of a query language for temporal OBDA in data-intensive applications that require fast processing, such as context recognition.

Organisationen wie Feuerwehren, med. Rettungsdienste oder Spezialeinheiten der Polizei zeigen auch in Situationen, die durch Unsicherheit, unvollständige Informationen oder eine sehr hohe Dynamik gekennzeichnet sind, eine effiziente und effektive Leistungsfähigkeit. Sie können deswegen als Hochleistungssysteme (HLS) bezeichnet werden. Im vorliegenden Artikel wird dargestellt wie sich Erkenntnisse der High Reliabilty Theory auf die Untersuchung von Hochleistungssystemen auswirken und welche Gemeinsamkeiten und Parallelen bestehen. Dabei wird insbesondere ein Schwerpunkt auf die Thematik des Lernens gelegt.

Inevitably, it becomes more and more important to verify that the systems surrounding us have certain properties. This is indeed unavoidable for safety-critical systems such as power plants and intensive-care units. We refer to the term system in a broad sense: it may be man-made (e.g. a computer system) or natural (e.g. a patient in an intensive-care unit). Whereas in Model Checking it is assumed that one has complete knowledge about the functioning of the system, we consider an open-world scenario and assume that we can only observe the behaviour of the actual running system by sensors. Such an abstract sensor could sense e.g. the blood pressure of a patient or the air traffic observed by radar. Then the observed data are preprocessed appropriately and stored in a fact base. Based on the data available in the fact base, situation-awareness tools are supposed to help the user to detect certain situations that require intervention by an expert. Such situations could be that the heart-rate of a patient is rather high while the blood pressure is low, or that a collision of two aeroplanes is about to happen. Moreover, the information in the fact base can be used by monitors to verify that the system has certain properties. It is not realistic, however, to assume that the sensors always yield a complete description of the current state of the observed system. Thus, it makes sense to assume that information that is not present in the fact base is unknown rather than false. Moreover, very often one has some knowledge about the functioning of the system. This background knowledge can be used to draw conclusions about the possible future behaviour of the system. Employing description logics (DLs) is one way to deal with these requirements. In this thesis, we tackle the sketched problem in three different contexts: (i) runtime verification using a temporalised DL, (ii) temporalised query entailment, and (iii) verification in DL-based action formalisms.

Aus der Einleitung: "In der Entwicklung technischer Produkte sind die Entwickler mit einer zunehmenden Komplexität der Produkte konfrontiert. Die Komplexität hat unterschiedliche Ursachen, wie bspw. eine höhere Anzahl an Anforderungen, eine steigende Anzahl an unterschiedlichen beteiligten Domänen oder eine kürzere Entwicklungszeit. Zusätzlich muss bereits während der Entwicklung der gesamte Lebenszyklus des Produkts bis zur Entsorgung beachtet werden. ..."

Constraint satisfaction problems (CSPs) form a large class of decision problems that con- tains numerous classical problems like the satisfiability problem for propositional formulas and the graph colourability problem. Feder and Vardi [52] gave the following logical for- malization of the class of CSPs: every finite relational structure A, the template, gives rise to the decision problem of determining whether there exists a homomorphism from a finite input structure B to A. In their seminal paper, Feder and Vardi recognised that CSPs had a particular status in the landscape of computational complexity: despite the generality of these problems, it seemed impossible to construct NP-intermediate problems `a la Ladner [72] within this class. The authors thus conjectured that the class of CSPs satisfies a complexity dichotomy , i.e., that every CSP is solvable in polynomial time or is NP-complete. The Feder-Vardi dichotomy conjecture was the motivation of an intensive line of research over the last two decades. Some of the landmarks of this research are the confirmation of the conjecture for special classes of templates, e.g., for the class of undi- rected graphs [55], for the class of smooth digraphs [5], and for templates with at most three elements [43, 84]. Finally, after being open for 25 years, Bulatov [44] and Zhuk [87] independently proved that the conjecture of Feder and Vardi indeed holds. The success of the research program on the Feder-Vardi conjecture is based on the con- nection between constraint satisfaction problems and universal algebra. In their seminal paper, Feder and Vardi described polynomial-time algorithms for CSPs whose template satisfies some closure properties. These closure properties are properties of the polymor- phism clone of the template and similar properties were later used to provide tractability or hardness criteria [61, 62]. Shortly thereafter, Bulatov, Jeavons, and Krokhin [46] proved that the complexity of the CSP depends only on the equational properties of the poly- morphism clone of the template. They proved that trivial equational properties imply hardness of the CSP, and conjectured that the CSP is solvable in polynomial time if the polymorphism clone of the template satisfies some nontrivial equation. It is this conjecture that Bulatov and Zhuk finally proved, relying on recent developments in universal algebra. As a by-product of the fact that the delineation between polynomial-time tractability and NP-hardness can be stated algebraically, we also obtain that the meta-problem for finite- domain CSPs is decidable. That is, there exists an algorithm that, given a finite relational structure A as input, decides the complexity of the CSP of A.

