Towards an embedding of Graph Transformation in Intuitionistic Linear Logic

Linear logics have been shown to be able to embed both rewriting-based approaches and process calculi in a single, declarative framework. In this paper we are exploring the embedding of double-pushout graph transformations into quantified linear logic, leading to a Curry-Howard style isomorphism between graphs and transformations on one hand, formulas and proof terms on the other. With linear implication representing rules and reachability of graphs, and the tensor modelling parallel composition of graphs and transformations, we obtain a language able to encode graph transformation systems and their computations as well as reason about their properties

Evidence-based lean logic profiles for conceptual data modelling languages

Multiple logic-based reconstruction of conceptual data modelling languages such as EER, UML Class Diagrams, and ORM exists. They mainly cover various fragments of the languages and none are formalised such that the logic applies simultaneously for all three modelling language families as unifying mechanism. This hampers interchangeability, interoperability, and tooling support. In addition, due to the lack of a systematic design process of the logic used for the formalisation, hidden choices permeate the formalisations that have rendered them incompatible. We aim to address these problems, first, by structuring the logic design process in a methodological way. We generalise and extend the DSL design process to apply to logic language design more generally and, in particular, by incorporating an ontological analysis of language features in the process. Second, availing of this extended process, of evidence gathered of language feature usage, and of computational complexity insights from Description Logics (DL), we specify logic profiles taking into account the ontological commitments embedded in the languages. The profiles characterise the minimum logic structure needed to handle the semantics of conceptual models, enabling the development of interoperability tools. There is no known DL language that matches exactly the features of those profiles and the common core is small (in the tractable ALNI). Although hardly any inconsistencies can be derived with the profiles, it is promising for scalable runtime use of conceptual data models

Reasoning About the Transfer of Control

We present DCL-PC: a logic for reasoning about how the abilities of agents and coalitions of agents are altered by transferring control from one agent to another. The logical foundation of DCL-PC is CL-PC, a logic for reasoning about cooperation in which the abilities of agents and coalitions of agents stem from a distribution of atomic Boolean variables to individual agents -- the choices available to a coalition correspond to assignments to the variables the coalition controls. The basic modal constructs of DCL-PC are of the form coalition C can cooperate to bring about phi. DCL-PC extends CL-PC with dynamic logic modalities in which atomic programs are of the form agent i gives control of variable p to agent j; as usual in dynamic logic, these atomic programs may be combined using sequence, iteration, choice, and test operators to form complex programs. By combining such dynamic transfer programs with cooperation modalities, it becomes possible to reason about how the power of agents and coalitions is affected by the transfer of control. We give two alternative semantics for the logic: a direct semantics, in which we capture the distributions of Boolean variables to agents; and a more conventional Kripke semantics. We prove that these semantics are equivalent, and then present an axiomatization for the logic. We investigate the computational complexity of model checking and satisfiability for DCL-PC, and show that both problems are PSPACE-complete (and hence no worse than the underlying logic CL-PC). Finally, we investigate the characterisation of control in DCL-PC. We distinguish between first-order control -- the ability of an agent or coalition to control some state of affairs through the assignment of values to the variables under the control of the agent or coalition -- and second-order control -- the ability of an agent to exert control over the control that other agents have by transferring variables to other agents. We give a logical characterisation of second-order control

Optimal methods for reasoning about actions and plans in multi-agent systems

Cet travail présente une solution au problème du décor inférenciel. Nous réalisons cela en donnant une éducation polynomiale d'un fragment du calcul des situations vers la logique épistémique dynamique (DEL). En suite, une nouvelle méthode de preuve pour DEL, dont la complexité algorithmique est inférieure à celle de la méthode de Reiter pour le calcul de situations, est proposée. Ce travail présente aussi une nouvelle logique pour raisonner sur les actions. Cette logique permet d'exprimer formellement "qu'il existe une suite d'action conduisant au but". L'idée étant que, avec la quantification sur les actions, la planification devient un problème de validité. Une axiomatisation et quelques résultats d'expressivité sont donnés, ainsi qu'une méthode de preuve basée sur les tableaux sémantiques.This work presents a solution to the inferential frame problem. We do so by providing a polynomial reduction from a fragment of situation calculus to espistemic dynamic logic (DEL). Then, a novel proof method for DEL, such that the computational complexity is much lower than that of Retier's proof method for situation caluculs, is proposed. This work also presents a new logic for reasoning about actions. This logic allows to formally express that "there exists a sequence of actions that leads to the goal". The idea is that, with quantification over actions, planning can become a validity problem. An axiomatisation and some expressivity results are provided, as well as a proof method based on sematic tableaux

