Programmers don't just have to write programs, they are have to reason about them. Programming languages aren't just tools for instructing computers what to do, they are tools for reasoning. And, it isn't just programmers who reason about programs: compilers and other tools reason similarly as they transform from one language into another one, or as they optimize an inefficient program into a better one. Languages, both surface languages and intermediate ones, need therefore to be both efficiently implementable and to support effective logical reasoning. However, these goals often seem to be in conflict. This dissertation studies programming language calculi inspired by the Curry-Howard correspondence, relating programming languages to proof systems. Our focus is on calculi corresponding logically to classical sequent calculus and connected computationally to abstract machines. We prove that these calculi have desirable properties to help bridge the gap between reasoning and implementation. Firstly, we explore a persistent conflict between extensionality and effects for lazy functional programs that manifests in a loss of confluence. Building on prior work, we develop a new rewriting theory for lazy functions and control which we first prove corresponds to the desired equational theory and then prove, by way of reductions into a smaller system, to be confluent. Next, we turn to the inconsistency between weak-head normalization and extensionality. Using ideas from our study of confluence, we develop a new operational semantics and series of abstract machines for head reduction which show us how to retain weak-head reduction's ease of implementation. After demonstrating the limitations of the above approach for call-by-value or types other than functions, we turn to typed calculi, showing how a type system can be used not only for mixing different kinds of data, but also different evaluation strategies in a single program. Building on variations of the reducibility candidates method such as biorthogonality and symmetric candidates, we present a uniform proof of strong normalization for our mixed-strategy system which works so long as all the strategies used satisfy criteria we isolate. This dissertation includes previously published co-authored material.

I study sequent calculus of combined logics in this thesis. Two specific logics are looked at-Logic BI that combines intuitionistic logic and multiplicative intuitionistic linear logic and Logic BBI that combines classical logic and multiplicative linear logic. A proof-theoretical study into logical combinations themsel ves then follows. To consolidate intuition about what this thesis is all about, let us suppose that we know about two different logics, Logic A developed for reasoning about Purpose A and Logic B developed for reasoning about Purpose B. Logic A serves Purpose A very well, but not Purpose B. Logic B serves Purpose B very well but not Purpose A. We wish to fulfill both Purpose A and Purpose B, but presently we can only afford to let one logic guide through our reasoning. What shall we do? One option is to be content with having Logic A with which we handle Purpose A efficiently and Purpose B rather inefficiently. Another option is to choose Logic B instead. But there is yet another option: we combine Logic A and Logic B to derive a new logic Logic C which is still one logic but which serves both Purpose A and Purpose B efficiently. The combined logic is synthetic of the strengths in more basic logics (Logic A and Logic B). As it nicely takes care of our requirements, it may be the best choice among all that have been so far considered. Yet this is not the end of the story. Depending on the manner Logic A and Logic B combine, Logic C may have extensions serving more purposes than just Purpose A and Purpose B. Ensuing is the following problem: we know about Logic A and Logic B, but we may not know about combined logics of the base logics. To understand the combined logics, we need to understand the extensions in which base logics interact each other. Analysis on the interesting parts tends to be non-trivial, however. The mentioned two specific combined logics BI and BBI do not make an exception, for which proof-theoretical development has been particularly slow. It has remained in obscurity how to properly handle base-logic interactions of the combined logics as appearing syntactically. As one objective of this thesis, I provide analysis on the syntactic phenomena of the BI and BBI base-logic interactions within sequent calculus, to augment the knowledge. For BI, I deliver, through appropriate methodologies to reason about the syntactic phenomena of the base-logic interactions, the first BI sequent calculus free of any structural rules. Given its positive consequence to efficient proof searches, this is a significant step forward in further maturity of BI proof theory. Based on the calculus, I prove decidability of a fragment of BI purely syntactically. For BBI which is closely connected to application via separation logic, I develop adequate sequent calculus conventions and consider the implication of the underlying semantics onto syntax. Sound BBI sequent calculi result with a closer syntax-semantics correspondence than previously envisaged. From them, adaptation to separation logic is also considered. To promote the knowledge of combined logics in general within computer science, it is also important that we be able to study logical combinations themselves. Towards this direction of generalisation, I present the concept of phased sequent calculus - sequent calculus which physically separates base logics, and in which a specific manner of logical combination to take place between them can be actually developed and analysed. For a demonstration, the said decidable BI fragment is formulated in phased sequent calculus, and the sense of logical combination in effect is analysed. A decision procedure is presented for the fragment.