A network design problem defines an infinite set whose elements, called instances, describe relationships and network constraints. It asks for an algorithm that, given an instance of this set, designs a network that respects the given constraints and at the same time optimizes some given criterion. In my thesis, I develop algorithms whose solutions are optimum or close to an optimum value within some guaranteed bound. I also examine the computational complexity of these problems. Problems from two vast areas are considered: graphs and the Euclidean plane. In the Maximum Edge Disjoint Paths problem, we are given a graph and a subset of vertex pairs that are called terminal pairs. We are asked for a set of paths where the endpoints of each path form a terminal pair. The constraint is that any two paths share at most one inner vertex. The optimization criterion is to maximize the cardinality of the set. In the hard-capacitated k-Facility Location problem, we are given an integer k and a complete graph where the distances obey a given metric and where each node has two numerical values: a capacity and an opening cost. We are asked for a subset of k nodes, called facilities, and an assignment of all the nodes, called clients, to the facilities. The constraint is that the number of clients assigned to a facility cannot exceed the facility's capacity value. The optimization criterion is to minimize the total cost which consists of the total opening cost of the facilities and the total distance between the clients and the facilities they are assigned to. In the Stabbing problem, we are given a set of axis-aligned rectangles in the plane. We are asked for a set of horizontal line segments such that, for every rectangle, there is a line segment crossing its left and right edge. The optimization criterion is to minimize the total length of the line segments. In the k-Colored Non-Crossing Euclidean Steiner Forest problem, we are given an integer k and a finite set of points in the plane where each point has one of k colors. For every color, we are asked for a drawing that connects all the points of the same color. The constraint is that drawings of different colors are not allowed to cross each other. The optimization criterion is to minimize the total length of the drawings. In the Minimum Rectilinear Polygon for Given Angle Sequence problem, we are given an angle sequence of left (+90°) turns and right (-90°) turns. We are asked for an axis-parallel simple polygon where the angles of the vertices yield the given sequence when walking around the polygon in counter-clockwise manner. The optimization criteria considered are to minimize the perimeter, the area, and the size of the axis-parallel bounding box of the polygon. / Ein Netzwerk-Design-Problem definiert eine unendliche Menge, deren Elemente, als Instanzen bezeichnet, Beziehungen und Beschränkungen in einem Netzwerk beschreiben. Die Lösung eines solchen Problems besteht aus einem Algorithmus, der auf die Eingabe einer beliebigen Instanz dieser Menge ein Netzwerk entwirft, welches die gegebenen Beschränkungen einhält und gleichzeitig ein gegebenes Kriterium optimiert. In meiner Dissertation habe ich Algorithmen entwickelt, deren Netzwerke stets optimal sind oder nachweisbar nahe am Optimum