Type Two Theory of Effectivity and Real PCF

Abstract

Listen

Translate article

There have been many suggestions for what should be a computable real number or function. Some of these exhibited pathological properties [16]. Presently, research concentrates on domain theoretic approaches [4, 9] based on an idea of Scott [15], an application of Weihrauch's Type Two Theory of Effectivity [19, 20, 21] and an approach introduced by Pour-El and Richards for the more general case of Banach spaces [13].All these approaches are claimed to be equivalent, but only in a few cases proofs have been given [3].In this paper we show that a real number as well as a function on the reals are computable in Weihrauch's approach if and only if they are definable in Real PCF [5,6], an extension of the functional language PCF [12] by new ground type representing intervals which approximate real numbers.