Orientador: Marcelo Esteban Coniglio / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Filosofia e Ciencias Humanas / Made available in DSpace on 2018-08-12T04:24:47Z (GMT). No. of bitstreams: 1 Testa_RafaelRodrigues_M.pdf: 526495 bytes, checksum: 51a1bb2b8ec08b5fab8947cdac7f3ae4 (MD5) Previous issue date: 2008 / Resumo: Nosso objetivo neste trabalho é apresentar uma proposta de solução a paradoxos relacionados à lógica deôntica presentes na literatura, reunidos sob o que é chamado de dilemas deônticos - situações nas quais duas obrigações conflitantes estão presentes num mesmo sistema normativo. Situações deste tipo, quando formalizadas (em SDL - standard deontic logic - ou em outras lógicas relacionadas), levam a uma inconsistência. Nossa proposta baseia-se em relações de preferência que geram uma ferramenta de escolha dentre as duas soluções normativas conflitantes, o que evita a inconsistência e permite o pleno cumprimento do sistema. Justificativas filosóficas são fornecidas as ferramentas lógicas, bem como as suas implicações. / Abstract: The main purpouse of this dissertation is the proposal of a solution to some paradoxes related to deontic logic presented in the literature, also known as deontic dilemmas - situations in which two conflicting obligations are present in the same normative system. Such situations, when formalized (in SDL - standard deontic logic - or in other related logic), lead to inconsistency. Our proposal is based on preference relations that generate a tool of choice between the two conflicting normative solutions, which avoids the inconsistency and allows the full implementation of the system. Philosophical justifications are given for the logical tools as well as for their implications. / Mestrado / Filosofia / Mestre em Filosofia

Orientador: Itala Maria Loffredo D'Ottaviano / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Filosofia e Ciências Humanas / Made available in DSpace on 2018-08-19T03:32:32Z (GMT). No. of bitstreams: 1 Almeida_DanteCardosoPintode_M.pdf: 571204 bytes, checksum: 5d6a5ced0609a9ec16f84bc5badb751d (MD5) Previous issue date: 2011 / Resumo: Esta dissertação tem como objetivo a análise de um resultado em lógica aléticoepistêmica, divulgado por Frederic Fitch em 1963, conhecido como Paradoxo da Cognoscibilidade. Segundo este resultado, se todas verdades podem ser conhecidas, então todas verdades são conhecidas. Isto sugere que há alguma verdade impossível de ser conhecida. Descrevemos, nesta dissertação, a lógica modal alética e a epistêmica, que consistem em recursos formais requeridos para a análise do Paradoxo. Também esclarecemos o papel deste no debate filosófico entre as correntes de pensamento realistas e antirealistas. Apontamos e analisamos duas propostas de solução do Paradoxo mais discutidas na literatura. Como principal ojetivo desta dissertação, investigamos o Paradoxo da Cognoscibilidade em sistemas multiagentes. Demonstramos que, apesar de em tais sistemas o Paradoxo ser minimizado, ele ainda não é completamente resolvido. Por fim, também apresentamos várias formas de obter a contraparte doxástica do Resultado, conhecida como Paradoxo da Credibilidade / Abstract: This text studies a result in epistemic-alethic logic, published by Frederic Fitch in 1963, known as Knowability Paradox. According to this result, if all truths are knowable, then all truths are known. This suggests there are unknowable truths. We describe alethic and epistemic modal logics, which are formal resources required in order to study the paradox. Also, we examine its role in the philosophical debate between realists and anti-realists. We point out and analize two attempts to solve the Paradox. The main aim of this text is to explore the Knowability Paradox in multi-agents systems. We shoe that, although in these systems the Paradox is weaker, it's not entirely solved. We also show many ways to derive the doxastic counterpart of the result, known as Belivability Paradox / Mestrado / Filosofia / Mestre em Filosofia

