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Finiteness and rational sequences, constructively*

Part of: JFP Research Articles

Published online by Cambridge University Press:  05 April 2017

TARMO UUSTALU  and
NICCOLÒ VELTRI
TARMO UUSTALU
Affiliation:
Institute of Cybernetics, Tallinn University of Technology, Akadeemia tee 21, 12618 Tallinn, Estonia (e-mails: tarmo@cs.ioc.ee, niccolo@cs.ioc.ee)
NICCOLÒ VELTRI
Affiliation:
Institute of Cybernetics, Tallinn University of Technology, Akadeemia tee 21, 12618 Tallinn, Estonia (e-mails: tarmo@cs.ioc.ee, niccolo@cs.ioc.ee)
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Abstract

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Rational sequences are possibly infinite sequences with a finite number of distinct suffixes. In this paper, we present different implementations of rational sequences in Martin–Löf type theory. First, we literally translate the above definition of rational sequence into the language of type theory, i.e., we construct predicates on possibly infinite sequences expressing the finiteness of the set of suffixes. In type theory, there exist several inequivalent notions of finiteness. We consider two of them, listability and Noetherianness, and show that in the implementation of rational sequences the two notions are interchangeable. Then we introduce the type of lists with backpointers, which is an inductive implementation of rational sequences. Lists with backpointers can be unwound into rational sequences, and rational sequences can be truncated into lists with backpointers. As an example, we see how to convert the fractional representation of a rational number into its decimal representation and vice versa.

Type
Articles
Information
Journal of Functional Programming , Volume 27 , 2017 , e13
Copyright
Copyright © Cambridge University Press 2017 

Footnotes

*

This work was supported by the ERDF funded project Coinduction, the Estonian Ministry of Education and Research institutional research grant no. IUT33-13 and the Estonian Science Foundation grant no. 9475.

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