Ph.D. (Mathematics) / Algebraic methods have been largely ignored within the eld of hybrid logics. A main theme of this thesis is to illustrate the usefulness of algebraic methods in this eld. It is a well-known fact that certain properties of a logic correspond to properties of particular classes of algebras, and that we therefore can use these classes of algebras to answer questions about the logic. The rst aim of this thesis is to identify a class of algebras corresponding to hybrid logics. In particular, we introduce hybrid algebras as algebraic semantics for the better known hybrid languages in the literature. The second aim of this thesis is to use hybrid algebras to solve logical problems in the eld of hybrid logic. Specically, we will focus on proving general completeness results for some well-known hybrid logics with respect to hybrid algebras. Next, we study Sahlqvist theory for hybrid logics. We introduce syntactically de ned classes of hybrid formulas that have rst-order frame correspondents, which are preserved under taking Dedekind MacNeille completions of atomic hybrid algebras, and which are preserved under canonical extensions of permeated hybrid algebras. Finally, we investigate the nite model property (FMP) for several hybrid logics. In particular, we give analogues of Bull's theorem for the hybrid logics under consideration in this thesis. We also show that if certain syntactically de ned classes of hybrid formulas are added to the normal modal logic S4 as axioms, we obtain hybrid logics with the nite model property.

CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / NÃs investigamos a teoria dos modelos de LÃgicas Preferenciais, LÃgica HÃbrida e fragmentos da LÃgica de Segunda-Ordem com relaÃÃo a modelos finitos. A semÃnticas dessas lÃgicas diferem da abordagem clÃssica pelo uso de relaÃÃes entre modelos ou por restringir a cardinalidade dos modelos a cardinais finitos. Este trabalho tem trÃs partes. Na primeira parte deste trabalho nÃs estudamos a teoria dos modelos de lÃgicas preferenciais. LÃgicas preferenciais surgem no contexto do raciocÃnio nÃo-monotÃnico em InteligÃncia Artificial. A principal caracterÃstica dessas lÃgicas à a existÃncia de uma relaÃÃo entre modelos. Isso permite a definiÃÃo de uma relaÃÃo de consequÃncia nÃo monotÃnica considerando-se os modelos minimais de um conjunto de sentenÃas. Usando a abordagem da Teoria dos Modelos Abstrata, nÃs generalizamos alguns resultados de expressividade para classes de lÃgicas preferenciais. NÃs mostramos que sempre que uma classe de modelos minimais de um conjunto finito de sentenÃas à axiomatizÃvel, entÃo tal classe à finitamente axiomatizÃvel. NÃs mostramos que se tal classe define implicitamente um sÃmbolo do vocabulÃrio, existe uma axiomatizaÃÃo finita de uma forma particular, a saber, o conjunto finito de sentenÃas inicial mais uma definiÃÃo explÃcita para o sÃmbolo definido. Na segunda parte desse trabalho, nÃs investigamos a teoria dos modelos finitos da LÃgica HÃbrida. LÃgicas HÃbridas sÃo extensÃes da lÃgica modal atravÃs de termos hÃbridos que se referem a estados individuais em um modelo de Kripke. NÃs estudamos a complexidade computacional dos problemas de model- e frame- checking para a LÃgica HÃbrida. NÃs mostramos que para cada problema de grafos na Hierarquia Polinomial e cada nÃmero n, existe uma fÃrmula que exprime esse problema para grafos de cardinalidade n. NÃs mostramos que o tamanho das fÃrmulas à limitado por um polinÃmio em n. NÃs mostramos que podemos abrir mÃo das modalidades globais se nos limitarmos a grafos conexos com loops. NÃs definimos fragmentos da LÃgica HÃbrida que correspondem a cada nÃvel da Hierarquia Polinomial. Isso nos leva a uma prova alternativa da NP-dificuldade do problema de model-checking para um fragmento especÃfico de da LÃgica HÃbrida. Na Ãltima parte desse trabalho, nÃs exploramos a complexidade descritiva da lÃgica obtida ao restringirmos a quantificaÃÃo de segunda-ordem a relaÃÃes de grau limitado. Baseados em trabalhos anteriores de Schwentick et al. e de Grandjean e Olive, nÃs introduzimos a LÃgica de Segunda-Ordem de Grau Limitado e mostramos que ela captura a classe ALIN de classes de estruturas unÃrias aceitas por uma mÃquina de acesso randÃmico em tempo linear e um nÃmero fixo de alternÃncias dependente apenas do problema. NÃs estendemos essa lÃgica com o operador de fecho transitivo sobre relaÃÃes de ordem superior sobre relaÃÃes de grau limitado. NÃs mostramos que a LÃgica de Segunda- Ordem de Grau Limitado com Fecho Transitivo captura quantidade linear de registradores em uma mÃquina de acesso