An interesting problem in proof theory is to find representations of proof that do not distinguish between proofs that are ‘morally’ the same. For many logics, the presentation of proofs in a traditional formalism, such as Gentzen’s sequent calculus, introduces artificial syntactic structure called ‘bureaucracy’; e.g., an arbitrary ordering of freely permutable inferences. A proof system that is free of bureaucracy is called canonical for a logic. In this dissertation two canonical proof systems are presented, for two logics: a notion of proof nets for additive linear logic with units, and ‘classical proof forests’, a graphical formalism for first-order classical logic. Additive linear logic (or sum–product logic) is the fragment of linear logic consisting of linear implication between formulae constructed only from atomic formulae and the additive connectives and units. Up to an equational theory over proofs, the logic describes categories in which finite products and coproducts occur freely. A notion of proof nets for additive linear logic is presented, providing canonical graphical representations of the categorical morphisms and constituting a tractable decision procedure for this equational theory. From existing proof nets for additive linear logic without units by Hughes and Van Glabbeek (modified to include the units naively), canonical proof nets are obtained by a simple graph rewriting algorithm called saturation. Main technical contributions are the substantial correctness proof of the saturation algorithm, and a correctness criterion for saturated nets. Classical proof forests are a canonical, graphical proof formalism for first-order classical logic. Related to Herbrand’s Theorem and backtracking games in the style of Coquand, the forests assign witnessing information to quantifiers in a structurally minimal way, reducing a first-order sentence to a decidable propositional one. A similar formalism ‘expansion tree proofs’ was presented by Miller, but not given a method of composition. The present treatment adds a notion of cut, and investigates the possibility of composing forests via cut-elimination. Cut-reduction steps take the form of a rewrite relation that arises from the structure of the forests in a natural way. Yet reductions are intricate, and initially not well-behaved: from perfectly ordinary cuts, reduction may reach unnaturally configured cuts that may not be reduced. Cutelimination is shown using a modified version of the rewrite relation, inspired by the game-theoretic interpretation of the forests, for which weak normalisation is shown, and strong normalisation is conjectured. In addition, by a more intricate argument, weak normalisation is also shown for the original reduction relation.

The thesis introduces a logic which combines intuitionistic implication with de Morgan negation in a way which conservatively extends both classical and intuitionistic logic. This logic is the intuitionistic counterpart of the four-valued Belnap-Dunn logic. In relation to this logic, we study de Morgan algebras and their expansions, in particular their expansion with a constant representing inconsistency. We prove a duality for such algebras extending the Priestley duality. We also introduce a weak notion of modal algebra and prove a duality for such algebras. We then define analytic sequent calculi for various logics of de Morgan negation. Powered by TCPDF (www.tcpdf.org)