liegen. Zusätzlich habe ich die Berechnungskomplexität dieser Probleme untersucht. Dabei wurden Probleme aus zwei weiten Gebieten betrachtet: Graphen und der Euklidische Ebene. Im Maximum-Edge-Disjoint-Paths-Problem besteht die Eingabe aus einem Graphen und einer Teilmenge von Knotenpaaren, die wir mit Terminalpaare bezeichnen. Gesucht ist eine Menge von Pfaden, die Terminalpaare verbinden. Die Beschränkung ist, dass keine zwei Pfade einen gleichen inneren Knoten haben dürfen. Das Optimierungskriterium ist die Maximierung der Kardinalität dieser Menge. Im Hard-Capacitated-k-Facility-Location-Problem besteht die Eingabe aus einer Ganzzahl k und einem vollständigen Graphen, in welchem die Distanzen einer gegebenen Metrik unterliegen und in welchem jedem Knoten sowohl eine numerische Kapazität als auch ein Eröffnungskostenwert zugeschrieben ist. Gesucht ist eine Teilmenge von k Knoten, Facilities genannt, und eine Zuweisung aller Knoten, Clients genannt, zu den Facilities. Die Beschränkung ist, dass die Anzahl der Clients, die einer Facility zugewiesen sind, nicht deren Kapazität überschreiten darf. Das Optimierungskriterium ist die Minimierung der Gesamtkosten bestehend aus den Gesamteröffnungskosten der Facilities sowie der Gesamtdistanz zwischen den Clients und den ihnen zugewiesenen Facilities. Im Stabbing-Problem besteht die Eingabe aus einer Menge von achsenparallelen Rechtecken in der Ebene. Gesucht ist eine Menge von horizontalen Geradenstücken mit der Randbedingung, dass die linke und rechte Seite eines jeden Rechtecks von einem Geradenstück verbunden ist. Das Optimierungskriterium ist die Minimierung der Gesamtlänge aller Geradenstücke. Im k-Colored-Non-Crossing-Euclidean-Steiner-Forest-Problem besteht die Eingabe aus einer Ganzzahl k und einer endlichen Menge von Punkten in der Ebene, wobei jeder Punkt in einer von k Farben gefärbt ist. Gesucht ist für jede Farbe eine Zeichnung, in welcher alle Punkte der Farbe verbunden sind. Die Beschränkung ist, dass Zeichnungen verschiedener Farben sich nicht kreuzen dürfen. Das Optimierungskriterium ist die Minimierung des Gesamtintenverbrauchs, das heißt, der Gesamtlänge der Zeichnungen. Im Minimum-Rectilinear-Polygon-for-Given-Angle-Sequence-Problem besteht die Eingabe aus einer Folge von Links- (+90°) und Rechtsabbiegungen (-90°). Gesucht ist ein achsenparalleles Polygon dessen Eckpunkte die gegebene Folge ergeben, wenn man das Polygon gegen den Uhrzeigersinn entlangläuft. Die Optimierungskriterien sind die Minimierung des Umfangs und der inneren Fläche des Polygons sowie der Größe des notwendigen Zeichenblattes, d.h., des kleinsten Rechteckes, das das Polygon einschließt. / Given points in the plane, connect them using minimum ink. Though the task seems simple, it turns out to be very time consuming. In fact, scientists believe that computers cannot efficiently solve it. So, do we have to resign? This book examines such NP-hard network-design problems, from connectivity problems in graphs to polygonal drawing problems on the plane. First, we observe why it is so hard to optimally solve these problems. Then, we go over to attack them anyway. We develop fast algorithms that find approximate solutions that are very close to the optimal ones. Hence, connecting points with slightly more ink is not hard.

