In this dissertation we prove the Completeness, Soundness and Compactness of the Monadic Hybrid Calculus MHC and we prove its expressive equivalence to the Monadic Predicate Calculus MPC. The Monadic Hybrid Calculus MHC is a new system that is based on the (propositional) modal logic S5. It is “Hybrid” in the sense that it includes quantifier free MPC and therefore, unlike S5, allows free individual constants. The main innovation in this system is the elimination of bound variables. In MHC, upper case letters denote properties and lower case letters denote individuals. Universal quantification is represented by square brackets, [], and existential quantification is represented by angled brackets, 〈〉. Thus, All Athenians are Greek and mortal is formalized as [A](G∧M), Some mortal Greeks are Athenians as 〈M∧G〉A, and Socrates is mortal and Athenian as s(M∧A). We give the formal syntax and the formal semantics of [MHC] and give Beth-style Tableau Rules (Inference Rules). In these rules, if [P]Q is on the right then we select a new constant [v] and we add [vP] on left, vQ on the right, and we cancel the formula. If [P]Q is on the left then we select a pre-used constant p and split the tree. We add pP on the right of one branch and pQ on the left of the other branch. We treat 〈P〉Q similarly. Our Completeness proof uses induction on formulas down a path in the proof tree. Our Soundness proof uses induction up a path. To prove that MPC is logically equivalent to the Monadic Predicate Calculus, we present algorithms that transform formulas back and forth between these two systems. Compactness follows immediately. Finally, we examine the pragmatic usage of the Monadic Hybrid Calculus and we compare it with the Monadic Predicate Calculus using natural language examples. We also examine the novel notions of the Hybrid Predicate Calculus along with their pragmatic implications. / Graduate / 0800 / 0984

In this thesis we will introduce multioperator weighted monadic datalog (mwmd), a formal model for specifying tree series, tree transformations, and tree languages. This model combines aspects of multioperator weighted tree automata (wmta), weighted monadic datalog (wmd), and monadic datalog tree transducers (mdtt). In order to develop a rich theory we will define multiple versions of semantics for mwmd and compare their expressiveness. We will study normal forms and decidability results of mwmd and show (by employing particular semantic domains) that the theory of mwmd subsumes the theory of both wmd and mdtt. We conclude this thesis by showing that mwmd even contain wmta as a syntactic subclass and present results concerning this subclass.

In this thesis we will introduce multioperator weighted monadic datalog (mwmd), a formal model for specifying tree series, tree transformations, and tree languages. This model combines aspects of multioperator weighted tree automata (wmta), weighted monadic datalog (wmd), and monadic datalog tree transducers (mdtt). In order to develop a rich theory we will define multiple versions of semantics for mwmd and compare their expressiveness. We will study normal forms and decidability results of mwmd and show (by employing particular semantic domains) that the theory of mwmd subsumes the theory of both wmd and mdtt. We conclude this thesis by showing that mwmd even contain wmta as a syntactic subclass and present results concerning this subclass.

Cost automata are traditional finite state automata enriched with a finite set of counters that can be manipulated on each transition. Based on the evolution of counter values, a cost automaton defines a function from the set of structures under consideration to the natural numbers extended with infinity, modulo a boundedness relation that ignores exact values but preserves boundedness properties. Historically, variants of cost automata have been used to solve problems in language theory such as the star height problem. They also have a rich theory in their own right as part of the theory of regular cost functions, which was introduced by Colcombet as an extension to the theory of regular languages. It subsumes the classical theory since a language can be associated with the function that maps every structure in the language to 0 and everything else to infinity; it is a strict extension since cost functions can count some behaviour within the input. Regular cost functions have been previously studied over finite words and trees. This thesis extends the theory to infinite trees, where classical parity automata are enriched with a finite set of counters. Weak cost automata, which have priorities {0,1} or {1,2} and an additional restriction on the structure of the transition function, are shown to be equivalent to a weak cost monadic logic. A new notion of quasi-weak cost automata is also studied and shown to arise naturally in this cost setting. Moreover, a decision procedure is given to determine whether or not functions definable using weak or quasi-weak cost automata are equivalent up to the boundedness relation, which also proves the decidability of the weak cost monadic logic over infinite trees. The semantics of these cost automata over infinite trees are defined in terms of cost-parity games which are two-player infinite games where one player seeks to minimize the counter values and satisfy the parity condition, and the other player seeks to maximize the counter values or sabotage the parity condition. The main contributions and key technical results involve proving that certain cost-parity games admit positional or finite-memory strategies. These results also help settle the decidability of some special cases of long-standing open problems in the classical theory. In particular, it is shown that it is decidable whether a regular language of infinite trees is recognizable using a nondeterministic co-Büchi automaton. Likewise, given a Büchi or co-Büchi automaton as input, it is decidable whether or not there is a weak automaton recognizing the same language.

