PSPACE Bounds for Rank-1 Modal Logics

For lack of general algorithmic methods that apply to wide classes of logics, establishing a complexity bound for a given modal logic is often a laborious task. The present work is a step towards a general theory of the complexity of modal logics. Our main result is that all rank-1 logics enjoy a shallow model property and thus are, under mild assumptions on the format of their axiomatisation, in PSPACE. This leads to a unified derivation of tight PSPACE-bounds for a number of logics including K, KD, coalition logic, graded modal logic, majority logic, and probabilistic modal logic. Our generic algorithm moreover finds tableau proofs that witness pleasant proof-theoretic properties including a weak subformula property. This generality is made possible by a coalgebraic semantics, which conveniently abstracts from the details of a given model class and thus allows covering a broad range of logics in a uniform way

Coalgebras and Their Logics

Transition systems pervade much of computer science. This article outlines the beginnings of a general theory of specification languages for transition systems. More specifically, transition systems are generalised to coalgebras. Specification languages together with their proof systems, in the following called (logical or modal) calculi, are presented by the associated classes of algebras (e.g., classical propositional logic by Boolean algebras). Stone duality will be used to relate the logics and their coalgebraic semantics

Composition Semantics of the Rosetta Specification Language

The Rosetta specification language aims to enable system designers to abstractly design complex heterogeneous systems. To this end, Rosetta allows for compositional design to facilitate modularity, separation of concerns, and specification reuse. The behavior of Rosetta components and facets can be viewed as systems, which are well suited for coalgebraic denotation. The previous semantics of Rosetta lacked detail in the denotational work, and had no firm semantic basis for the composition operators. This thesis refreshes previous work on the coalgebraic denotation of Rosetta. It then goes on to define the denotation of the composition operators. A real-world Rosetta example using all types of composition serves as a demonstration of the power of composition as well as the clean, modular abstractness it affords the designer

Global semantic typing for inductive and coinductive computing

Inductive and coinductive types are commonly construed as ontological (Church-style) types, denoting canonical data-sets such as natural numbers, lists, and streams. For various purposes, notably the study of programs in the context of global semantics, it is preferable to think of types as semantical properties (Curry-style). Intrinsic theories were introduced in the late 1990s to provide a purely logical framework for reasoning about programs and their semantic types. We extend them here to data given by any combination of inductive and coinductive definitions. This approach is of interest because it fits tightly with syntactic, semantic, and proof theoretic fundamentals of formal logic, with potential applications in implicit computational complexity as well as extraction of programs from proofs. We prove a Canonicity Theorem, showing that the global definition of program typing, via the usual (Tarskian) semantics of first-order logic, agrees with their operational semantics in the intended model. Finally, we show that every intrinsic theory is interpretable in a conservative extension of first-order arithmetic. This means that quantification over infinite data objects does not lead, on its own, to proof-theoretic strength beyond that of Peano Arithmetic. Intrinsic theories are perfectly amenable to formulas-as-types Curry-Howard morphisms, and were used to characterize major computational complexity classes Their extensions described here have similar potential which has already been applied