Unification has been introduced in Description Logic (DL) as a means to detect redundancies in ontologies. In particular, it was shown that testing unifiability in the DL EL is an NP-complete problem, and this result has been extended in several directions. Surprisingly, it turned out that the complexity increases to PSpace if one disallows the use of the top concept in concept descriptions. Motivated by features of the medical ontology SNOMED CT, we extend this result to a setting where the top concept is disallowed, but there is a background ontology consisting of restricted forms of concept and role inclusion axioms. We are able to show that the presence of such axioms does not increase the complexity of unification without top, i.e., testing for unifiability remains a PSpace-complete problem.

Einem Konsumenten werden in verschiedensten Situationen Rabatte angeboten. In dieser Dissertation wird die Frage untersucht, wie solche Rabattsituationen aus konsumentensicht formalisiert werden können und wie Kaufentscheidungen getroffen werden können. Um diese Frage zu beantworten, wird ein formaler Rahmen für Rabattsituationen angegeben und zur Analyse einer neuen Gruppe von acht Problemen, die auf alltäglichen Erfahrungen mit Rabattaktionen basieren, angewendet. Diese Probleme werden hinsichtlich der Rabattgrundlage (Stempel / Punkte), dem Kartentyp (Einzelkarte, Gruppenkarte) und der Frage, ob Stempel/Punkte für Käufe mit Rabatt gesammelt werden, unterschieden. Der inhärenten Planungsunsicherheit für Konsumentenentscheidungen wird explizit durch die Betrachtung jedes Problems als eine Onlinesituation Rechnung getragen. Für die Onlineprobleme wird eine zugeschnittene Methode zur Güteabschätzung präsentiert. Jedes der acht Probleme wird als Entscheidungs-, Optimierungs- und Onlineproblem analysiert. Für alle Entscheidungsprobleme wird NP-Vollständigkeit nachgewiesen. Jedes Optimierungsproblem wird mit ganzzahliger linearer Programmierung und einige stempelbasierte Probleme zusätzlich mit dynamischer Programmierung gelöst. Für die Onlineprobleme wird jeweils eine untere Güteschranke gezeigt und für drei Gruppen von Onlinealgorithmen die Güte abgeschätzt.:1. Einleitung 2. Vorbetrachtungen 3. Problemformulierung und Analysemethodik 4. Die Probleme im Detail 5. Zusammenfassung 6. Ausblick A. Implementationen / A consumer is offered discounts in a variety of situations. The central question investigated in this dissertation is how to formalize such discount situations from a consumer perspective and what methods for deducing purchase decisions are possible. To answer this question a formal framework for discount situations is established and used to explore a new group of eight discount problems based on everyday experience with loyalty programs. These problems are distinguished by discount basis (stamps / points), card type (single / group) and whether stamps/-points are collectable if a discount is granted. The inherent uncertainty in consumer decisions is explicitly taken into account by considering each of these problems as an online situation as well. Regarding the online problems, a method for competitive analysis is presented. Each of the eight problems is examined as a decision, an optimization and an online problem. For all decision problems N P-completeness is shown. Each optimization problem is solved via linear integer programming and some stamp based optimization problems are furthermore solved with dynamic programming. For each online problem a lower bound on the competitive ratio is presented together with three groups of online algorithms and the respective bounds on the competitive ratio.:1. Einleitung 2. Vorbetrachtungen 3. Problemformulierung und Analysemethodik 4. Die Probleme im Detail 5. Zusammenfassung 6. Ausblick A. Implementationen

Recently, verification and analysis of data words have gained a lot of interest. Metric temporal logic (MTL) and timed propositional temporal logic (TPTL) are two extensions of Linear time temporal logic (LTL). In MTL, the temporal operator are indexed by a constraint interval. TPTL is a more powerful logic that is equipped with a freeze formalism. It uses register variables, which can be set to the current data value and later these register variables can be compared with the current data value. For monotonic data words, Alur and Henzinger proved that MTL and TPTL are equally expressive and the satisfiability problem is decidable. We study the expressive power, satisfiability problems and path checking problems for MLT and TPTL over all data words. We introduce Ehrenfeucht-Fraisse games for MTL and TPTL. Using the EF-game for MTL, we show that TPTL is strictly more expressive than MTL. Furthermore, we show that the MTL definability problem that whether a TPTL-formula is definable in MTL is not decidable. When restricting the number of register variables, we are able to show that TPTL with two register variables is strictly more expressive than TPTL with one register variable. For the satisfiability problem, we show that for MTL, the unary fragment of MTL and the pure fragment of MTL, SAT is not decidable. We prove the undecidability by reductions from the recurrent state problem and halting problem of two-counter machines. For the positive fragments of MTL and TPTL, we show that a positive formula is satisfiable if and only it is satisfied by a finite data word. Finitary SAT and infinitary SAT coincide for positive MTL and positive TPTL. Both of them are r.e.-complete. For existential TPTL and existential MTL, we show that SAT is NP-complete. We also investigate the complexity of path checking problems for TPTL and MTL over data words. These data words can be either finite or infinite periodic. For periodic words without data values, the complexity of LTL model checking belongs to the class AC^1(LogDCFL). For finite monotonic data words, the same complexity bound has been shown for MTL by Bundala and Ouaknine. We show that path checking for TPTL is PSPACE-complete, and for MTL is P-complete. If the number of register variables allowed is restricted, we obtain path checking for TPTL with only one register variable is P-complete over both infinite and finite data words; for TPTL with two register variables is PSPACE-complete over infinite data words. If the encoding of constraint numbers of the input TPTL-formula is in unary notation, we show that path checking for TPTL with a constant number of variables is P-complete over infinite unary encoded data words. Since the infinite data word produced by a deterministic one-counter machine is periodic, we can transfer all complexity results for the infinite periodic case to model checking over deterministic one-counter machines.

