Polarities & Focussing: a journey from Realisability to Automated Reasoning

This dissertation explores the roles of polarities and focussing in various aspects of Computational Logic.These concepts play a key role in the the interpretation of proofs as programs, a.k.a. the Curry-Howard correspondence, in the context of classical logic. Arising from linear logic, they allow the construction of meaningful semantics for cut-elimination in classical logic, some of which relate to the Call-by-Name and Call-by-Value disciplines of functional programming. The first part of this dissertation provides an introduction to these interpretations, highlighting the roles of polarities and focussing. For instance: proofs of positive formulae provide structured data, while proofs of negative formulae consume such data; focussing allows the description of the interaction between the two kinds of proofs as pure pattern-matching. This idea is pushed further in the second part of this dissertation, and connected to realisability semantics, where the structured data is interpreted algebraically, and the consumption of such data is modelled with the use of an orthogonality relation. Most of this part has been proved in the Coq proof assistant.Polarities and focussing were also introduced with applications to logic programming in mind, where computation is proof-search. In the third part of this dissertation, we push this idea further by exploring the roles that these concepts can play in other applications of proof-search, such as theorem proving and more particularly automated reasoning. We use these concepts to describe the main algorithm of SAT-solvers and SMT-solvers: DPLL. We then describe the implementation of a proof-search engine called Psyche. Its architecture, based on the concept of focussing, offers a platform where smart techniques from automated reasoning (or a user interface) can safely and trustworthily be implemented via the use of an API

Synthesis of Event-Based Controllers for Software Engineering

Behavioural modelling has been widely used to aid in the design of concurrent systems. Behaviour models have shown to be useful to uncover design errors in early stages of the development process. However, building correct behaviour models is costly and requires significant experience. Controller synthesis offers a way to build models that are correct by construction. Existing software engineering techniques for synthesising controllers have various limitations. Such limitations can be seen as restrictions in the expressiveness of the controller goals and environment model, or in the relation between the controllable and monitored actions. The main aim of this thesis is the development of novel techniques overcoming known limitations of previous approaches and methodological guidelines for synthesising useful controllers. This thesis establishes the framework for controller synthesis techniques that support event-based models, expressive goal specifications, distinguish controllable from monitored actions and guarantee achievement of the desired goals. Together with these techniques, methodological guidelines are proposed to help in building more accurate descriptions of the environment and more effective controllers. In addition, this thesis presents a tool that implements the proposed techniques. Evaluation of the techniques has been conducted using the tool to model known case studies from the literature, showing that by allowing more expressive controller goals and environment models, and explicitly distinguishing controllable and monitored actions such case studies can be more accurately modelled and solutions guaranteeing satisfaction of the goals can be achieved

Model and Proof Theory of Constructive ALC, Constructive Description Logics

Description logics (DLs) represent a widely studied logical formalism with a significant impact in the field of knowledge representation and the Semantic Web. However, they are equipped with a classical descriptive semantics that is characterised by a platonic notion of truth, being insufficiently expressive to deal with evolving and incomplete information, as from data streams or ongoing processes. Such partially determined and incomplete knowledge can be expressed by relying on a constructive semantics. This thesis investigates the model and proof theory of a constructive variant of the basic description logic ALC, called cALC. The semantic dimension of constructive DLs is investigated by replacing the classical binary truth interpretation of ALC with a constructive notion of truth. This semantic characterisation is crucial to represent applications with partial information adequately, and to achieve both consistency under abstraction as well as robustness under refinement, and on the other hand is compatible with the Curry-Howard isomorphism in order to form the cornerstone for a DL-based type theory. The proof theory of cALC is investigated by giving a sound and complete Hilbert-style axiomatisation, a Gentzen-style sequent calculus and a labelled tableau calculus showing finite model property and decidability. Moreover, cALC can be strengthened towards normal intuitionistic modal logics and classical ALC in terms of sound and complete extensions and hereby forms a starting point for the systematic investigation of a constructive correspondence theory.Beschreibungslogiken (BLen) stellen einen vieluntersuchten logischen Formalismus dar, der den Bereich der Wissensrepräsentation und das Semantic Web signifikant geprägt hat. Allerdings basieren BLen meist auf einer klassischen deskriptiven Semantik, die gekennzeichnet ist durch einen idealisierten Wahrheitsbegriff nach Platons Ideenlehre, weshalb diese unzureichend ausdrucksstark sind, um in Entwicklung befindliches und unvollständiges Wissen zu repräsentieren, wie es beispielsweise durch Datenströme oder fortlaufende Prozesse generiert wird. Derartiges partiell festgelegtes und unvollständiges Wissen lässt sich auf der Basis einer konstruktiven Semantik ausdrücken. Diese Arbeit untersucht die Model- und Beweistheorie einer konstruktiven Variante der Basis-BL ALC, die im Folgenden als cALC bezeichnet wird. Die Semantik dieser konstruktiven Beschreibungslogik resultiert daraus, die traditionelle zweiwertige Interpretation logischer Aussagen des Systems ALC durch einen konstruktiven Wahrheitsbegriff zu ersetzen. Eine derartige Interpretation ist die Voraussetzung dafür, um einerseits Anwendungen mit partiellem Wissen angemessen zu repräsentieren, und sowohl die Konsistenz logischer Aussagen unter Abstraktion als auch ihre Robustheit unter Verfeinerung zu gewährleisten, und andererseits um den Grundstein für eine Beschreibungslogik-basierte Typentheorie gemäß dem Curry-Howard Isomorphismus zu legen. Die Ergebnisse der Untersuchung der Beweistheorie von cALC umfassen eine vollständige und korrekte Hilbert Axiomatisierung, einen Gentzen Sequenzenkalkül, und ein semantisches Tableaukalkül, sowie Beweise zur endlichen Modelleigenschaft und Entscheidbarkeit. Darüber hinaus kann cALC zu normaler intuitionistischer Modallogik und klassischem ALC durch vollständige und korrekte Erweiterungen ausgebaut werden, und bildet damit einen Startpunkt für die systematische Untersuchung einer konstruktiven Korrespondenztheorie