Proof support for hybridised logics

Dissertação de mestrado em Engenharia InformáticaFormal methods are mathematical techniques used to certify safe systems. Such methods abound and have been successfully used in classical Engineering domains, yet informatics is the exception. There, they are still immature and costly; furthermore, software engineers frequently view them with "fear". Thus, the use of formal methods is typically restricted to cases where they are essential. In other words, they are mostly used in the class of systems where safety is imperative, as the lack of it can lead to significant losses (material or human). We denote such systems critical. The present is leading us to a future where critical systems are ubiquitous. Recent research in the Mondrian project emphasises the need for expressive logics to formally specify reconfigurable systems, i.e., systems capable of evolving in order to adapt to the different contexts induced by the dynamics of their surroundings. In the same project, theoretical foundations for the formal specification of reconfigurable systems, were developed in a sound, generic, and systematic way, resorting for this to hybrid logics – their intrinsic properties make them natural candidates for such job. From those foundations a methodology for specifying reconfigurable systems was built and proposed: Instead of choosing a logic for the specification, build an hybrid ad-hoc one, by taking into account the particular characteristics of each reconfigurable system to be specified. The purpose of this dissertation is to bring the proposed methodology into practice, by creating suitable tools for it, and by illustrating its application to relevant case studies.Métodos formais são técnicas matemáticas usadas para certificar sistemas fiáveis. Tais métodos são comuns e usados com sucesso nas engenharias clássicas. No entanto, informática é a excepção. No que respeita este campo, os métodos formais são prematuros e relativamente dispendiosos; para além disso, os engenheiros de software vêem estas técnicas com alguma apreensão. Assim, o emprego de métodos formais está tipicamente restrito a casos onde são absolutamente essenciais. Por outras palavras, são maioritariamente usados na classe de sistemas, cujas falhas têm o potencial de tragédia, seja ela material ou humana; tais sistemas têm a denominação de críticos. O presente leva-nos para um futuro em que os sistemas críticos são ubíquos. Investigação recente no project Mondrian enfatiza a necessidade de lógicas expressivas, para especificar formalmente sistemas reconfiguráveis, i.e., sistemas que evoluem de modo a se adaptarem aos diferentes contextos, induzidos pela dinâmica do meio que os rodeia. No mesmo projecto, bases teóricas para a especificação formal de sistemas reconfiguráveis foram establecidas de forma sólida, genérica e sistemática, recorrendo-se para isso às lógicas híbridas – as suas propriedades intrínsecas, fazem delas candidatos naturais para a especificação de sistemas reconfiguráveis. Dessas teorias foi inferida e proposta uma metodologia para especificar sistemas reconfiguráveis: Em vez de escolher uma lógica para a especificação, construir uma outra, híbrida ad-hoc, tendo em conta as características particulares de cada sistema reconfigurável a especificar. O propósito desta dissertação é de trazer a metodologia proposta à práctica, criando-se para isso, ferramentas que a suportem, e ilustrando a sua aplicação a casos de estudo relevantes

Coalgebras of topological types

In This work, we focus on developing the basic theory of coalgebras over the category Top (the category of topological spaces with continuous maps). We argue that, besides Set, the category Top is an interesting base category for coalgebras. We study some endofunctors on Top, in particular, Vietoris functor and P-Vietoris Functor (where P is a set of propositional letters) that due to Hofmann et. al. [42] can be considered as the topological versions of the powerset functor and P-Kripke functor, respectively. We define the notion of compact Kripke structures and we prove that Kripke homomorphisms preserve compactness. Our definition of "compact Kripke structure" coincides with the notion of "modally saturated structures" introduced in Fine [27]. We prove that the class of compact Kripke structures has Hennessy-Milner property. As a consequence we show that in this class of Kripke structures, bihavioral equivalence, modal equivalence and Kripke bisimilarity all coincide.Furthermore, we generalize the notion of descriptive structures defined in Venema et. al. [11] by introducing a notion Vietoris models. We identify Vietoris models as coalgebras for the P-Vietoris functor on the category Top. One can see that each compact Kripke model can be modified to a Vietoris model. This yields an adjunction between the category of Vietoris structures (VS) and the category of compact Kripke structurs (CKS). Moreover, we will prove that the category of Vietoris models has a terminal object. We study the concept of a Vietoris bisimulation between Vietoris models, and we will prove that the closure of a Kripke bisimulation between underlying Kripke models of two Vietoris models is a Vietoris bisimulation. In the end, it will be shown that in the class of Vietoris models, Vietoris bisimilarity, bihavioral equivalence, modal equivalence, all coincide

Architectural Refinement in HETS

The main objective of this work is to bring a number of improvements to the Heterogeneous Tool Set HETS, both from a theoretical and an implementation point of view. In the first part of the thesis we present a number of recent extensions of the tool, among which declarative specifications of logics, generalized theoroidal comorphisms, heterogeneous colimits and integration of the logic of the term rewriting system Maude. In the second part we concentrate on the CASL architectural refinement language, that we equip with a notion of refinement tree and with calculi for checking correctness and consistency of refinements. Soundness and completeness of these calculi is also investigated. Finally, we present the integration of the VSE refinement method in HETS as an institution comorphism. Thus, the proof manangement component of HETS remains unmodified