63 research outputs found

    Two-Dimensional Tableaux

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    We present two-dimensional tableau systems for the actuality, fixedly, and up-arrow operators. All systems are proved sound and complete with respect to a two-dimensional semantics. In addition, a decision procedure for the actuality logics is discussed

    Modal Hybrid Logic

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    This is an extended version of the lectures given during the 12-th Conference on Applications of Logic in Philosophy and in the Foundations of Mathematics in Szklarska Poręba (7–11 May 2007). It contains a survey of modal hybrid logic, one of the branches of contemporary modal logic. In the first part a variety of hybrid languages and logics is presented with a discussion of expressivity matters. The second part is devoted to thorough exposition of proof methods for hybrid logics. The main point is to show that application of hybrid logics may remarkably improve the situation in modal proof theory

    Tableau methods for formal verification of multi-agent distributed systems

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    A general proof certification framework for modal logic

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    One of the main issues in proof certification is that different theorem provers, even when designed for the same logic, tend to use different proof formalisms and to produce outputs in different formats. The project ProofCert promotes the usage of a common specification language and of a small and trusted kernel in order to check proofs coming from different sources and for different logics. By relying on that idea and by using a classical focused sequent calculus as a kernel, we propose here a general framework for checking modal proofs. We present the implementation of the framework in a prolog-like language and show how it is possible to specialize it in a simple and modular way in order to cover different proof formalisms, such as labeled systems, tableaux, sequent calculi and nested sequent calculi. We illustrate the method for the logic K by providing several examples and discuss how to further extend the approach

    Incremental decision procedures for modal logics with nominals and eventualities

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    This thesis contributes to the study of incremental decision procedures for modal logics with nominals and eventualities. Eventualities are constructs that allow to reason about the reflexive-transitive closure of relations. Eventualities are an essential feature of temporal logics and propositional dynamic logic (PDL). Nominals extend modal logics with the possibility to reason about state equality. Modal logics with nominals are often called hybrid logics. Incremental procedures are procedures that can potentially solve a problem by performing only the reasoning steps needed for the problem in the underlying calculus. We begin by introducing a class of syntactic models called demos and showing how demos can be used for obtaining nonincremental but worst-case optimal decision procedures for extensions of PDL with nominals, converse and difference modalities. We show that in the absence of nominals, such nonincremental procedures can be refined into incremental demo search procedures, obtaining a worst-case optimal decision procedure for modal logic with eventualities. We then develop the first incremental decision procedure for basic hybrid logic with eventualities, which we eventually extend to deal with hybrid PDL. The approach in the thesis suggests a new principled design of modular, incremental decision procedures for expressive modal logics. In particular, it yields the first incremental procedures for modal logics containing both nominals and eventualities.Diese Dissertation untersucht inkrementelle Entscheidungsverfahren für Modallogiken mit Nominalen und Eventualities. Eventualities sind Konstrukte, die erlauben, über den reflexiv-transitiven Abschluss von Relationen zu sprechen. Sie sind ein Schlüsselmerkmal von Temporallogiken und dynamischer Aussagenlogik (PDL). Nominale erweitern Modallogik um die Möglichkeit, über Gleichheit von Zuständen zu sprechen. Modallogik mit Nominalen nennt man Hybridlogik. Inkrementell ist ein Verfahren dann, wenn es ein Problem so lösen kann, dass für die Lösung nur solche Schritte in dem zugrundeliegenden Kalkül gemacht werden, die für das Problem relevant sind. Wir führen zunächst eine Klasse syntaktischer Modelle ein, die wir Demos nennen. Wir nutzen Demos um nichtinkrementelle aber laufzeitoptimale Entscheidungsverfahren für Erweiterungen von PDL zu konstruieren. Wir zeigen, dass im Fall ohne Nominale solche Verfahren durch algorithmische Verfeinerung zu inkrementellen Verfahren ausgebaut werden können. Insbesondere erhalten wir so ein optimales Verfahren für Modallogik mit Eventualities. Anschließend entwickeln wir das erste inkrementelle Verfahren für Hybridlogik mit Eventualities, welches wir schließlich auf hybrides PDL erweitern. Die Dissertation vermittelt einen neuen Ansatz zur Konstruktion modularer, inkrementeller Entscheidungsverfahren für expressive Modallogiken. Insbesondere liefert der Ansatz die ersten inkrementellen Verfahren für Modallogiken mit Nominalen und Eventualities