randÃmico nÃo-determinÃstica onde os valores armazenados em cada registrador durante a computaÃÃo sÃo limitados por uma funÃÃo linear na cardinalidade da estrutura de entrada. / We investigate the model theory of Preferential Logics, Hybrid Logic and fragments of Second-Order Logic with respect to finite models. The semantics of these logics differ from the semantics of classical logics either by using relations between models or by restricting the cardinality of the models considered. This work has three main parts. In the first part of this work we study the model theory of preferential logics. Preferential logics arise in the context of nonmonotonic reasoning in Artificial Intelligence. The main characteristic of those logics is the existence of a relation between models. It allows the definition of a nonmonotonic consequence relation by considering the minimal models of a set of sentences. Using the approach of Abstract Model Theory we generalize some expressiveness results to classes of preferential logics. We show that whenever a class of minimal models of a finite set of sentences is axiomatizable, without considering the preference relation, then it is finitely axiomatizable. We also show that when such class of minimal models implicitly defines a symbol, then the finite axiomatization can be put in a very specic form, namely, the initial set of sentences plus a explicit definition for the symbol. In the second part of this work, we investigate the finite model theory of Hybrid Logic. Hybrid Logics are extensions of modal logics with hybrid terms which refer to single states in a Kripke model. We study the complexity of the model- and frame-checking problems for Hybrid Logic. We show that for each graph problem in the Polynomial Hierarchy and each natural number n there is a formula which expresses this problem for graphs of cardinality n. We also show that the size of such formulas is bounded by a polynomial in n. We show that one can disregard the global modalities if one consider only connected graphs with loops. We define fragments which correspond to each degree of the Polynomial Hierarchy. This leads to an alternative proof of the NP-hardness of the model-checking problem for an specic fragment of Full Hybrid Logic. In the last part of this work, we explore the descriptive complexity of the logic obtained by restricting second-order quantication to relations of bounded degree. Based on previous work from Schwentick et al. and Grandjean and Olive, we introduce the Bounded-Degree Second-Order Logic and show that it captures the class ALIN of classes of unary structures accepted by a alternating random access machine in linear time and bounded number of alternations. We also extend this logic with the transitive closure operator on high-order relations on bounded-degree relations. We show that the Bounded-Degree Second-Order Logic with Transitive Closure Operator captures linear number of registers in a nondeterministic random access machine provided that registers store values bounded by a linear function in the cardinality of the input structure.

This is a study of the semantics and proof theory of the logic of bunched implications (BI), which is promoted as a logic of (computational) resources, and is a foundational component of separation logic, an approach to program analysis. BI combines an additive, or intuitionistic, fragment with a multiplicative fragment. The additive fragment has full use of the structural rules of weakening and contraction, and the multiplicative fragment has none. Thus it contains two conjunctive and two implicative connectives. At various points, we illustrate a resource view of BI based upon the Kripke resource semantics. Our first original contribution is the formulation of a proof system for BI in the newly developed proof-theoretical formalism of the calculus of structures. The calculus of structures is distinguished by its employment of deep inference, but we already see deep inference in a limited form in the established proof theory for BI. We show that our system is sound with respect to the elementary Kripke resource semantics for BI, and complete with respect to a formulation of the partially-defined Kripke resource semantics. Our second contribution is the development from a semantic standpoint of preliminary ideas for a hybrid logic of bunched implications (HBI). We give a Kripke semantics for HBI in which nominal propositional atoms can be seen as names for resources, rather than as names for locations, as is the case with related proposals for BI-Loc and for intuitionistic hybrid logic.

