Generating the algebraic theory of C(X)C(X): the case of partially ordered compact spaces

It is known since the late 1960's that the dual of the category of compact Hausdorff spaces and continuous maps is a variety -- not finitary, but bounded by $\aleph_1$. In this note we show that the dual of the category of partially ordered compact spaces and monotone continuous maps is a $\aleph_1$-ary quasivariety, and describe partially its algebraic theory. Based on this description, we extend these results to categories of Vietoris coalgebras and homomorphisms. We also characterise the $\aleph_1$-copresentable partially ordered compact spaces

A Complete V-Equational System for Graded lambda-Calculus

Modern programming frequently requires generalised notions of program equivalence based on a metric or a similar structure. Previous work addressed this challenge by introducing the notion of a V-equation, i.e. an equation labelled by an element of a quantale V, which covers inter alia (ultra-)metric, classical, and fuzzy (in)equations. It also introduced a V-equational system for the linear variant of lambda-calculus where any given resource must be used exactly once. In this paper we drop the (often too strict) linearity constraint by adding graded modal types which allow multiple uses of a resource in a controlled manner. We show that such a control, whilst providing more expressivity to the programmer, also interacts more richly with V-equations than the linear or Cartesian cases. Our main result is the introduction of a sound and complete V-equational system for a lambda-calculus with graded modal types interpreted by what we call a Lipschitz exponential comonad. We also show how to build such comonads canonically via a universal construction, and use our results to derive graded metric equational systems (and corresponding models) for programs with timed and probabilistic behaviour

An Internal Language for Categories Enriched over Generalised Metric Spaces

Programs with a continuous state space or that interact with physical processes often require notions of equivalence going beyond the standard binary setting in which equivalence either holds or does not hold. In this paper we explore the idea of equivalence taking values in a quantale ?, which covers the cases of (in)equations and (ultra)metric equations among others. Our main result is the introduction of a ?-equational deductive system for linear ?-calculus together with a proof that it is sound and complete (in fact, an internal language) for a class of enriched autonomous categories. In the case of inequations, we get an internal language for autonomous categories enriched over partial orders. In the case of (ultra)metric equations, we get an internal language for autonomous categories enriched over (ultra)metric spaces. We use our results to obtain examples of inequational and metric equational systems for higher-order programs that contain real-time and probabilistic behaviour

Limits in categories of Vietoris coalgebras

Motivated by the need to reason about hybrid systems, we study limits in categories of coalgebras whose underlying functor is a Vietoris polynomial one - intuitively, the topological analogue of a Kripke polynomial functor. Among other results, we prove that every Vietoris polynomial functor admits a final coalgebra if it respects certain conditions concerning separation axioms and compactness. When the functor is restricted to some of the categories induced by these conditions the resulting categories of coalgebras are even complete. As a practical application, we use these developments in the specification and analysis of non-deterministic hybrid systems, in particular to obtain suitable notions of stability, and behaviour.publishe

A Semantics for Hybrid Iteration

The recently introduced notions of guarded traced (monoidal) category and guarded (pre-)iterative monad aim at unifying different instances of partial iteration whilst keeping in touch with the established theory of total iteration and preserving its merits. In this paper we use these notions and the corresponding stock of results to examine different types of iteration for hybrid computations. As a starting point we use an available notion of hybrid monad restricted to the category of sets, and modify it in order to obtain a suitable notion of guarded iteration with guardedness interpreted as progressiveness in time - we motivate this modification by our intention to capture Zeno behaviour in an arguably general and feasible way. We illustrate our results with a simple programming language for hybrid computations and interpret it over the developed semantic foundations

Hybrid automata as coalgebras

Publicado em "Theoretical aspects of computing - ICTAC 2016: 13th International Colloquium, Taipei, Taiwan, ROC, October 24–31, 2016, Proceedings". ISBN 978-3-319-46749-8Able to simultaneously encode discrete transitions and continuous behaviour, hybrid automata are the de facto framework for the formal specification and analysis of hybrid systems. The current paper revisits hybrid automata from a coalgebraic point of view. This allows to interpret them as state-based components, and provides a uniform theory to address variability in their definition, as well as the corresponding notions of behaviour, bisimulation, and observational semantics.FCT grants SFRH/BD/52234/2013, SFRH/BSAB/ 113890/2015ERDF - European Regional Development Fund, through the COMPETE Programme, and by National Funds through FCT within project PTDC/EEI-CTP/4836/201

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