| A Type of Partial Recursive Functions (2009) | |||||||||||||||
Abstract | |||||||||||||||
| 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. In both cases, 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 one evaluates them, and the last one 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: a partial recursive function is created by giving the first and third of the above components. As in one of our previous methods, every partial recursive function is associated with an inductive domain predicate and the evaluation of the function requires a proof that the predicate holds on the input values. We give a constructive justification for the new construct by means of an interpretation from the extended type theory into the base one. This shows that the extended theory is consistent and constructive. 1 | |||||||||||||||
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