# A Type of Partial Recursive Functions

• Ana Bove
• Venanzio Capretta
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5170)

## Abstract

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.

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## Copyright information

© Springer-Verlag Berlin Heidelberg 2008

## Authors and Affiliations

• Ana Bove
• 1
• Venanzio Capretta
• 2
1. 1.Department of Computer Science and EngineeringChalmers University of TechnologyGöteborgSweden
2. 2.Computer Science Institute (ICIS)Radboud University NijmegenThe Netherlands