A Framework for the Extended Notion of Computation Implied by the Limiting Recursive Functions

Abstract

The halting property as such is not guaranteed for limitimg recursion, since its evaluation process requires infinitely many steps. Nevertheless, as limiting recursive functions are useful in the study of computational aspects of mathematics on the continuum, it is desirable that some extended notion of computation be bestowed upon limiting recursion. We will here present a framework which general recursion and limiting recursion can share in characterizing their functions, and in which the halting property of the former and the identification in the limit of the latter can be expressed in terms of a formula stating a kind of compactness of a sequence of sets. We regard this fact as claiming that limiting recursion is conjunctive to general recursion, and hence can be viewed as a computation in an extended sense.