From 2-sequents and Linear Nested Sequents to Natural Deduction for Normal Modal Logics

We extend to natural deduction the approach of Linear Nested Sequents and 2-sequents. Formulas are decorated with a spatial coordinate, which allows a formulation of formal systems in the original spirit of natural deduction---only one introduction and one elimination rule per connective, no additional (structural) rule, no explicit reference to the accessibility relation of the intended Kripke models. We give systems for the normal modal logics from K to S4. For the intuitionistic versions of the systems, we define proof reduction, and prove proof normalisation, thus obtaining a syntactical proof of consistency. For logics K and K4 we use existence predicates (following Scott) for formulating sound deduction rules

Countermodel Construction via Optimal Hypersequent Calculi for Non-normal Modal Logics

International audienceWe develop semantically-oriented calculi for the cube of non-normal modal logics and some deontic extensions. The calculi manipulate hypersequents and have a simple semantic interpretation. Their main feature is that they allow for direct countermodel extraction. Moreover they provide an optimal decision procedure for the respective logics. They also enjoy standard proof-theoretical properties, such as a syntactical proof of cut-admissibility

Hypersequent calculi for non-normal modal and deontic logics: Countermodels and optimal complexity

We present some hypersequent calculi for all systems of the classical cube and their extensions with axioms $T$, $P$, $D$, and, for every $n\geq 1$, rule $RD^+_n$. The calculi are internal as they only employ the language of the logic, plus additional structural connectives. We show that the calculi are complete with respect to the corresponding axiomatisation by a syntactic proof of cut elimination. Then we define a terminating root-first proof search strategy based on the hypersequent calculi and show that it is optimal for coNP-complete logics. Moreover, we obtain that from every saturated leaf of a failed proof it is possible to define a countermodel of the root hypersequent in the bi-neighbourhood semantics, and for regular logics also in the relational semantics. We finish the paper by giving a translation between hypersequent rule applications and derivations in a labelled system for the classical cube

