146 research outputs found

    Certificates for decision problems in temporal logic using context-based tableaux and sequent calculi.

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    115 p.Esta tesis trata de resolver problemas de Satisfactibilidad y Model Checking, aportando certificados del resultado. En ella, se trabaja con tres lógicas temporales: Propositional Linear Temporal Logic (PLTL), Computation Tree Logic (CTL) y Extended Computation Tree Logic (ECTL). Primero se presenta el trabajo realizado sobre Certified Satisfiability. Ahí se muestra una adaptación del ya existente método dual de tableaux y secuentes basados en contexto para satisfactibilidad de fórmulas PLTL en Negation Normal Form. Se ha trabajado la generación de certificados en el caso en el que las fórmulas son insactisfactibles. Por último, se aporta una prueba de soundness del método. Segundo, se ha optimizado con Sat Solvers el método de Certified Satisfiability para el contexto de Certified Model Checking. Se aportan varios ejemplos de sistemas y propiedades. Tercero, se ha creado un nuevo método dual de tableaux y secuentes basados en contexto para realizar Certified Satisfiability para fórmulas CTL yECTL. Se presenta el método y un algoritmo que genera tanto el modelo en el caso de que las fórmulas son satisfactibles como la prueba en el caso en que no lo sean. Por último, se presenta una implementación del método para CTL y una experimentación comparando el método propuesto con otro método de similares características

    Category Theory in Isabelle/HOL as a Basis for Meta-logical Investigation

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    This paper presents meta-logical investigations based on category theory using the proof assistant Isabelle/HOL. We demonstrate the potential of a free logic based shallow semantic embedding of category theory by providing a formalization of the notion of elementary topoi. Additionally, we formalize symmetrical monoidal closed categories expressing the denotational semantic model of intuitionistic multiplicative linear logic. Next to these meta-logical-investigations, we contribute to building an Isabelle category theory library, with a focus on ease of use in the formalization beyond category theory itself. This work paves the way for future formalizations based on category theory and demonstrates the power of automated reasoning in investigating meta-logical questions.Comment: 15 pages. Preprint of paper accepted for CICM 2023 conferenc

    Through and beyond classicality: analyticity, embeddings, infinity

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    Structural proof theory deals with formal representation of proofs and with the investigation of their properties. This thesis provides an analysis of various non-classical logical systems using proof-theoretic methods. The approach consists in the formulation of analytic calculi for these logics which are then used in order to study their metalogical properties. A specific attention is devoted to studying the connections between classical and non-classical reasoning. In particular, the use of analytic sequent calculi allows one to regain desirable structural properties which are lost in non-classical contexts. In this sense, proof-theoretic versions of embeddings between non-classical logics - both finitary and infinitary - prove to be a useful tool insofar as they build a bridge between different logical regions

    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

    Proof theoretic criteria for logical constancy

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    Logic concerns inference, and some inferences can be distinguished from others by their holding as a matter of logic itself, rather than say empirical factors. These inferences are known as logical consequences and have a special status due to the strong level of confidence they inspire. Given this importance, this dissertation investigates a method of separating the logical from the non-logical. The method used is based on proof theory, and builds on the work of Prawitz, Dummett and Read. Requirements for logicality are developed based on a literature review of common philosophical use of the term, with the key factors being formality, and the absolute generality / topic neutrality of interpretations of logical constants. These requirements are used to generate natural deduction criteria for logical constancy, resulting in the classification of certain predicates, truth functional propositional operators, first order quantifiers, second order quantifiers in sound and complete formal systems using Henkin semantics, and modal operators from the systems K and S5 as logical constants. Semantic tableaux proof systems are also investigated, resulting in the production of semantic tableaux-based criteria for logicality

    Defining Logical Systems via Algebraic Constraints on Proofs

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    We comprehensively present a program of decomposition of proof systems for non-classical logics into proof systems for other logics, especially classical logic, using an algebra of constraints. That is, one recovers a proof system for a target logic by enriching a proof system for another, typically simpler, logic with an algebra of constraints that act as correctness conditions on the latter to capture the former; for example, one may use Boolean algebra to give constraints in a sequent calculus for classical propositional logic to produce a sequent calculus for intuitionistic propositional logic. The idea behind such forms of reduction is to obtain a tool for uniform and modular treatment of proof theory and provide a bridge between semantics logics and their proof theory. The article discusses the theoretical background of the project and provides several illustrations of its work in the field of intuitionistic and modal logics. The results include the following: a uniform treatment of modular and cut-free proof systems for a large class of propositional logics; a general criterion for a novel approach to soundness and completeness of a logic with respect to a model-theoretic semantics; and a case study deriving a model-theoretic semantics from a proof-theoretic specification of a logic.Comment: submitte

