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Well-founded coalgebras, revisited

Published online by Cambridge University Press:  09 February 2016

JEAN-BAPTISTE JEANNIN ,
DEXTER KOZEN  and
ALEXANDRA SILVA
JEAN-BAPTISTE JEANNIN
Affiliation:
Computer Science, Cornell University, Ithaca, New York 14853-7501, U.S.A. Email: kozen@cs.cornell.edu
DEXTER KOZEN
Affiliation:
Computer Science, Cornell University, Ithaca, New York 14853-7501, U.S.A. Email: kozen@cs.cornell.edu
ALEXANDRA SILVA
Affiliation:
Intelligent Systems, Radboud University Nijmegen, Postbus 9010, 6500 GL Nijmegen, the Netherlands Email: alexandra@cs.ru.nl
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Abstract

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Theoretical models of recursion schemes have been well studied under the names well-founded coalgebras, recursive coalgebras, corecursive algebras and Elgot algebras. Much of this work focuses on conditions ensuring unique or canonical solutions, e.g. when the coalgebra is well founded.

If the coalgebra is not well founded, then there can be multiple solutions. The standard semantics of recursive programs gives a particular solution, typically the least fixpoint of a certain monotone map on a domain whose least element is the totally undefined function; but this solution may not be the desired one. We have recently proposed programming language constructs to allow the specification of alternative solutions and methods to compute them. We have implemented these new constructs as an extension of OCaml.

In this paper, we prove some theoretical results characterizing well-founded coalgebras, along with several examples for which this extension is useful. We also give several examples that are not well founded but still have a desired solution. In each case, the function would diverge under the standard semantics of recursion, but can be specified and computed with the programming language constructs we have proposed.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 27 , Special Issue 7: Special Issue: Coalgebraic Logic , October 2017 , pp. 1111 - 1131
Copyright
Copyright © Cambridge University Press 2016 

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