    Relation-changing modal logics

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    Tesis (Doctor en Cs. de la Computación)--Universidad Nacional de Córdoba, Facultad de Matemática, Astronomía y Física, 2014.En esta tesis investigamos operadores modales dinámicos que pueden cambiar el modelo durante la evaluación de una fórmula. En particular, extendemos el lenguaje modal básico con modalidades que son capaces de invertir, borrar o agregar pares de elementos relacionados. Estudiamos la versión local de los operadores (es decir,la realización de modificaciones desde el punto de evaluación) y la versión global(cambiar arbitrariamente el modelo). Investigamos varias propiedades de los lenguajes introducidos, desde un punto de vista abstracto. En primer lugar, se introduce la semántica formal de los modificadores de modelo, e inmediatamente se introduce una noción de bisimulación. Las bisimulaciones son una herramienta importante para investigar el poder expresivo de los lenguajes introducidos en esta tesis. Se demostró que todas los lenguajes son incomparables entre sí en términos de poder expresivo (a excepción de los dos versiones de swap, aunque conjeturamos que también ́en son incomparables). Continuamos por investigar el comportamiento computacional de este tipo de operadores. En primer lugar, demostramos que el problema de satisfactibilidad para las versiones locales de las lógicas que cambian la relación que investigamos es indecidible. También demostramos que el problema de model checking es PSPACE-completo para las seis lógicas. Finalmente, investigamos model checking fijando el modelo y fijando la fórmula (problemas conocidos como complejidad de fórmula y complejidad del programa, respectivamente). Es posible también definir métodos para comprobar satisfactibilidad que no necesariamente terminan. Introducimos métodos de tableau para las lógicas que cambian las relaciones y demostramos que todos estos métodos son correctos y completos y mostramos algunos aplicaciones. En la última parte de la tesis, se discute un contexto concreto en el que pueden aplicarse las lógicas modales que cambian la relación: Lógicas Dinámicas Epistémicas (DEL, por las siglas en inglés). Definimos una lógica que cambia la relación capaz de codificar DEL, e investigamos su comportamiento computacional.In this thesis we study dynamic modal operators that can change the model during the evaluation of a formula. In particular, we extend the basic modal language with modalities that are able to swap, delete or add pairs of related elements of the domain. We call the resulting logics Relation-Changing Modal Logics. We study local version of the operators (performing modifications from the evaluation point) and global version (changing arbitrarily edges in the model). We investigate several properties of the given languages, from an abstract point of view. First, we introduce the formal semantics of the model modifiers, afterwards we introduce a notion of bisimulation. Bisimulations are an important tool to investigate the expressive power of the languages introduced in this thesis. We show that all the languages are incomparable among them in terms of expressive power (except for the two versions of swap, which we conjecture are also incomparable). We continue by investigating the computational behaviour of this kind of operators. First, we prove that the satisfiability problem for some of the relation-changing modal logics we investigate is undecidable. Then, we prove that the model checking problem is PSpace-complete for the six logics. Finally, we investigate model checking fixing the model and fixing the formula (problems known as formula and program complexity, respectively). We show that it is possible to define complete but non-terminating methods to check satisfiability. We introduce tableau methods for relation-changing modal logics and we prove that all these methods are sound and complete, and we show some applications. In the last part of the thesis, we discuss a concrete context in which we can apply relation-changing modal logics: Dynamic Epistemic Logics (DEL). We motivate the use of the kind of logics that we investigate in this new framework, and we introduce some examples of DEL. Finally, we define a new relation-changing modal logic that embeds DEL and we investigate its computational behaviour.Fil: Fervari, Raúl Alberto. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía, Física; Argentina