    Asking and Answering

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    Questions are everywhere and the ubiquitous activities of asking and answering, as most human activities, are susceptible to failure - at least from time to time. This volume offers several current approaches to the systematic study of questions and the surrounding activities and works toward supporting and improving these activities. The contributors formulate general problems for a formal treatment of questions, investigate specific kinds of questions, compare different frameworks with regard to how they regulate the activities of asking and answering of questions, and situate these activities in a wider framework of cognitive/epistemic discourse. From the perspectives of logic, linguistics, epistemology, and philosophy of language emerges a report on the state of the art of the theory of questions

    CLASS: A Logical Foundation for Typeful Programming with Shared State

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    Software construction depends on imperative state sharing and concurrency, which are naturally present in several application domains and are also exploited to improve the structure and efficiency of computer programs. However, reasoning about concurrency and shared mutable state is hard, error-prone and the source of many programming bugs, such as memory leaks, data corruption, deadlocks and non-termination. In this thesis, we develop CLASS: a core session-based language with a lightweight substructural type system, that results from a principled extension of the propositions-astypes correspondence with second-order classical linear logic. More concretely, CLASS offers support for session-based communication, mutex-protected first-class reference cells, dynamic state sharing, generic polymorphic algorithms, data abstraction and primitive recursion. CLASS expresses and types significant realistic programs, that manipulate memoryefficient linked data structures (linked lists, binary search trees) with support for updates in-place, shareable concurrent ADTs (counters, stacks, functional and imperative queues), resource synchronisation methods (fork-joins, barriers, dining philosophers, generic corecursive protocols). All of these examples are guaranteed to be safe, a result that follows by the logical approach. The linear logical foundations guarantee that well-typed CLASS programs do not go wrong: they never deadlock on communication or reference cell acquisition, do not leak memory and always terminate, even if they share complex data structures protected by synchronisation primitives. Furthermore, since we follow a propositions-as-types approach, we can reason about the behaviour of concurrent stateful processes by algebraic program manipulation. The feasibility of our approach is witnessed by the implementation of a type checker and interpreter for CLASS, which validates and guides the development of many realistic programs. The implementation is available with an open-source license, together with several examples.A construção de software depende de estado partilhado imperativo e concorrência, que estão naturalmente presentes em vários domínios de aplicação e que também são explorados para melhorar o a estrutura e o desempenho dos programas. No entanto, raciocinar sobre concorrência e estado mutável partilhado é difícil e propenso à introdução de erros e muitos bugs de programação, tais como fugas de memória, corrupção de dados, programas bloqueados e programas que não terminam a sua execução. Nesta tese, desenvolvemos CLASS: uma linguagem baseada em sessões, com um sistema de tipos leve e subestrutural, que resulta de uma extensão metodológica da correspondência proposições-como-tipos com a lógica linear clássica de segunda ordem. Mais concretamente, a linguagem CLASS oferece suporte para comunicação baseada em sessões, células de memória protegidas com mutexes de primeira classe, partilha dinâmica de estado, algoritmos polimórficos genéricos, abstração de dados e recursão primitiva. A linguagem CLASS expressa e tipifica programas realistas significativos, que manipulam estruturas de dados ligadas eficientes (listas ligadas, árvores de pesquisa binária) suportando actualização imperativa local, TDAs partilhados e concorrentes (contadores, pilhas, filas funcionais e imperativas), métodos de sincronização e partilha de recursos (bifurcar-juntar, barreiras, jantar de filósofos, protocolos genéricos corecursivos). Todos estes exemplos são seguros, uma garantia que resulta da nossa abordagem lógica. Os fundamentos, baseados na lógica linear, garantem que programas em CLASS bem tipificados não incorrem em erros: nunca bloqueiam, quer na comunicação, quer na aquisição de células de memória, nunca causam fugas de memória e terminam sempre, mesmo que compartilhem estruturas de dados complexas protegidas por primitivas de sincronização. Além disso, uma vez que seguimos uma abordagem de proposições-comotipos, podemos raciocinar sobre o comportamento de processos concorrentes, que usam estado, através de manipulação algébrica. A viabilidade da nossa abordagem é evidenciada pela implementação de um verificador de tipos e interpretador para a linguagem CLASS, que valida e orienta o desenvolvimento de vários programs realistas. A implementação está disponível com uma licença de acesso livre, juntamente com inúmeros exemplos

    Automated Reasoning

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    This volume, LNAI 13385, constitutes the refereed proceedings of the 11th International Joint Conference on Automated Reasoning, IJCAR 2022, held in Haifa, Israel, in August 2022. The 32 full research papers and 9 short papers presented together with two invited talks were carefully reviewed and selected from 85 submissions. The papers focus on the following topics: Satisfiability, SMT Solving,Arithmetic; Calculi and Orderings; Knowledge Representation and Jutsification; Choices, Invariance, Substitutions and Formalization; Modal Logics; Proofs System and Proofs Search; Evolution, Termination and Decision Prolems. This is an open access book
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