Predicate conjoining in Hadiyya: a head driven PS grammar

In examining certain structures of the East Cushitic language Hadiyya, this thesis, in keeping with recent trends, adopts a mono-stratal framework, framed in terms of the mathematical operation of Unification; namely Head-driven Phrase Structure Grammar (HPSG). Chapter 1 is devoted to an exposition of the model employing situation semantics. Chapter 2 discusses the categories of noun, noun phrase, and verb. The discussion centres on the basic morphological categories of Person, Number, Gender and Case, and the variety of verbal forms which are relevant to an appreciation of following chapters, and a tentative (partial) feature system is set out. Chapter 3 deals with the mono-clausal sentence, briefly expounding basic sentence types, with the focus of the chapter on the issues of subcategorisation, constituent order, "pro-drop", and agreement. Several revisions of the formalism are proposed, and a general goal formulated. Part II deals with nexus mechanisms. First is a short chapter, 4, on canonical coordination as it occurs in Hadiyya, in which an attempt is made to formalise resolution rules, and a broader, cross-linguistic look is taken at the categories of Person, Gender and Number in coordinate phrases. Some of Hadiyya's other lexical connectors are also briefly considered. In the two final chapters, both subordinative and coordinative systems are reviewed, and these chapters provide an end-focus to the study. Chapter 5, discusses the adverbial clause, and the complementation system, while Chapter 6 covers clause chaining/ serialisation, switch reference, and the encoding of simultaneous events, in which agreement and control questions are addressed. A short final chapter brings together some of the major theoretical suggestions arising.

Towards formalisation of situation-specific computations in pervasive computing environments

We have categorised the characteristics and the content of pervasive computing environments (PCEs), and demonstrated why a non-dynamic approach to knowledge conceptualisation in PCEs does not fulfil the expectations we may have from them. Consequently, we have proposed a formalised computational model, the FCM, for knowledge representation and reasoning in PCEs which, secures the delivery of situation and domain specific services to their users. The proposed model is a user centric model, materialised as a software engineering solution, which uses the computations generated from the FCM, stores them within software architectural components, which in turn can be deployed using modern software technologies. The model has also been inspired by the Semantic Web (SW) vision and provision of SW technologies. Therefore, the FCM creates a semantically rich situation-specific PCE based on SWRL-enabled OWL ontologies that allows reasoning about the situation in a PCE and delivers situation specific service. The proposed FCM model has been illustrated through the example of remote patient monitoring in the healthcare domain. Numerous software applications generated from the FCM have been deployed using Integrated Development Environments and OWL-API

Relational reasoning for effects and handlers

This thesis studies relational reasoning techniques for FRANK, a strict functional language supporting algebraic effects and their handlers, within a general, formalised approach for completely characterising observational equivalence. Algebraic effects and handlers are an emerging paradigm for representing computational effects where primitive operations, which give rise to an effect, are primary, and given semantics through their interpretation by effect handlers. FRANK is a novel point in the design space because it recasts effect handling as part of a generalisation of call-by-value function application. Furthermore, FRANK generalises unary effect handlers to the n-ary notion of multihandlers, supporting more elegant expression of certain handlers. There have been recent efforts to develop sound reasoning principles, with respect to observational equivalence, for languages supporting effects and handlers. Such techniques support powerful equational reasoning about code, such as substitution of equivalent sub-terms (‘equals for equals’) in larger programs. However, few studies have considered a complete characterisation of observational equivalence, and its implications for reasoning techniques. Furthermore, there has been no account of reasoning principles for FRANK programs. Our first contribution is a formal reconstruction of a general proof technique, triangulation, for proving completeness results for observational equivalence. The technique brackets observational equivalence between two structural relations, a logical and an applicative notion. We demonstrate the triangulation proof method for a pure simply-typed λ-calculus. We show that such results are readily formalisable in an implementation of type theory, specifically AGDA, using state-of-the-art technology for dealing with syntaxes with binding. Our second contribution is a calculus, ELLA, capturing the essence of FRANK’s novel design. In particular, ELLA supports binary handlers and generalises function application to incorporate effect handling. We extend our triangulation proof technique to this new setting, completely characterising observational equivalence for this calculus. We report on our partial progress in formalising our extension to ELLA in AGDA. Our final contribution is the application of sound reasoning principles, inspired by existing literature, to a variety of ELLA programs, including a proof of associativity for a canonical pipe multihandler. Moreover, we show how leveraging completeness leads, in certain instances, to simpler proofs of observational equivalence

Indeterminacy and the law of the excluded middle

This thesis is an investigation into indeterminacy in the foundations of mathematics and its possible consequences for the applicability of the law of the excluded middle (LEM). It characterises different ways in which the natural numbers as well as the sets may be understood to be indeterminate, and asks in what sense this would cease to support applicability of LEM to reasoning with them. The first part of the thesis reviews the indeterminacy phenomena on which the argument is based and argues for a distinction between two notions of indeterminacy: a) indeterminacy as applied to domains and b) indefiniteness as applied to concepts. It then addresses possible attempts to secure determinacy in both cases. The second part of the thesis discusses the advantages that an argument from indeterminacy has over traditional intuitionistic arguments against LEM, and it provides the framework in which conditions for the applicability of LEM can be explicated in the setting of indeterminacy. The final part of the thesis then applies these findings to concrete cases of indeterminacy. With respect to indeterminacy of domains, I note some problems for establishing a rejection of LEM based on the indeterminacy of the height of the set theoretic hierarchy. I show that a coherent argument can be made for the rejection of LEM based on the indeterminacy of its width, and assess its philosophical commitments. A final chapter addresses the notion of indefiniteness of our concepts of set and number and asks how this might affect the applicability of LEM

Proof-theoretic Semantics for Intuitionistic Multiplicative Linear Logic

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