Nominal Models of Linear Logic

PhD thesisMore than 30 years after the discovery of linear logic, a simple fully-complete model has still not been established. As of today, models of logics with type variables rely on di-natural transformations, with the intuition that a proof should behave uniformly at variable types. Consequently, the interpretations of the proofs are not concrete. The main goal of this thesis was to shift from a 2-categorical setting to a first-order category. We model each literal by a pool of resources of a certain type, that we encode thanks to sorted names. Based on this, we revisit a range of categorical constructions, leading to nominal relational models of linear logic. As these fail to prove fully-complete, we revisit the fully-complete game-model of linear logic established by Melliès. We give a nominal account of concurrent game semantics, with an emphasis on names as resources. Based on them, we present fully complete models of multiplicative additive tensorial, and then linear logics. This model extends the previous result by adding atomic variables, although names do not play a crucial role in this result. On the other hand, it provides a nominal structure that allows for a nominal relationship between the Böhm trees of the linear lambda-terms and the plays of the strategies. However, this full-completeness result for linear logic rests on a quotient. Therefore, in the final chapter, we revisit the concurrent operators model which was first developed by Abramsky and Melliès. In our new model, the axiomatic structure is encoded through nominal techniques and strengthened in such a way that full completeness still holds for MLL. Our model does not depend on any 2-categorical argument or quotient. Furthermore, we show that once enriched with a hypercoherent structure, we get a static fully complete model of MALL

Art-ificial: The Philosophy of AI Art

This thesis aims to contribute to a novel area of philosophical work: the philosophy of AI art. AI art is proliferating online and increasingly in the world of art. The growing presence of works made by (or with) artificial intelligence has led to a clamour of questions such as 'Is AI art really art?' and 'can AI be truly creative?'. As yet, these questions have barely been tackled in the philosophical literature, especially in aesthetics. This thesis aims to address this gap. This thesis starts by establishing what we mean by 'AI art' by examining examples of AI works and the technological underpinnings of these systems. Existing work on the topic of AI art is explicated. In particular, Mark Coeckelbergh's three questions on AI art scaffold the first three chapters of the thesis: 'can machines create art?', 'can machines create art?' and 'can machines create art?' Chapter 1 aims to answer the question of whether AI can make art through the evaluation of AI works against three definitions of art: the institutional account, the historical account, and the cluster account. Chapter 2 focusses on the question of whether AI can be creative. Three accounts of creativity are utilised: a Darwinian theory of creativity, Margaret Boden's account of creativity, and Berys Gaut's agential account of creativity. It is argued that some AI systems can meet the requirements of each of these, aside from Gaut's necessary criterion of agency. After chapters 1 and 2, questions about the limitations of AI systems in meeting the requirements of different accounts of art and creativity remain; chapter 3 aims to address some of these. The possibility of AI (extended) mind is investigated, followed by AI embodiment. Finally, an argument for the possibility of AI agency is put forward. This minimal account of agency allows for the possibility of AI creativity under the agential account. The latter part of this thesis begins with chapter 4, which examines the possibility that AI systems will not share aesthetic or artistic values with humans, and whether this is cause for concern. Finally, Chapter 5 examines two qualities of AI images: weirdness and convincingness, showing that AI art can offer interesting aesthetic qualities worthy of investigation. Through this thesis, I put forward a first step in developing a philosophy of AI art

Concepts as Correlates of Lexical Labels. A Cognitivist Perspective, 274 s.

This is a submitted manuscript version. The publisher should be contacted for permission to re-use or reprint the material in any form. Final published version, copyright Peter Lang: https://doi.org/10.3726/978-3-653-05287-9The study of language becomes particularly attractive when it is not practised as an isolated descriptive enterprise, but when it has wide-ranging implications for the study of the human mind. Such is the spirit of this book. While categorisation may be the single most basic cognitive process in organisms, and as an area of inquiry, it is fundamental to Cognitive Science as a whole, at the other end of the spectrum, high-level cognition is organised and permeated by language, giving rise to categories that count and function as concepts. Working from considering the philosophical assumptions of the cognitivist perspective, this study offers an argument for a very productive understanding of the relation between concepts, categories, and their theoretical models

Programming Languages and Systems

This open access book constitutes the proceedings of the 28th European Symposium on Programming, ESOP 2019, which took place in Prague, Czech Republic, in April 2019, held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2019