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

Modal Interface Theories for Specifying Component-based Systems

Large software systems frequently manifest as complex, concurrent, reactive systems and their correctness is often crucial for the safety of the application. Hence, modern techniques of software engineering employ incremental, component-based approaches to systems design. These are supported by interface theories which may serve as specification languages and as semantic foundations for software product lines, web-services, the internet of things, software contracts and conformance testing. Interface theories enable a systems designer to express communication requirements of components on their environments and to reason about the mutual compatibility of these requirements in order to guarantee the communication safety of the system. Further, interface theories enrich traditional operational specification theories by declarative aspects such as conjunction and disjunction, which allow one to specify systems heterogeneously. However, substantial practical aspects of software verification are not supported by current interface theories, e.g., reusing components, adapting components to changed operational environments, reasoning about the compatibility of more than two components, modelling software product lines or tracking erroneous behaviour in safety-critical systems. The goal of this thesis is to investigate the theoretical foundations for making interface theories more practical by solving the above issues. Although partial solutions to some of these issues have been presented in the literature, none of them succeeds without sacrificing other desired features. The particular challenge of this thesis is to solve these problems simultaneously within a single interface theory. To this end, the arguably most general interface theory Modal Interface Automata (MIA) is extended, yielding the interface theory Error-preserving Modal Interface Automata (EMIA). The above problems are addressed as follows. Quotient operators are adjoint to composition and, therefore, support component reuse. Such a quotient operator is introduced to both MIA and EMIA. It is the first one that considers nondeterministic dividends and compatibility. Alphabet extension operators for MIA and EMIA allow for the change of operational environment by permitting one to adapt system components to new interactions without breaking previously satisfied requirements. Erroneous behavior is identified as a common source of problems with respect to the compatibility of more than two components, the modelling of software product lines and erroneous behaviour in safety-critical systems. EMIA improves on previous interface theories by providing a more precise semantics with respect to erroneous behaviour based on error-preservation. The relation between error-preservation and the usual error-abstraction employed in previous interface theories is investigated, establishing a Galois insertion from MIA into EMIA that is relevant at the levels of specifications, composition operations and proofs. The practical utility of interface theories is demonstrated by providing a software implementation of MIA and EMIA that is applied to two case studies. Further, an outlook is given on the relation between type checking and refinement checking. As a proof of concept, the simple interface theory Interface Automata is extended to a behavioural type theory where type checking is a syntactic approximation of refinement checking.Große Softwaresysteme bilden häufig komplexe, nebenläufige, reaktive Systeme, deren Korrektheit für die Sicherheit der Anwendung entscheidend ist. Daher setzen moderne Verfahren der Softwaretechnik inkrementelle, komponentenbasierte Ansätze zum Software-Entwurf ein. Diese werden von Interface-Theorien unterstützt, die als Spezifikationssprachen und semantische Grundlagen für Softwareproduktlinien, Web-Services, das Internet der Dinge, Softwarekontrakte und Konformanztests dienen können. Interface-Theorien ermöglichen es, Kommunikationsanforderungen von Komponenten an ihre Umgebung auszudrücken, um die gegenseitige Kompatibilität dieser Anforderungen zu überprüfen und die Kommunikationssicherheit des Systems zu garantieren. Zudem erweitern Interface-Theorien traditionelle operationale Spezifikationstheorien um deklarative Aspekte wie beispielsweise Konjunktion und Disjunktion, die heterogenes Spezifizieren ermöglichen. Allerdings werden wesentliche praktische Aspekte der Softwareverifikation von Interface-Theorien nicht unterstützt, z.B. das Wiederverwenden von Komponenten, das Anpassen von Komponenten an geänderte operationale Umgebungen, die Kompatibilitätsprüfung von mehr als zwei Komponenten, das Modellieren von Softwareproduktlinien oder das Zurückverfolgen von Fehlverhalten sicherheitskritischer Systeme. Diese Arbeit untersucht die theoretischen Grundlagen von Interface-Theorien mit dem Ziel, die oben genannten praktischen Probleme zu lösen. Obwohl es in der Literatur Teillösungen zu manchen dieser Probleme gibt, erreicht keine davon ihr Ziel, ohne andere wünschenswerte Eigenschaften aufzugeben. Die besondere Herausforderung dieser Arbeit besteht darin, diese Probleme innerhalb einer einzigen Interface-Theorie zugleich zu lösen. Zu diesem Zweck wurde die wohl allgemeinste Interface-Theorie Modal Interface Automata (MIA) zu der Interface-Theorie Error-preserving Modal Interface Automata (EMIA) weiterentwickelt. Die obigen Probleme werden wie folgt gelöst. Ein zur Komposition adjungierter Quotientenoperator, der das Wiederverwenden von Komponenten ermöglicht, wurde für MIA und EMIA eingeführt. Es handelt sich dabei um den ersten Quotientenoperator, der nichtdeterministische Dividenden und Kompatibilität betrachtet. Alphabeterweiterungsoperatoren erlauben eine Änderung der operationalen Umgebung, indem sie es ermöglichen, Komponenten an neue Interaktionen anzupassen, ohne zuvor erfüllte Anforderungen zu missachten. Fehlerhaftes Verhalten wird als eine gemeinsame Ursache von Problemen bezüglich der Kompatibilität von mehr als zwei Komponenten, der Modellierung von Softwareproduktlinien und des Fehlverhaltens sicherheitskritischer Systeme erkannt. EMIA verbessert bisherige Interface-Theorien durch eine präzisere Fehlersemantik, die auf dem Erhalten von Fehlern beruht. Als Beziehung zwischen diesem Fehlererhalt und der in bisherigen Interface-Theorien üblichen Fehlerabstraktion ergibt sich eine Galois-Einbettung von MIA in EMIA, die auf den Ebenen der Spezifikationen, Operatoren und Beweise relevant ist. Die praktische Anwendbarkeit von Interface-Theorien wird mittels einer Implementierung von MIA und EMIA als Software und deren Anwendung auf zwei Fallstudien demonstriert. Zudem wird das Verhältnis zwischen Verfeinerung und Typprüfung diskutiert. In einer Machbarkeitsstudie wurde die einfache Interface-Theorie Interface Automata zu einer Verhaltenstyptheorie erweitert, bei der die Typprüfung eine syntaktische Approximation der Verfeinerung ist

Parameterized monads in linguistics

A thesis submitted in partial fulfilment of the requirements of the University of Wolverhampton for the degree of Doctor of Philosophy.This dissertation follows the formal semantics approach to linguistics. It applies recent developments in computing theories to study theoretical linguistics in the area of the interaction between semantics and pragmatics and analyzes several natural language phenomena by parsing them in these theories. Specifically, this dissertation uses parameterized monads, a particular theoretical framework in category theory, as a dynamic semantic framework to reinterpret the compositional Discourse Representation Theory(cDRT), and to provide an analysis of donkey anaphora. Parameterized monads are also used in this dissertation to interpret information states as lists of presuppositions, and as dot types. Alternative interpretations for demonstratives and imperatives are produced, and the conventional implicature phenomenon in linguistics substantiated, using the framework. Interpreting donkey anaphora shows that parameterized monads is able to handle the sentential dependency. Therefore, this framework shows an expressive power equal to that of related frameworks such as the typed logical grammar and the dynamic predicate logic. Interpreting imperatives via parameterized monads also provides a compositional dynamic semantic analysis which is one of the main approaches to analysing imperatives