    Proof-theoretic Semantics for Intuitionistic Multiplicative Linear Logic

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    This work is the first exploration of proof-theoretic semantics for a substructural logic. It focuses on the base-extension semantics (B-eS) for intuitionistic multiplicative linear logic (IMLL). The starting point is a review of Sandqvist’s B-eS for intuitionistic propositional logic (IPL), for which we propose an alternative treatment of conjunction that takes the form of the generalized elimination rule for the connective. The resulting semantics is shown to be sound and complete. This motivates our main contribution, a B-eS for IMLL , in which the definitions of the logical constants all take the form of their elimination rule and for which soundness and completeness are established

    Lógicas modales con memoria

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    Desde la antigüedad hasta hoy, el campo de la lógica ha ido ganando fuerza y actualmente contribuye activamente en muchas áreas distintas, como filosofía, matemática, lingüística, ciencias de la computación, inteligencia artificial, fabricaci ón de hardware, etc. Cada uno de estos escenarios tiene necesidades espec íficas, que van desde requerimientos muy concretos, como un método de inferencia eficiente, hasta propiedades teóricas más abstractas, como un sistema axiomático elegante. Durante muchos años, los lenguajes clásicos (principalmente la lógica de primer orden) eran la alternativa a utilizar, pero esta gran variedad de aplicaciones hizo que otro tipo de lógicas empezaran a resultar atractivas en muchas situaciones. Supongamos que llega el momento de elegir una lógica para una tarea en particular. ¿Cómo podemos decidir cuál es la más adecuada? ¿Qué propiedades deberíamos buscar? ¿Cómo podemos "medir" una lógica con respecto a otras? Éstas no son preguntas sencillas, y no hay una receta general que uno pueda seguir. En esta tesis vamos a restringir estas cuestiones a una familia de lógicas en particular, y en ese contexto vamos a investigar algunos aspectos teóricos que nos van a ayudar a responder parte de estas inquietudes. Podemos aprender mucho estudiando casos particularmente interesantes, y nuestra contribución se va a desarrollar teniendo esa filosofía en mente. Las lógicas modales proposicionales ofrecen una alternativa a los lenguajes tradicionales. Pueden ser pensadas como un conjunto de herramientas que permiten diseñar lógicas específicamente construidas para una tarea en particular, posibilitando tener un control fino en su expresividad. Más aún, las lógicas modales resultaron tener un buen comportamiento computacional que probó ser bastante robusto frente a extensiones. Estas características, entre otras, ubicaron a las lógicas modales como una alternativa atractiva con respecto a los lenguajes clásicos. En este trabajo vamos a presentar una nueva familia de lógicas modales llamada memory logics. Las lógicas modales tradicionales posibilitan describir estructuras relacionales desde una perspectiva local, ¿pero cuál será el resultado si permitimos que una fórmula cambie la estructura en la que está siendo evaluada? Queremos explorar el efecto de agregar a las lógicas modales clásicas una estructura de almacenamiento explícita, una memoria, que permita modelar comportamiento dinámico a través de operadores que permitan almacenar y recuperar información de la memoria. Naturalmente, dependiendo del tipo de estructura de almacenamiento que utilicemos, y de qué operadores tengamos disponibles, la lógica resultante va a tener distintas propiedades que valen la pena investigar. Esta tesis está organizada de la siguiente manera. En el Capítulo 1 empezamos dando un breve resumen de cómo nacieron las lógicas modales, mostrando las diferentes perspectivas con las que históricamente se miró a la lógica modal. Luego presentamos formalmente a la lógica modal básica, y a un conjunto extendido de operadores que permiten apreciar el "estilo" modal de algunos lenguajes más ricos. Este capítulo ánaliza con un primer bosquejo de las memory logics, mostrando cómo pueden ayudar a modelar la noción de estado cuando dejamos a un conjunto como estructura de almacenamiento. El capítulo 2 está dedicado a presentar a las memory logics con detalle. En dicho capítulo mostramos algunos ejemplos que pueden ser descriptos agregando un conjunto a las estructuras relacionales estándar, junto con los operadores usuales sobre conjuntos que permiten agregar elementos y verificar pertenencia. A continuación mostramos otros operadores sobre conjuntos que pueden ser considerados, y discutimos la posibilidad de agregar restricciones a la interacción entre los operadores que trabajan sobre la memoria y los operadores modales. Estas restricciones pueden ser pensadas como una manera de lograr un control más úno en la expresividad. Dado que realizamos cambios en las lógicas modales clásicas, estamos interesados en analizar el impacto que esos cambios causaron en las lógicas resultantes. Por lo tanto, el resto del capítulo presenta un conjunto de herramientas a través de las cuales analizamos esta nueva familia de lógicas. Este conjunto de herramientas puede verse como un esquema que organiza el resto de la tesis, y que permite analizar a las memory logics en téerminos de expresividad, complejidad, interpolación y teoría de prueba. El resto de los capítulos investigan cada uno de estos aspectos en detalle. En los Capítulos 3 y 4 exploramos el poder expresivo de varias memory logics y estudiamos para cada fragmento la decidibilidad del problema de determinar la satisfactibilidad de una fórmula. En