[pt] A história recente da Lógica Matemática foi marcada por alguns conflitos entre diferentes correntes filosóficas, cada uma buscando contextualizar a atividade matemática a partir de seu próprio prisma analítico e, por meio disso, tentando conquistar para si mesma o pódio fundacional das Ciências Formais Tais discussões, perenes o bastante para ainda quedarem sem solução, foram fortemente impactadas pela apropriação semântica de alguns resultados técnicos obtidos no campo da teoria da prova, o que redefiniu a relação existente entre as abordagens clássica e intuicionista na matemática. Neste contexto, a presente dissertação tem por finalidade realizar uma descrição da emergente literatura de propostas integrativas entre diferentes sistemas lógicos e matemáticos (apelidadas por Dag Prawitz de ecumenismo lógico), além de investigar alguns impactos que mudanças formais poderiam ocasionar nas concepções filosóficas de algumas teorias matemáticas. No capítulo introdutório, traçamos um panorama geral desta nova proposta ecumênica e analisamos com mais atenção o conflito entre as lógicas Clássica, Intuicionista e Minimal, considerado por muitos como um dos mais influentes na literatura contemporânea. No segundo capítulo, este trabalho fornece uma contribuição original para a literatura ao criar uma nova abordagem ecumênica, além de provar algumas equivalências no interior do sistema Clássico-Intuicionista recentemente criado por Prawitz e compará-lo com uma lógica que criamos usando esta nova abordagem. No terceiro capítulo, contribuímos tanto com a abordagem tradicional quanto com nossa abordagem original ao criar e comparar dua lógicas ecumênicas Minimal-Intuicionistas. Por fim, realizamos uma breve revisão do tímido estado da arte no último capítulo, oferecendo um novo esquema conceitual de interpretação dos sistemas ecumênicos e comentando alguns aspectos promissores do campo, que poderão vir a ser melhor trabalhados no futuro. / [en] The recent history of Mathematical Logic was marked by some conficts between different philosophical positions, each trying to contextualize mathematical activity from its own analytical viewpoint and, with this, trying to conquer the foundational podium of the formal sciences for itself. Such discussions, lasting enough to remain without a solution, were strongly impacted by the semantical appropriation of some technical results obtained in the field of proof theory, which redefined the relation between the classical and intuitionistic approaches to mathematics. In this context, the present dissertation aims to describe the emergent literature about the integration of different logical and mathematical systems (nicknamed logical ecumenism by Dag Prawitz), in addition to investigating some impacts that those formal changes could have on the philosophical conceptions of some mathematical theories. In the introductory chapter, we have outlined a general overview of this new ecumenical proposal and analysed in greater depht the conflicts between Classical, Intuitionistic and Minimal logic, considered by many as one of the most influent on the contemporary literature. In the second chapter, this work provides an original contribution to the literature by creating a new ecumenical approach, in addition to proving some equivalencies within Prawitz s recently created Classical-Intuitionist system, and compares it with the logical system we have created using this new approach. In the third chapter, we contribute both to the traditional approach and our original approach by creating and comparing two Minimal-Intuitionist ecumenical logics. Finally, we briefly review the timid state of the art in the last chapter, offering a new conceptual framework for interpreting ecumenical systems, as well as commenting on some promising aspects of the field, which may be better analyzed in the future.

Le thème général de cette thèse est l’exploitation de l’interaction entre la sémantique dénotationnelle et la syntaxe. Des sémantiques satisfaisantes ont été découvertes pour les preuves en logique intuitionniste et linéaire, mais dans le cas de la logique classique, la solution du problème est connue pour être particulièrement difficile. Ce travail commence par l’étude d’une interprétation concrète des preuves classiques dans la catégorie des ensembles ordonnés et bimodules, qui mène à l’extraction d’invariants significatifs. Suit une généralisation de cette sémantique concrète, soit l’interprétation des preuves classiques dans une catégorie compacte fermée où chaque objet est doté d’une structure d’algèbre de Frobenius. Ceci nous mène à une définition de réseaux de démonstrations pour la logique classique. Le concept de correction, l’élimination des coupures et le problème de la “full completeness” sont abordés au moyen d’un enrichissement naturel dans les ordres sur la catégorie de Frobenius, produisant une catégorie pour l'élimination des coupures et un concept de ressources pour la logique classique. Revenant sur notre première sémantique concrète, nous montrons que nous avons une représentation fidèle de la catégorie de Frobenius dans la catégorie des ensembles ordonnés et bimodules. / The general theme of this thesis is the exploitation of the fruitful interaction between denotational semantics and syntax. Satisfying semantics have been discovered for proofs in intuitionistic and certain linear logics, but for the classical case, solving the problem is notoriously difficult.This work begins with investigations of concrete interpretations of classical proofs in the category of posets and bimodules, resulting in the definition of meaningful invariants of proofs. Then, generalizing this concrete semantics, classical proofs are interpreted in a free symmetric compact closed category where each object is endowed with the structure of a Frobenius algebra. The generalization paves a way for a theory of proof nets for classical proofs. Correctness, cut elimination and the issue of full completeness are addressed through natural order enrichments defined on the Frobenius category, yielding a category with cut elimination and a concept of resources in classical logic. Revisiting our initial concrete semantics, we show we have a faithful representation of the Frobenius category in the category of posets and bimodules.

