A Type of Partial Recursive Functions
Paper i proceeding, 2008

Our goal is to define a type of partial recursive functions in constructive type theory. In a series of previous articles, we studied two different formulations of partial functions and general recursion. We could obtain a type only by extending the theory with either an impredicative universe or with coinductive definitions. Here we present a new type constructor that eludes such entities of dubious constructive credentials. We start by showing how to break down a recursive function definition into three components: the first component generates the arguments of the recursive calls, the second evaluates them, and the last computes the output from the results of the recursive calls. We use this dissection as the basis for the introduction rule of the new type constructor. Every partial recursive function is associated with an inductive domain predicate; evaluation of the function requires a proof that the input values satisfy the predicate. We give a constructive justification for the new construct by interpreting it into the base type theory. This shows that the extended theory is consistent and constructive.

type theory

partial functions

general recursion


Ana Bove

Chalmers, Data- och informationsteknik, Datavetenskap

Venanzio Capretta

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

03029743 (ISSN) 16113349 (eISSN)

Vol. LNCS 5170 102--117-
978-3-540-71065-3 (ISBN)


Datavetenskap (datalogi)



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