Continuous Functions on Final Coalgebras

Authors:
Neil Ghani;Peter Hancock;Dirk Pattinson
Affiliations:
Department of Computer and Information Sciences, University of Strathclyde, Glasgow, UK;Department of Computer and Information Sciences, University of Strathclyde, Glasgow, UK;Department of Computer Science, Imperial College, London, UK
Venue:
Electronic Notes in Theoretical Computer Science (ENTCS)
Year:
2009

Citing 5
Cited 2

Mathematical theory of domains

Mathematical theory of domains
Refinement Calculus: A Systematic Introduction

Refinement Calculus: A Systematic Introduction
On Continuous Normalization

CSL '02 Proceedings of the 16th International Workshop and 11th Annual Conference of the EACSL on Computer Science Logic
Containers: constructing strictly positive types

Theoretical Computer Science - Applied semantics: Selected topics
∂ for Data: Differentiating Data Structures

Fundamenta Informaticae - Typed Lambda Calculi and Applications 2003, Selected Papers

Mixed inductive/coinductive types and strong normalization

APLAS'07 Proceedings of the 5th Asian conference on Programming languages and systems
Proofs, programs, processes

CiE'10 Proceedings of the Programs, proofs, process and 6th international conference on Computability in Europe

Quantified Score

Hi-index	0.00

Visualization

Abstract

In a previous paper we gave a representation of, and simultaneously a way of programming with, continuous functions on streams, whether discrete-valued functions, or functions between streams. We also defined a combinator on the representations of such continuous functions that reflects composition. Streams are one of the simplest examples of non-trivial final coalgebras. Here we extend our previous results to cover the case of final coalgebras for a broader class of functors than that giving rise to streams. Among the functors we can deal with are those that arise from countable signatures of finite-place untyped operators. These have many applications. The topology we put on the final coalgebra for such a functor is that induced by taking for basic neighbourhoods the set of infinite objects which share a common 'prefix', a la Baire space. The datatype of prefixes is defined together with the set of 'growth points' in a prefix, simultaneously. This we call beheading. To program and reason about representations of continuous functions requires a language whose type system incorporates the dependent function and pair types, inductive definitions at types Set, I-Set and (@SI:Set)Set^I, coinductive definitions at types Set and I-Set, as well as universal arrows for such definitions.