Nesta tese apresentamos o projeto e a implementação do KEMS, um provador de teoremas multi-estratégia baseado no método de tablôs KE. Um provador de teoremas multi-estratégia é um provador de teoremas onde podemos variar as estratégias utilizadas sem modificar o núcleo da implementação. Além de multi-estratégia, o KEMS é capaz de provar teoremas em três sistemas lógicos: lógica clássica proposicional, mbC e mCi. Listamos abaixo algumas das contribuições deste trabalho: * um sistema KE para mbC que é analítico, correto e completo; * um sistema KE para mCi que é correto e completo; * um provador de teoremas multi-estratégia com as seguintes características: - aceita problemas em três sistemas lógicos: lógica clássica proposicional, mbC e mCi; - tem seis estratégias implementadas para lógica clássica proposicional, duas para mbC e duas para mCi; - tem treze ordenadores que são usados em conjunto com as estratégias; - implementa regras simplificadoras para lógica clássica proposicional; - possui uma interface gráfica que permite a visualização de provas; - é de código aberto e está disponível na Internet em http://kems.iv.fapesp.br; * benchmarks obtidos através da comparação das estratégias para lógica clássica proposicional resolvendo várias famílias de problemas; - sete famílias de problemas para avaliar provadores de teoremas paraconsistentes; * os primeiros benchmarks para as famílias de problemas para avaliar provadores de teoremas paraconsistentes. / In this thesis we present the design and implementation of KEMS, a multi-strategy theorem prover based on the KE tableau inference system. A multi-strategy theorem prover is a theorem prover where we can vary the strategy without modifying the core of the implementation. Besides being multi-strategy, KEMS is capable of proving theorems in three logical systems: classical propositional logic, mbC and mCi. We list below some of the contributions of this work: * an analytic, correct and complete KE system for mbC; * a correct and complete KE system for mCi; * a multi-strategy prover with the following characteristics: - accepts problems in three logical systems: classical propositional logic, mbC and mCi; - has 6 implemented strategies for classical propositional logic, 2 for mbC and 2 for mCi; - has 13 sorters to be used alongside with the strategies; - implements simplification rules of classical propositional logic; - provides a proof viewer with a graphical user interface; - it is open source and available on the internet at http://kems.iv.fapesp.br; * benchmark results obtained by KEMS comparing its classical propositional logic strategies with several problem families; * seven problem families designed to evaluate provers for logics of formal inconsistency; * the first benchmark results for the problem families designed to evaluate provers for logics of formal inconsistency.

In this thesis we see deductive systems for classical propositional and predicate logic which use deep inference, i.e. inference rules apply arbitrarily deep inside formulas, and a certain symmetry, which provides an involution on derivations. Like sequent systems, they have a cut rule which is admissible. Unlike sequent systems, they enjoy various new interesting properties. Not only the identity axiom, but also cut, weakening and even contraction are reducible to atomic form. This leads to inference rules that are local, meaning that the effort of applying them is bounded, and finitary, meaning that, given a conclusion, there is only a finite number of premises to choose from. The systems also enjoy new normal forms for derivations and, in the propositional case, a cut elimination procedure that is drastically simpler than the ones for sequent systems.