Category theory is proving a useful tool in programming and program specification - not only as a descriptive language but as directly applicable to programming and specification tasks. Category theory achieves a level of generality of description at which computation is still possible. We show that theorems from category theory often have constructive proofs in the sense that they may be encoded as programs. In particular we look at the computation of colimits in categories showing that general theorems give rise to routines which considerably simplify the rather awkward computation of colimits. The general routines arising from categorical constructions can be used to build programs in the 'combinatorial' style of programming. We show this with an example - a program to implement the semantics of a specification language. More importantly, the intimate relationship between these routines and algebraic specifications allows us to develop programs from certain forms of specifications. Later we turn to algebraic specifications themselves and look at properties of "monadic theories". We establish that, under suitable conditions: 1. Signatures and presentations may be defined for monadic theories and free theories on a signature may be constructed. 2. Theory morphisms give rise to ad junctions between categories of algebras and moreover a collection of algebras of a theory give rise to a new theory with certain properties. 3. Finite colimits and certain factorisations exist in categories of monadic theories. 4. Many-sorted, order-sorted and even category-sorted theories may be handled by somewhat extending the notion of monadic theories. These results show that monadic theories are sufficiently well-behaved to be used in the semantics of algebraic specification languages. Some of the constructions can be encoded as programs by the techniques mentioned above.

Les automates distribués sont des machines à états finis qui opèrent sur des graphes orientés finis. Fonctionnant comme des algorithmes distribués synchrones, ils utilisent leur graphe d'entrée comme un réseau dans lequel des processeurs identiques communiquent entre eux pendant un certain nombre (éventuellement infini) de rondes synchrones. Pour la variante locale de ces automates, où le nombre de rondes est borné par une constante, Hella et al. (2012, 2015) ont établi une caractérisation logique par des formules de la logique modale de base. Dans le cadre de cette thèse, nous présentons des caractérisations logiques similaires pour deux classes d'automates distribués plus expressives.La première classe étend les automates locaux avec une condition d'acceptation globale et la capacité d'alterner entre des modes de calcul non-déterministes et parallèles. Nous montrons qu'elle est équivalente à la logique monadique du second ordre sur les graphes.En nous restreignant à des transitions non-déterministes ou déterministes, nous obtenons également deux variantes d'automates strictement plus faibles pour lesquelles le problème du vide est décidable.Notre seconde classe adapte la notion standard d'algorithme asynchrone au cadre des automates distribués non-locaux. Les machines résultantes sont prouvées équivalentes à un petit fragment de la logique de point fixe, et plus précisément, à une variante restreinte du μ-calcul modal qui autorise les plus petits points fixes mais interdit les plus grands points fixes. Profitant du lien avec la logique, nous montrons aussi que la puissance expressive de ces automates asynchrones est indépendante du fait que des messages puissent être perdus ou non.Nous étudions ensuite la décidabilité du problème du vide pour plusieurs classes d'automates non-locaux. Nous montrons que le problème est indécidable en général, en simulant une machine de Turing par un automate distribué qui échange les rôles de l'espace et du temps. En revanche, le problème s'avère décidable en LOGSPACE pour une classe d'automates oublieux, où les nœuds voient les messages reçus de leurs voisins, mais ne se souviennent pas de leur propre état. Finalement, à titre de contribution mineure, nous donnons également de nouvelles preuves de séparation pour plusieurs hiérarchies d'alternance de quantificateurs basées sur la logique modale. / Distributed automata are finite-state machines that operate on finitedirected graphs. Acting as synchronous distributed algorithms, they use their input graph as a network in which identical processors communicate for a possibly infinite number of synchronous rounds. For the local variant of those automata, where the number of rounds is bounded by a constant, Hella et al. (2012, 2015) have established a logical characterization in terms of basic modal logic. In this thesis, we provide similar logical characterizations for two more expressive classes of distributed automata.The first class extends local automata with a global acceptance condition and the ability to alternate between non deterministic and parallel computations. We show that it is equivalent to monadic second-order logic on graphs. By restricting transitions to be non deterministic or deterministic, we also obtain two strictly weaker variants for which the emptiness problem is decidable.Our second class transfers the standard notion of asynchronous algorithm to the setting of non local distributed automata. There sulting machines are shown to be equivalent to a small fragment of least fixpoint logic, and more specifically, to a restricted variantof the modal μ -calculus that allows least fixpoints but forbids greatest fixpoints. Exploiting the connection with logic, we additionally prove that the expressive power of those asynchronous automata is independent of whether or not messages can be lost.We then investigate the decidability of the emptiness problem forseveral classes of nonlocal automata. We show that the problem isundecidable in general, by simulating a Turing machine with adistributed automaton that exchanges the roles of space and time. Onthe other hand, the problem is found to be decidable in logspace for a class of forgetful automata, where the nodes see the messages received from their neighbors but cannot remember their own state. As a minor contribution, we also give new proofs of the strictness of several set quantifier alternation hierarchies that are based on modallogic.

