This thesis examines so called folding neural networks as a mechanism for machine learning. Folding networks form a generalization of partial recurrent neural networks such that they are able to deal with tree structured inputs instead of simple linear lists. In particular, they can handle classical formulas - they were proposed originally for this purpose. After a short explanation of the neural architecture we show that folding networks are well suited as a learning mechanism in principle. This includes three parts: the proof of their universal approximation ability, the aspect of information theoretical learnability, and the examination of the complexity of training. Approximation ability: It is shown that any measurable function can be approximated in probability. Explicit bounds on the number of neurons result if only a finite number of points is dealt with. These bounds are new results in the case of simple recurrent networks, too. Several restrictions occur if a function is to be approximated in the maximum norm. Afterwards, we consider briefly the topic of computability. It is shown that a sigmoidal recurrent neural network can compute any mapping in exponential time. However, if the computation is subject to noise almost the capability of tree automata arises. Information theoretical learnability: This part contains several contributions to distribution dependent learnability: The notation of PAC and PUAC learnability, consistent PAC/ PUAC learnability, and scale sensitive versions are considered. We find equivalent characterizations of these terms and examine their respective relation answering in particular an open question posed by Vidyasagar. It is shown at which level learnability only because of an encoding trick is possible. Two approaches from the literature which can guarantee distribution dependent learnability if the VC dimension of the concept class is infinite are generalized to function classes: The function class is stratified according to the input space or according to a so-called luckiness function which depends on the output of the learning algorithm and the concrete training data. Afterwards, the VC, pseudo-, and fat shattering dimension of folding networks are estimated: We improve some lower bounds for recurrent networks and derive new lower bounds for the pseudodimension and lower and upper bounds for folding networks in general. As a consequence, folding architectures are not distribution independent learnable. Distribution dependent learnability can be guaranteed. Explicit bounds on the number of examples which guarantee valid generalization can be derived using the two approaches mentioned above. We examine in which cases these bounds are polynomial. Furthermore, we construct an explicit example for a learning scenario where an exponential number of examples is necessary. Complexity: It is shown that training a fixed folding architecture with perceptron activation function is polynomial. Afterwards, a decision problem, the so-called loading problem, which is correlated to neural network training is examined. For standard multilayer feed-forward networks the following situations turn out to be NP-hard: Concerning the perceptron activation function, a classical result from the literature, the NP-hardness for varying input dimension, is generalized to arbitrary multilayer architectures. Additionally, NP-hardness can be found if the input dimension is fixed but the number of neurons may vary in at least two hidden layers. Furthermore, the NP-hardness is examined if the number of patterns and number of hidden neurons are correlated. We finish with a generalization of the classical NP result as mentioned above to the sigmoidal activation function which is used in practical applications.

Matching concept descriptions against concept patterns was introduced as a new inference task in Description Logics (DLs) almost 20 years ago, motivated by applications in the Classic system. For the DL EL, it was shown in 2000 that the matching problem is NP-complete. It then took almost 10 years before this NP-completeness result could be extended from matching to unification in EL. The next big challenge was then to further extend these results from matching and unification without a TBox to matching and unification w.r.t. a general TBox, i.e., a finite set of general concept inclusions. For unification, we could show some partial results for general TBoxes that satisfy a certain restriction on cyclic dependencies between concepts, but the general case is still open. For matching, we solve the general case in this paper: we show that matching in EL w.r.t. general TBoxes is NP-complete by introducing a goal-oriented matching algorithm that uses non-deterministic rules to transform a given matching problem into a solved form by a polynomial number of rule applications. We also investigate some tractable variants of the matching problem.

