Deriving animations from recursive definitions

This paper describes a generic method to derive an animation from a recursive definition, with the objective of debugging and understanding this definition by expliciting its control structure. This method is based on a well known algorithm of factorizing a recursive function into the composition of the producer and the consumer of its call tree. We developed a systematic method to transform both the resulting functions in order to draw the tree step by step. The theory of data types as fixed points of functors, generic recursion patterns, and monads, are fundamental to our work and are brie y presented. Using polytypic implementations of monadic recursion patterns and an application to manipulate and generate graph layouts we developed a prototype that, given a recursive function written in a subset of Haskell, returns a function whose execution yields the desired animation

Continuity as a computational effect

The original purpose of component-based development was to provide techniques to master complex software, through composition, reuse and parametrisation. However, such systems are rapidly moving towards a level in which software becomes prevalently intertwined with (continuous) physical processes. A possible way to accommodate the latter in component calculi relies on a suitable encoding of continuous behaviour as (yet another) computational effect. This paper introduces such an encoding through a monad which, in the compositional development of hybrid systems, may play a role similar to the one played by 1+, powerset, and distribution monads in the characterisation of partial, nondeterministic and probabilistic components, respectively. This monad and its Kleisli category provide a universe in which the effects of continuity over (different forms of) composition can be suitably studied.This work is financed by the ERDF - 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 - Fundacao para a Ciencia e a Tecnologia within project POCI-01-0145-FEDER-016692.The first author is also sponsored by FCT grant SFRH/BD/52234/2013, and the second one by FCT grant SFRH/BSAB/113890/2015. Moreover, D. Hofmann and M. Martins are supported by the EU FP7 Marie Curie PIRSES-GA-2012-318986 project GeTFun: Generalizing Truth-Functionality and FCT project UID/MAT/04106/2013 through CIDMA

Proving the unique fixed-point principle correct: an adventure with category theory

Abstract Say you want to prove something about an infinite data-structure, such as a stream or an infinite tree, but you would rather not subject yourself to coinduction. The unique fixed-point principle is an easyto-use, calculational alternative. The proof technique rests on the fact that certain recursion equations have unique solutions; if two elements of a coinductive type satisfy the same equation of this kind, then they are equal. In this paper we precisely characterize the conditions that guarantee a unique solution. Significantly, we do so not with a syntactic criterion, but with a semantic one that stems from the categorical notion of naturality. Our development is based on distributive laws and bialgebras, and draws heavily on Turi and Plotkin's pioneering work on mathematical operational semantics. Along the way, we break down the design space in two dimensions, leading to a total of nine points. Each gives rise to varying degrees of expressiveness, and we will discuss three in depth. Furthermore, our development is generic in the syntax of equations and in the behaviour they encode-we are not caged in the world of streams

Functional Query Languages with Categorical Types

We study three category-theoretic types in the context of functional query languages (typed lambda-calculi extended with additional operations for bulk data processing). The types we study are:Engineering and Applied Science