Real-world computers and recursive function theory

Abstract

Is there a connection between the classical theory of computation based on the mathematics of recursive function theory, and todays real-world electronic computers? While recursive functions are usually defined as functions of natural numbers, computers, as we know, transform tapes and sequences of binary data. The purpose of this paper is to discuss the linkage between these two worlds. A topological approach is applied here to model real-world electronic computers as functions mapping one topological space into another. The domain and range of the computer function is the set of all infinite binary sequences, and it is viewed here as a countable extension of the real-line. Using a simple order topology on this space, it is shown that computers can actually be described as continuous functions on these spaces.