Cette thèse étudie les aspects combinatoires, algorithmiques et la complexité de problèmes de théorie des graphes, et tout spécialement de problèmes liés aux notions de marches, de transitions et de distance dans les graphes. Nous nous intéressons d’abord au problème de traffic monitoring, qui consiste à placer aussi peu de capteurs que possible sur les arcs d’un graphe de façon à pouvoir reconstituer des marches d’objets. La caractérisation d’instances intéressantes dans la pratique nous amène à la notion de transitions interdites, qui renforce le modèle de graphe. Notre travail sur les graphes à transitions interdites comprend aussi l’étude de la notion d’ensemble de transitions connectant, que l’on peut voir comme l’analogue en terme de transitions de la notion d’arbre couvrant. Une partie importante de cette thèse porte sur les graphes géométriques, qui sont des graphes dont les sommets sont des points de l’espace réel et dont les arêtes sont déterminées par les distances géométriques entre les sommets. Ces graphes sont au coeur du célèbre problème de Hadwiger-Nelson et nous sont d’une grande aide dans notre étude de la densité des ensembles qui évitent la distance 1 dans plusieurs types d’espaces normés. Nous développons des outils pour étudier ces problèmes et les utilisons pour prouver la conjecture de Bachoc-Robins sur plusieurs paralléloèdres. Nous nous penchons aussi sur le cas du plan euclidien et améliorons les bornes sur la densité des ensembles évitant la distance 1 et sur son nombre chromatique fractionnaire. Enfin, nous étudions la complexité de problèmes d’homomorphismes de graphes et établissons des théorèmes de dichotomie sur la complexité des homomorphismes localement injectifs vers les tournois réflexifs. / This thesis studies combinatorial, algorithmic and complexity aspects of graph theory problems, and especially of problems related to the notions of walks, transitions and distances in graphs. We first study the problem of traffic monitoring, in which we have to place as few censors as possible on the arcs of a graph to be able to retrace walks of objects. The characterization of instances of practical interests brings us to the notion of forbidden transitions, which strengthens the model of graphs. Our work on forbidden-transition graphs also includes the study of connecting transition sets, which can be seen as a translation to forbidden-transition graphs of the notion of spanning trees. A large part of this thesis focuses on geometric graphs, which are graphs whose vertices are points of the real space and whose edges are determined by geometric distance between the vertices. This graphs are at the core of the famous Hadwiger- Nelson problem and are of great help in our study of the density of sets avoiding distance 1 in various normed spaces. We develop new tools to study these problems and use them to prove the Bachoc-Robins conjecture on several parallelohedra. We also investigate the case of the Euclidean plane and improve the bounds on the density of sets avoiding distance 1 and on its fractional chromatic number. Finally, we study the complexity of graph homomorphism problems and establish dichotomy theorems for the complexity of locally-injective homomorphisms to reflexive tournaments.

Given a set of n points P ⊂ R2, the Delaunay graph of P for a family of geometric objects C is a graph defined as follows: the vertex set is P and two points p, p' ∈ P are connected by an edge if and only if there exists some C ∈ C containing p, p' but no other point of P. Delaunay graph of circle is often called as Delaunay triangulation as each of its inner face is a triangle if no three points are co-linear and no four points are co-circular. The dual of the Delaunay triangulation is the Voronoi diagram, which is a well studied structure. The study of graph theoretic properties on Delaunay graphs was motivated by its application to wireless sensor networks, meshing, computer vision, computer graphics, computational geometry, height interpolation, etc. The problem of finding an optimal vertex cover on a graph is a classical NP-hard problem. In this thesis we focus on the vertex cover problem on Delaunay graphs for geometric objects like axis-parallel slabs and circles(Delaunay triangulation). 1. We consider the vertex cover problem on Delaunay graph of axis-parallel slabs. It turns out that the Delaunay graph of axis-parallel slabs has a very special property — its edge set is the union of two Hamiltonian paths. Thus, our problem reduces to solving vertex cover on the class of graphs whose edge set is simply the union of two Hamiltonian Paths. We refer to such a graph as a braid graph. Despite the appealing structure, we show that deciding k-vertex cover on braid graphs is NP-complete. This involves a rather intricate reduction from the problem of finding a vertex cover on 2-connected cubic planar graphs. 2. Having established the NP-hardness of the vertex cover problem on braid graphs, we pursue the question of improved fixed parameter algorithms on braid graphs. The best-known algorithm for vertex cover on general graphs has a running time of O(1.2738k + kn) [CKX10]. We propose a branching based fixed parameter tractable algorithm with running time O⋆(1.2637k) for graphs with maximum degree bounded by four. This improves the best known algorithm for this class, which surprisingly has been no better than the algorithm for general graphs. Note that this implies faster algorithms for the class of braid graphs (since they have maximum degree at most four). 3. A triangulation is a 2-connected plane graph in which all the faces except possibly the outer face are triangles, we often refer to such graphs as triangulated graphs. A chordless-NST is a triangulation that does not have chords or separating triangles (non-facial triangles). We focus on the computational problem of optimal vertex covers on triangulations, specifically chordless-NST. We call a triangulation Delaunay realizable if it is combinatorially equivalent to some Delaunay triangulation. Characterizations of Delaunay triangulations have been well studied in graph theory. Dillencourt and Smith [DS96] showed that chordless-NSTs are Delaunay realizable. We show that for chordless-NST, deciding the vertex cover problem is NP-complete. We prove this by giving a reduction from vertex cover on 3-connected, triangle free planar graph to an instance of vertex cover on a chordless-NST. 4. If the outer face of a triangulation is also a triangle, then it is called a maximal planar graph. We prove that the vertex cover problem is NP-complete on maximal planar graphs by reducing an instance of vertex cover on a triangulated graph to an instance of vertex cover on a maximal planar graph.