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Cette thèse se concentre autour du problème de l'évaluation des requêtes. Étant donné une requête q et une base de données D, l'objectif est de calculer l'ensemble q(D) des nuplets résultant de l'évaluation de q sur D. Toutefois, l'ensemble q(D) peut être plus grand que la base de données elle-même car elle peut avoir une taille de la forme n^l où n est la taille de la base de données et l est l'arité de la requête. Calculer entièrement q(D) peut donc nécessiter plus que les ressources disponibles. L'objectif principal de cette thèse est une solution particulière à ce problème: une énumération de q(D) avec un délai constant. Intuitivement, cela signifie qu'il existe un algorithme avec deux phases: une phase de pré-traitement qui fonctionne en temps linéaire dans la taille de la base de données, suivie d'une phase d'énumération produisant un à un tous les éléments de q(D) avec un délai constant (indépendant de la taille de la base de données) entre deux éléments consécutifs. En outre, quatre autres problèmes sont considérés: le model-checking (où la requête q est un booléen), le comptage (où on veut calculer la taille |q(D)|), les tests (où on s'intéresse à un test efficace pour savoir si un uplet donné appartient au résultat de la requête) et la j-ième solution (où on veut accéder directement au j-ième élément de q(D)). Les résultats présentés dans cette thèse portent sur les problèmes ci-dessus concernant: - les requêtes du premier ordre sur les classes de structures de degré borné, - les requêtes du second ordre monadique sur les classes de structures de largeur d'arborescente bornée, - les requêtes du premier ordre sur les classes de structures avec expansion bornée.

The problem with which this investigation is concerned is the determination of compositional styles in terms of manners of employing monadic and dyadic intervals in the music of the common practice period. An aspect for determining style is proposed by way of comparing the frequency of occurrence of dyads and monads in selected musical examples from the baroque, classical, and romantic periods. Chapter I is a discussion of the problem and methodology of the study. Chapters II, III, and IV present analytic comparison of examples in the baroque, classical, and romantic periods respectively. Chapter V presents a summary of the findings with references to the pedagogical applications of the two-part framework principle.

Dans cette thèse, nous étudions diverses largeurs de graphes autour de la largeur arborescente ainsi que de la largeur de clique. Nous commençons avec une étude comparative entre la largeur arborescente d’un graphe et la largeur de clique du graphe d’incidence associé, de laquelle nous extrayons des résultats algorithmiques encourageants. Puis nous présentons quelques propriétés structurelles liées à la largeur arborescente spéciale, largeur relativement récente qui est à mi-chemin entre les deux largeurs précédentes. Enfin nous nous intéressons à une notion plus générale connue sous le nom de fonction de partition sous-modulaire qui englobe, entre autres, les largeurs arborescentes « classique » et spéciale, la largeur de chemin ainsi que la largeur linéaire et les largeurs de branches de coupe et de découpe. Nous présentons alors un algorithme linéaire à paramètre fixé pour le calcul de ces différentes largeurs, lequel généralise un certain nombre de résultats propres à chacune de ces largeurs. / In this thesis, we study some width parameters on graphs, beyond tree-width and clique-width. Our first investigation is a comparative study between the tree-width of a graph and the clique-width of the associated incidence graph, from which we extract some strong algorithmic results. Then we present a few structural properties over a recently defined width called special tree-width and which takes its definition through both tree-width and clique-width. Finally, we end our journey with a more general notion named sub-modular partition fonction and which encompass both “classic” and special tree-widths, path-width, branch-width, linear-width, cut-width and carvingwidth among others. So, we introduce a fixed parameter tractable algorithm computing those widths parameters and thus we generalize a number of results specific to each of them.