los casos decidibles, determinamos la complejidad computacional de cada uno. Analizamos el impacto de los diferentes operadores, su interacción, y también estudiamos otras estructuras de almacenamiento, como la pila. Luego, en el Capítulo 5, analizamos las propiedades de interpolación de Craig y definibilidad de Beth para algunas memory logics. También nos interesa estudiar a las memory logics desde la perspectiva de la teoría de prueba. En los Capítulos 6 y 7 nos volcamos al estudio de axiomatizaciones á la Hilbert y sistemas de tableau, y caracterizamos varios fragmentos usando principalmente técnicas utilizadas en lógicas híbridas. En el Capítulo 8 presentamos nuestras conclusiones, mencionamos algunos problemas abiertos y futuras direcciones de investigación.From ancient times to the present day, the field of logic has gained significant strength and now it actively contributes to many different areas, such as philosophy, mathematics, linguistic, computer science, artificial intelligence, hardware manufacture, etc. Each of these scenarios has specific needs, that range from very concrete requirements, like an efficient inference method, to more abstract theoretical properties, like a neat axiomatic system. Given this wide diversity of uses, a motley collection of formal languages has been developed. For many years, classical languages (mainly classical first order logic) were the alternative, but this assortment of applications made other types of logics also attractive in many situations. Imagine that the time for choosing a logic for some specific task arrives. How can we decide which is the one that fits best? Which properties should we look for? How can we "measure" a logic with respect to others? These are not easy questions, and there is not a general recipe one can follow. In this thesis we are just going to restrict these questions to a particular family of logics, and in that context we will investigate theoretical aspects that help to answer some of these concerns. Much can be discovered by carefully analyzing appealing cases, and our contribution will be developed having that philosophy in mind. Propositional modal logics offer an alternative to traditional languages. They can be regarded as a set of tools that allow to design logics specially tailored for specific tasks, having a fine-grained control on their expressivity. Additionally, modal logics turned out to have a good computational behavior, which proved to be quite robust under extensions. These characteristics, among others, placed modal logics as an attractive alternative to classical languages. In this dissertation we are going to present a new family of modal logics called memory logics. Traditional modal logics enable us to describe relational structures from a local perspective. But what about changing the structure? We want to explore the addition of an explicit storage structure to modal logics, a memory, that allows to model dynamic behavior through explicit memory operators. These operators store or retrieve information to and from the memory. Naturally, depending on which type of storage structure we want, and which memory operators are available, the resulting logic will enjoy different properties that are worth investigating. The thesis is organized as follows. In Chapter 1 we start by giving a brief recap of how modal logic was born, showing the different historical perspectives used to look at modal logic. Then we formally present the basic modal logic and a set of extended operators that helps grasp the modal "flavor" of some richer languages. We finish this chapter by giving a first glance of memory logics, and showing how they can help to model state when we choose to use a set as storage structure. Chapter 2 is devoted to present memory logics in detail. We show some examples that can be described by adding a set to standard relational structures, and the usual set operators to add elements and test membership. We then show some other memory operators that can be considered, and we discuss the possibility of adding constraints to the interplay between memory and modal operators. These constraints can be regarded as a way to have a finer-grained control on the logic expressivity. Since we have made changes to classical modal logics, we are interested in analyzing the impact those changes cause in the resulting logics. Therefore, the rest of this chapter presents a basic logic toolkit through which we can analyze this new family of logics. This toolkit can be seen as an outline that organizes the rest of the thesis and that allows to analyze memory logics in terms of expressivity, complexity, interpolation and proof theory. The rest of the chapters investigate each of these aspects in detail. In Chapters 3 and 4 we explore the expressive power of several memory logics and we study the decidability of their satisfiability problem. In the decidable cases, we determine their computational complexity. We analyze the impact of the different memory operators we consider, and how they interact. We also study other memory containers, such as a stack. Then, in Chapter 5, we analyze Craig interpolation and Beth definability for some memory logic fragments. We also study memory logics from a proof-theoretic perspective. In Chapter 6 and 7 we turn to Hilbert-style axiomatizations and tableau systems, and we characterize several fragments of the memory logic family mostly using techniques borrowed from hybrid logics. We close in Chapter 8 with some concluding remarks, open problems and directions for further research.Fil:Mera, Sergio Fernando. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina

    Syntactic approaches to negative results in process algebras and modal logics

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    Concurrency as a phenomenon is observed in most of the current computer science trends. However the inherent complexity of analyzing the behavior of such a system is incremented due to the many different models of concurrency, the variety of applications and architectures, as well as the wide spectrum of specification languages and demanded correctness criteria. For the scope of this thesis we focus on state based models of concurrent computation, and on modal logics as specification languages. First we study syntactically the process algebras that describe several different concurrent behaviors, by analyzing their equational theories. Here, we use well-established techniques from the equational logic of processes to older and newer setups, and then transition to the use of more general and novel methods for the syntactical analysis of models of concurrent programs and specification languages. Our main contributions are several positive and negative axiomatizability results over various process algebraic languages and equivalences, along with some complexity results over the satisfiability of multi-agent modal logic with recursion, as a specification language.Samhliða sem fyrirbæri sést í flestum núverandi tölvunarfræði stefnur. Hins vegar er eðlislægt flókið að greina hegðun slíks kerfis- tem er aukið vegna margra mismunandi gerða samhliða, fjölbreytileikans af forritum og arkitektúr, svo og breitt svið forskrifta mælikvarða og kröfðust réttmætisviðmiða. Fyrir umfang þessarar ritgerðar leggjum við áherslu á ástandsbundin líkön af samhliða útreikningum og á formlegum rökfræði sem forskrift tungumálum. Fyrst skoðum við setningafræðilega ferlialgebrurnar sem lýsa nokkrum mismunandi samhliða hegðun, með því að greina jöfnukenningar þeirra. Hér notum við rótgróin tækni mynda jöfnunarrökfræði ferla til eldri og nýrri uppsetningar, og síðan umskipti yfir í notkun almennari og nýrra aðferða fyrir setningafræðileg greining á líkönum samhliða forrita og forskriftartungumála. Helstu framlög okkar eru nokkrar jákvæðar og neikvæðar niðurstöður um axiomatizability yfir ýmis ferli algebrumál og jafngildi, ásamt nokkrum samSveigjanleiki leiðir af því að fullnægjanleiki fjölþátta formrökfræði með endurkomu, sem a forskrift tungumál.RANNIS: `Open Problems in the Equational Logic of Processes’ (OPEL) (grant No 196050-051) Reykjavik University research fund: `Runtime and Equational Verification of Concurrent Programs' (ReVoCoP) (grant No 222021
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