Dans une composition de services Web basée protocole, un ensemble de services composants se collaborent pour donner lieu à un service Composite. Chaque service est représenté par un automate à états finis (AEF). Au sein d’un AEF, chaque transition exprime l’exécution d’une opération qui fait avancer le service vers un état suivant. Une exécution du composite correspond à une séquence de transitions où chacune est déléguée à un des composants. Lors de l’exécution du composite, un ou plusieurs composants peuvent devenir indisponibles. Ceci peut produire une exécution incomplète du composite, et de ce fait un recouvrement est nécessaire. Le recouvrement consiste à transformer l’exécution incomplète en une exécution alternative ayant encore la capacité d’aller vers un état final. La transformation s'effectue en compensant certaines transitions et exécutant d’autres. Cette thèse présente une étude formelle du problème de recouvrement dans une composition de service Web basée protocole. Le problème de recouvrement consiste à trouver une meilleure exécution alternative parmi celles disponibles. Une meilleure alternative doit être atteignable à partir de l’exécution incomplète avec un nombre minimal de compensations visibles (vis-à-vis le client). Pour une exécution alternative donnée, nous prouvons que le problème de décision associé au calcul du nombre de transitions invisiblement compensées est NP-Complet. De ce fait, nous concluons que le problème de décision associé au recouvrement appartient à la classe ΣP2. / In a protocol-based Web service composition, a set of available component services collaborate together in order to provide a new composite service. Services export their protocols as finite state machines (FSMs). A transition in the FSM represents a task execution that makes the service moving to a next state. An execution of the composite corresponds to a sequence of transitions where each task is delegated to a component service. During composite run, one or more delegated components may become unavailable due to hard or soft problems on the Network. This unavailability may result in a failed execution of the composite. We provide in this thesis a formal study of the automatic recovery problem in the protocol-based Web service composition. Recovery consists in transforming the failed execution into a recovery execution. Such a transformation is performed by compensating some transitions and executing some others. The recovery execution is an alternative execution of the composite that still has the ability to reach a final state. The recovery problem consists then in finding the best recovery execution(s) among those available. The best recovery execution is attainable from the failed execution with a minimal number of visible compensations with respect to the client. For a given recovery execution, we prove that the decision problem associated with computing the number of invisibly-compensated transitions is NP-complete. Thus, we conclude that deciding of the best recovery execution is in ΣP2.

Bibliography: pages 192-195 / The avoidability, or unavoidability of patterns in words over finite alphabets has been studied extensively. The word α over a finite set A is said to be unavoidable for an infinite set B+ of nonempty words over a finite set B if, for all but finitely many elements w of B+, there exists a semigroup morphism φ ∶ A+ → B+ such that φ(α) is a factor of w. In this treatise, we start by presenting a historical background of results that are related to unavoidability. We present and discuss the most important theorems surrounding unavoidability in detail. We present various complexity-related properties of unavoidable words. For words that are unavoidable, we provide a constructive upper bound to the lengths of words that avoid them. In particular, for a pattern α of length n over an alphabet of size r, we give a concrete function N(n, r) such that no word of length N(n, r) over the alphabet of size r avoids α. A natural subsequent question is how many unavoidable words there are. We show that the fraction of words that are unavoidable drops exponentially fast in the length of the word. This allows us to calculate an upper bound on the number of unavoidable patterns for any given finite alphabet. Subsequently, we investigate computational aspects of unavoidable words. In particular, we exhibit concrete algorithms for determining whether a word is unavoidable. We also prove results on the computational complexity of the problem of determining whether a given word is unavoidable. Specifically, the NP-completeness of the aforementioned problem is established. / Decision Sciences / D. Phil. (Operations Research)