FERREIRA, Francicleber Martins. Expressividade e complexidade em lógicas preferenciais, híbridas e de grau limitado. 2012. 130 f. Tese (Doutorado em ciência da computação)- Universidade Federal do Ceará, Fortaleza-CE, 2012. / Submitted by Elineudson Ribeiro (elineudsonr@gmail.com) on 2016-07-20T11:58:10Z No. of bitstreams: 1 2012_tese_fmferreira.pdf: 772301 bytes, checksum: b1e1a404221e38ffb8e0712513749a4c (MD5) / Approved for entry into archive by Rocilda Sales (rocilda@ufc.br) on 2016-07-25T11:47:07Z (GMT) No. of bitstreams: 1 2012_tese_fmferreira.pdf: 772301 bytes, checksum: b1e1a404221e38ffb8e0712513749a4c (MD5) / Made available in DSpace on 2016-07-25T11:47:07Z (GMT). No. of bitstreams: 1 2012_tese_fmferreira.pdf: 772301 bytes, checksum: b1e1a404221e38ffb8e0712513749a4c (MD5) Previous issue date: 2012 / We investigate the model theory of Preferential Logics, Hybrid Logic and fragments of Second-Order Logic with respect to finite models. The semantics of these logics differ from the semantics of classical logics either by using relations between models or by restricting the cardinality of the models considered. This work has three main parts. In the first part of this work we study the model theory of preferential logics. Preferential logics arise in the context of nonmonotonic reasoning in Artificial Intelligence. The main characteristic of those logics is the existence of a relation between models. It allows the definition of a nonmonotonic consequence relation by considering the minimal models of a set of sentences. Using the approach of Abstract Model Theory we generalize some expressiveness results to classes of preferential logics. We show that whenever a class of minimal models of a finite set of sentences is axiomatizable, without considering the preference relation, then it is finitely axiomatizable. We also show that when such class of minimal models implicitly defines a symbol, then the finite axiomatization can be put in a very specic form, namely, the initial set of sentences plus a explicit definition for the symbol. In the second part of this work, we investigate the finite model theory of Hybrid Logic. Hybrid Logics are extensions of modal logics with hybrid terms which refer to single states in a Kripke model. We study the complexity of the model- and frame-checking problems for Hybrid Logic. We show that for each graph problem in the Polynomial Hierarchy and each natural number n there is a formula which expresses this problem for graphs of cardinality n. We also show that the size of such formulas is bounded by a polynomial in n. We show that one can disregard the global modalities if one consider only connected graphs with loops. We define fragments which correspond to each degree of the Polynomial Hierarchy. This leads to an alternative proof of the NP-hardness of the model-checking problem for an specic fragment of Full Hybrid Logic. In the last part of this work, we explore the descriptive complexity of the logic obtained by restricting second-order quantication to relations of bounded degree. Based on previous work from Schwentick et al. and Grandjean and Olive, we introduce the Bounded-Degree Second-Order Logic and show that it captures the class ALIN of classes of unary structures accepted by a alternating random access machine in linear time and bounded number of alternations. We also extend this logic with the transitive closure operator on high-order relations on bounded-degree relations. We show that the Bounded-Degree Second-Order Logic with Transitive Closure Operator captures linear number of registers in a nondeterministic random access machine provided that registers store values bounded by a linear function in the cardinality of the input structure. / Nós investigamos a teoria dos modelos de Lógicas Preferenciais, Lógica Híbrida e fragmentos da Lógica de Segunda-Ordem com relação a modelos finitos. A semânticas dessas lógicas diferem da abordagem clássica pelo uso de relações entre modelos ou por restringir a cardinalidade dos modelos a cardinais finitos. Este trabalho tem três partes. Na primeira parte deste trabalho nós estudamos a teoria dos modelos de lógicas preferenciais. Lógicas preferenciais surgem no contexto do raciocínio não-monotônico em Inteligência Artificial. A principal característica dessas lógicas é a existência de uma relação entre modelos. Isso permite a definição de uma relação de consequência não monotônica considerando-se os modelos minimais de um conjunto de sentenças. Usando a abordagem da Teoria dos Modelos Abstrata, nós generalizamos alguns resultados de expressividade para classes de lógicas preferenciais. Nós mostramos que sempre que uma classe de modelos minimais de um conjunto finito de sentenças é axiomatizável, então tal classe é finitamente axiomatizável. Nós mostramos que se tal classe define implicitamente um símbolo do vocabulário, existe uma axiomatização finita de uma forma particular, a saber, o conjunto finito de sentenças inicial mais uma definição explícita para o símbolo definido. Na segunda parte desse trabalho, nós investigamos a teoria dos modelos finitos da Lógica Híbrida. Lógicas Híbridas são extensões da lógica modal através de termos híbridos que se referem a estados individuais em um modelo de Kripke. Nós estudamos a complexidade computacional dos problemas de model- e frame- checking para a Lógica Híbrida. Nós mostramos que para cada problema de grafos na Hierarquia Polinomial e cada número n, existe uma fórmula que exprime esse problema para grafos de cardinalidade n. Nós mostramos que o tamanho das fórmulas é limitado por um polinômio em n. Nós mostramos que podemos abrir mão das modalidades globais se nos limitarmos a grafos conexos com loops. Nós definimos fragmentos da Lógica Híbrida que correspondem a cada nível da Hierarquia Polinomial. Isso nos leva a uma prova alternativa da NP-dificuldade do problema de model-checking para um fragmento específico de da Lógica Híbrida. Na última parte desse trabalho, nós exploramos a complexidade descritiva da lógica obtida ao restringirmos a quantificação de segunda-ordem a relações de grau limitado. Baseados em trabalhos anteriores de Schwentick et al. e de Grandjean e Olive, nós introduzimos a Lógica de Segunda-Ordem de Grau Limitado e mostramos que ela captura a classe ALIN de classes de estruturas unárias aceitas por uma máquina de acesso randômico em tempo linear e um número fixo de alternâncias dependente apenas do problema. Nós estendemos essa lógica com o operador de fecho transitivo sobre relações de ordem superior sobre relações de grau limitado. Nós mostramos que a Lógica de Segunda- Ordem de Grau Limitado com Fecho Transitivo captura quantidade linear de registradores em uma máquina de acesso randômico não-determinística onde os valores armazenados em cada registrador durante a computação são limitados por uma função linear na cardinalidade da estrutura de entrada.

