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Structural types for systems of equations

Higher-Order and Symbolic Computation 2012年第2-4期25卷 275-310页

作者： Capper, John Nilsson, Henrik Functional Programming Lab School of Computer Science University of Nottingham NottinghamUK United Kingdom

Characterising a problem in terms of a system of equations is common to many branches of science and engineering. Due to their size, such systems are often described in a modular fashion by composition of individual equation system fragments. Checking the balance between the number of variables (unknowns) and equations is a common approach to early detection of mistakes that might render such a system unsolvable. However, current approaches to modular balance checking have a number of limitations. This paper investigates a more flexible approach that makes it possible to treat equation system fragments as true first-class entities. Furthermore, the approach handles so-called structurally dynamic systems, systems whose behaviour changes discretely and abruptly over time. The central idea is to record balance information in the type of an equation fragment. This information can then be used to determine if individual fragments are well formed, and if composing fragments preserves this property. The type system presented in this paper is developed in the context of functional Hybrid Modelling (FHM). However, the key ideas are in no way specific to FHM, but should be applicable to any language featuring a notion of modular systems of equations, including systems with first-class components and structural dynamism. © Springer Science+Business Media New York 2013

关键词： First-class components Linear constraints Non-causal Refinement types Structural analysis Structurally dynamic modelling Systems of equations

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Contractive functions on infinite data structures 28

Contractive functions on infinite data structures

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28th Symposium on the Implementation and Application of functional programming Languages, IFL 2016

作者： Capretta, Venanzio Hutton, Graham Jaskelioff, Mauro Functional Programming Lab University of Nottingham United Kingdom CIFASIS-CONICET Universidad Nacional de Rosario Argentina

ISBN: (纸本)9781450347679

Coinductive data structures, such as streams or infinite trees, have many applications in functional programming and type theory, and are naturally defined using recursive equations. But how do we ensure that such equations make sense, i.e. that they actually generate a productive infinite object? A standard means to achieve productivity is to use Banach's fixed-point theorem, which guarantees the unique existence of solutions to recursive equations on metric spaces under certain conditions. Functions satisfying these conditions are called contractions. In this article, we give a new characterization of contractions on streams in the form of a sound and complete representation theorem, and generalize this result to a wide class of non-well-founded structures, first to infinite binary trees, then to final coalgebras of container functors. These results have important potential applications in functional programming, where coinduction and corecursion are successfully deployed to model continuous reactive systems, dynamic interactivity, signal processing, and other tasks that require exible manipulation of non-well-founded data. Our representation theorems provide a definition paradigm to compactly compute with such data and easily reason about them. © 2016 ACM.

关键词： functional programming

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Synthetic fibered (∞,1)-category theory

arXiv

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arXiv 2021年

作者： Buchholtz, Ulrik Weinberger, Jonathan Functional Programming Lab School of Computer Science University of Nottingham Jubilee Campus NottinghamNG7 1BB United Kingdom Dept. of Mathematics Krieger School of Arts and Sciences Johns Hopkins University 3400 N Charles St. BaltimoreMD21218 United States

We study cocartesian fibrations in the setting of the synthetic (∞,1)-category theory developed in simplicial type theory introduced by Riehl and Shulman. Our development culminates in a Yoneda Lemma for cocartesian ... 详细信息

关键词： Topology

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Review of “Thinking with Types”* by Sandy Maguire, LeanPub, 2019

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Journal of functional programming 2020年 30卷 e4-e4页

作者： J. HACKETT Functional Programming Lab School of Computer Science University of Nottingham E-mail: jennifer.hackett@nottingham.ac.uk

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