Datalog ist die relationale Variante der logischen Programmierung und ist eine Standard-Abfragesprache in der Datenbankentheorie geworden. Die Programmkomplexität von Datalog im bisherigen Hauptanwendungsgebiet, auf endlichen Strukturen, ist bekanntermassen in EXPTIME. Wir untersuchen die Komplexität von Datalog auf unendlichen Strukturen, motiviert durch mögliche Anwendungen von Datalog auf unendlichen Strukturen (z.B. linearen Ordnungen) im zeitlichen und räumlichen Schliessen, aber auch durch das aufkommende Interesse an unendlichen Strukturen bei verwandten theoretischen Problemen, wie Constraint Satisfaction Problems (CSP): Im Gegensatz zu endlichen Strukturen können Datalog-Berechnungen auf unendlichen Strukturen unendlich lange dauern, was zur Unentscheidbarkeit von Datalog auf unendlichen Strukturen führen kann. Aber auch in den entscheidbaren Fällen kann die Komplexität von Datalog auf unendlichen Strukturen beliebig hoch sein. Im Hinblick auf dieses Ergebnis widmen wir uns dann unendlichen Strukturen mit der niedrigsten Komplexität von Datalog: Wir zeigen, dass Datalog auf linearen Ordnungen (auch dichte und diskrete, mit oder ohne Konstanten und sogar gefärbte) und Baumordnungen EXPTIME-vollständig ist. Für die Bestimmung der oberen Schranke werden Werkzeuge für Datalog auf Ordnungen eingeführt: Ordnungstypen, Abstandstypen und typdisjunkte Programme. Die Typkonzepte liefern eine endliche Beschreibung der unendlichen Programmergebnisse und könnten auch für praktische Anwendungen von Interesse sein. Wir erzeugen spezielle typdisjunkte Programme, die sich ohne Rekursion lösen lassen. Ein Transfer unserer Methoden auf CSPs zeigt, dass CSPs auf unendlichen Strukturen mit beliebig hoher Zeitkomplexität vorkommen, wie Datalog. / Datalog is the relational variant of logic programming and has become a standard query language in database theory. The (program) complexity of datalog in its main context so far, on finite databases, is well known to be in EXPTIME. We research the complexity of datalog on infinite databases, motivated by possible applications of datalog to infinite structures (e.g. linear orders) in temporal and spatial reasoning on one hand and the upcoming interest in infinite structures in problems related to datalog, like constraint satisfaction problems: Unlike datalog on finite databases, on infinite structures the computations may take infinitely long, leading to the undecidability of datalog on some infinite structures. But even in the decidable cases datalog on infinite structures may have arbitrarily high complexity, and because of this result, we research some structures with the lowest complexity of datalog on infinite structures: Datalog on linear orders (also dense or discrete, with and without constants, even colored) and tree orders has EXPTIME-complete complexity. To achieve the upper bound on these structures, we introduce a tool set specialized for datalog on orders: Order types, distance types and type disjoint programs. The type concept yields a finite representation of the infinite program results, which could also be of interest for practical applications. We create special type disjoint versions of the programs allowing to solve datalog without the recursion inherent in each datalog program. A transfer of our methods shows that constraint satisfaction problems on infinite structures occur with arbitrarily high time complexity, like datalog.