Le développement de nouveaux formalismes logiques est au cœur de nombreuses problématiques de méthodes formelles. Ces formalismes doivent répondre à la fois à des impératifs de modélisation (ils doivent permettre de décrire certains systèmes) et de calcul (ils doivent fournir des méthodes de calcul correctes et complètes). Dans ce contexte, nous nous intéressons aux logiques de ressources, en particulier les logiques BI et BBI qui traitent du partage et de la séparation de ressources et qui ont conduit aux diverses logiques de séparation dont les applications à la vérification de programmes se sont développées fortement ces dernières années. Nous proposons dans cette thèse d’étudier, à partir des logiques BI et BBI, des logiques de séparation modales et épistémiques en se focalisant sur leurs capacités de modélisation et leur expressivité mais aussi les nouveaux calculs de preuve pour ces logiques. Une première étude a porté sur la modélisation de propriétés dynamiques de ressources au travers d’une nouvelle logique LTBI, qui est une logique de séparation temporelle, fondée sur la logique BI et des modalités temporelles. Cette logique offre notamment des perspectives intéressantes de modélisation temporelle branchante, permettant par exemple de caractériser les processus multi-thread. Une étude complémentaire a porté sur la modélisation de l’accès par des agents à des propriétés sous conditions de posséder certaines ressources, au travers d’une nouvelle logique ERL, qui est une logique de séparation épistémique, fondée sur la logique BBI et des modalités épistémiques. Cette logique permet de nombreuses modélisations de systèmes de contrôle d’accès. En vue d’étendre l’expressivité de telles logiques de séparation, comme la logique BBI et ses variantes, une étude sur l’internalisation des symboles de ressources dans la syntaxe de la logique a été développée au travers des nouvelles logiques HRL et HBBI (version hybride de BBI). L’internalisation permet à la fois d’étendre l’expressivité des logiques et d’axiomatiser la logique BBI et certaines de ses variantes. Outre la conception de ces logiques, l’étude de leur sémantique et aussi de leurs capacités de modélisation, une partie de cette thèse a été consacrée à la définition de calculs de preuve, ici de tableaux, pour ces nouvelles logiques ainsi qu’à leurs preuves de correction et de complétude / The design of new logical formalisms is at the heart of several problems in formal methods. Those formalisms must respond to requirements both concerning modelling (they must be able to describe certain systems) and computing (they must provide complete and sound calculus methods). In this context, we look at resource logics, and in particular BI and BBI logics, that deal with the separation and sharing of resources and have led to several separation logics whose applications to software verification have been widely developped recently. We propose in this thesis, starting from BI and BBI logics, to study some modal and epistemic separation logics by focusing on their modelling capacities and their expresiveness, as well as on the new proof calculi for those logics. A first study deals with the modelling of dynamic resource properties through new logic LTBI, which is a temporal separation logic, based on BI logic and temporal modalities. This logic notably offers interesting perspectives in temporal branching modelling, allowing for instance to characterize multi-thread processes. A complementary study concerns the modelling of access by agents to properties under the conditions of posessing some resources, through a new logic ERL, which is an epistemic separation logic, based on BBI logic and epistemic modalities. This logic allows many modellings of access control systems. In order to extend the expressivity of such separation logics, like BBI logic and its variants, a study on the internalization of resources symbols in the logic’s syntax has been developed through the new logics HRL and HBBI (hybrid version of BBI). Internalization allows both the extension of the expressivity of logics and the axiomatisation of BBI logic and some of its variants. In addition to the conception of those logics, the study of their semantics and their modelling capacities, a part of this thesis is dedicated to the definition of proof calculs, here tableaux calculus, for those new logics, as well as their proof of soundness and completeness