Calculating invariants as coreflexive bisimulations

Invariants, bisimulations and assertions are the main ingredients of coalgebra theory applied to software systems. In this paper we reduce the first to a particular case of the second and show how both together pave the way to a theory of coalgebras which regards invariant predicates as types. An outcome of such a theory is a calculus of invariants’ proof obligation discharge, a fragment of which is presented in the paper. The approach has two main ingredients: one is that of adopting relations as “first class citizens” in a pointfree reasoning style; the other lies on a synergy found between a relational construct, Reynolds’ relation on functions involved in the abstraction theorem on parametric polymorphism and the coalgebraic account of bisimulations and invariants. This leads to an elegant proof of the equivalence between two different definitions of bisimulation found in coalgebra literature (due to B. Jacobs and Aczel & Mendler, respectively) and to their instantiation to the classical Park-Milner definition popular in process algebra.Partially supported by the Fundacao para a Ciencia e a Tecnologia, Portugal, under grant number SFRH/BD/27482/2006

On difunctions

The notion of a difunction was introduced by Jacques Riguet in 1948. Since then it has played a prominent role in database theory, type theory, program specification and process theory. The theory of difunctions is, however, less known in computing than it perhaps should be. The main purpose of the current paper is to give an account of difunction theory in relation algebra, with the aim of making the topic more mainstream. As is common with many important concepts, there are several different but equivalent characterisations of difunctionality, each with its own strength and practical significance. This paper compares different proofs of the equivalence of the characterisations. A well-known property is that a difunction is a set of completely disjoint rectangles. This property suggests the introduction of the (general) notion of the “core” of a relation; we use this notion to give a novel and, we believe, illuminating characterisation of difunctionality as a bijection between the classes of certain partial equivalence relations

Greedy and dynamic programming by calculation

Dissertação mestrado integrado em Informatics EngineeringThe mathematical study of the greedy algorithm provides a blueprint for the study of Dynamic Programming (DP), whose body of knowledge is largely unorganized, remaining obscure to a large part of the software engineering community. This study aims to structure this body of knowledge, narrowing the gap between a purely examplebased approach to DP and its scientific foundations. To that effect, matroid theory is leveraged through a pointfree relation algebra, which is applied to greedy and DP problems. A catalogue of such problems is compiled, and a broad characterization of DP algorithms is given. Alongside, the theory underlying the thinning relational operator is explored.O estudo matemático do algoritmo ganancioso («greedy») serve como guia para o estudo da programação dinâmica, cujo corpo de conhecimento permanece desorganizado e obscuro a uma grande parte da comunidade de engenharia de software. Este estudo visa estruturar esse corpo de conhecimento, fazendo a ponte entre a abordagem popular baseada em exemplos e os métodos mais teóricos da literatura científica. Para esse efeito, a teoria dos matroides é explorada pelo uso de uma álgebra de relações pointfree, e aplicada a problemas «greedy» e de programação dinâmica. Um catálogo de tais problemas é compilado, e é feita uma caraterização geral de algoritmos de programação dinâmica. Em paralelo, é explorada a teoria do combinador relacional de «thinning».This work is financed by National Funds through the Portuguese funding agency, FCT - Fundação para a Ciência e a Tecnologia, within project UIDB/50014/202