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

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

Hybrid programs

The MAP-i Doctoral Programme in Informatics, of the Universities of Minho, Aveiro and PortoThis thesis studies hybrid systems, an emerging family of devices that combine in their models digital computations and physical processes. They are very quickly becoming a main concern in software engineering, which is explained by the need to develop software products that closely interact with physical attributes of their environment e. g. velocity, time, energy, temperature – typical examples range from micro-sensors and pacemakers, to autonomous vehicles, transport infrastructures and district-wide electric grids. But even if already widespread, these systems entail different combinations of programs with physical processes, and this renders their development a challenging task, still largely unmet by the current programming practices. Our goal is to address this challenge at its core; we wish to isolate the basic interactions between discrete computations and physical processes, and bring forth the programming paradigm that naturally underlies them. In order to do so in a precise and clean way, we resort to monad theory, a well established categorical framework for developing program semantics systematically. We prove the existence of a monad that naturally encodes the aforementioned interactions, and use it to develop and examine the foundations of the paradigm alluded above, which we call hybrid programming: we show how to build, in a methodical way, different programming languages that accommodate amplifiers, differential equations, and discrete assignments – the basic ingredients of hybrid systems – we list all program operations available in the paradigm, introduce if-then-else constructs, abort operations, and different types of feedback. Hybrid systems bring several important aspects of control theory into computer science. One of them is the notion of stability, which refers to a system’s capacity of avoiding significant changes in its output if small variations in its state or input occur. We introduce a notion of stability to hybrid programming, explore it, and show how to analyse hybrid programs with respect to it in a compositional manner. We also introduce hybrid programs with internal memory and show that they form the basis of a component-based software development discipline in hybrid programming. We develop their coalgebraic theory, namely languages, notions of behaviour, and bisimulation. In the process, we introduce new theoretical results on Coalgebra, including improvements of well-known results and proofs on the existence of suitable notions of behaviour for non-deterministic transition systems with infinite state spaces.Esta tese estuda sistemas híbridos, uma família emergente de dispositivos que envolvem diferentes interações entre computações digitais e processos físicos. Estes sistemas estão rapidamente a tornar-se elementos-chave da engenharia de software, o que é explicado pela necessidade de desenvolver produtos que interagem com os atributos físicos do seu ambiente e. g. velocidade, tempo, energia, e temperatura – exemplos típicos variam de micro-sensores e pacemakers, a veículos autónomos, infra-estruturas de transporte, e redes eléctricas distritais. Mas ainda que amplamente usados, estes sistemas são geralmente desenvolvidos de forma pouco sistemática nas prácticas de programação atuais. O objetivo deste trabalho é isolar as interações básicas entre computações digitais e processos físicos, e subsequentemente desenvolver o paradigma de programação subjacente. Para fazer isto de forma precisa, a nossa base de trabalho irá ser a teoria das mónadas, uma estrutura categórica para o desenvolvimento sistemático de semânticas na programação. A partir desta base, provamos a existência de uma mónada que capta as interações acima mencionadas, e usamo-la para desenvolver e examinar os fundamentos do paradigma de programação correspondente a que chamamos programação híbrida: mostramos como construir, de maneira metódica, diferentes linguagens de programação que acomodam amplificadores, equações diferenciais, e atribuições - os ingredientes básicos dos sistemas híbridos - caracterizamos todas as operações sobre programas disponíveis, introduzimos construções if-then-else, operações para lidar com excepções, e diferentes tipos de feedback. Os sistemas híbridos trazem vários aspectos da teoria de controlo para a ciência da computação. Um destes é a noção de estabilidade, que se refere à capacidade de um sistema de evitar mudanças drásticas no seu output se pequenas variações no seu estado ou input ocorrerem. Neste trabalho, desenvolvemos uma noção composicional de estabilidade para a programação híbrida. Introduzimos também programas híbridos com memória interna, que formam a base de uma disciplina de desenvolvimento de software baseado em componentes. Desenvolvemos a sua teoria coalgébrica, nomeadamente linguagens, noções de comportamento e bisimulação. Neste processo, introduzimos também novos resultados teóricos sobre Coalgebra, incluindo melhorias a resultados conhecidos e provas acerca da existência de noções de comportamento para sistemas de transição não determinísiticos com espaço de estados infinitos.The present work was financed by FCT – Fundação para a Ciência e a Tecnologia – with the grant SFRH/BD/52234/2013. Additional support was provided by the PTFLAD Chair on Smart Cities & Smart Governance and by project Dalí (POCI-01-0145-FEDER-016692), the latter funder by ERDF – European Regional Development Fund – through COMPETE 2020 – Operational Programme for Competitiveness and Internationalisation – together with FCT

Languages and models for hybrid automata: A coalgebraic perspective

article in pressWe study hybrid automata from a coalgebraic point of view. We show that such a perspective supports a generic theory of hybrid automata with a rich palette of definitions and results. This includes, among other things, notions of bisimulation and behaviour, state minimisation techniques, and regular expression languages.POCI-01-0145-FEDER-016692. RDF — European Regional Development Fund through the Operational Programme for Competitiveness and Internationalisation — COMPETE 2020 Programme and by National Funds through the Portuguese funding agency, FCT — Fundação para a Ciência e a Tecnologia within project POCI-01-0145-FEDER-016692 and by the PT-FLAD Chair on Smart Cities & Smart Governance at Universidade do Minh