Recursion and topology on 2^⩽ω for possibly infinite computations

Abstract

In the context of possibly infinite computations yielding finite or infinite (binary) outputs, the space $\text{2}{}^{\mathrm{⩽\omega }}\text{=}\text{2}{}^{\mathrm{\ast }}\text{∪}\text{2}{}^{\mathrm{\omega }}$ appears to be one of the most fundamental spaces in Computer Science. Though underconsidered, next to 2^ω, this space can be viewed (Section 3.5.2) as the simplest compact space native to computer science.

In this paper we study some of its properties involving topology and computability.

Though 2^⩽ω can be considered as a computable metric space in the sense of computable analysis, a direct and self-contained study, based on its peculiar properties related to words, is much illuminating.

It is well known that computability for maps $\text{2}{}^{\mathrm{\omega }}\text{→}\text{2}{}^{\mathrm{\omega }}$ reduces to continuity with recursive modulus of continuity. With 2^⩽ω, things get less simple. Maps $\text{2}{}^{\mathrm{\omega }}\text{→}\text{2}{}^{\mathrm{⩽\omega }}$ or $\text{2}{}^{\mathrm{⩽\omega }}\text{→}\text{2}{}^{\mathrm{⩽\omega }}$ induced by input/output behaviours of Turing machines on finite or infinite words—which we call semicomputable maps—are not necessarily continuous but merely lower semicontinuous with respect to the prefix partial ordering on 2^⩽ω. Continuity asks for a stronger notion of computability.

We prove for (semi)continuous and (semi)computable maps $\text{F}\text{:}\text{I}\text{→}\text{O}$ with $\text{I}\text{,}\text{O}\text{∈{2}{}^{\mathrm{\omega }}\text{,2}{}^{\mathrm{⩽\omega }}\text{}}$ a detailed representation theorem (Theorem 81) via functions $\text{f}\text{:}\text{2}{}^{\mathrm{\ast }}\text{→}\text{2}{}^{\mathrm{\ast }}$ following two approaches: bottom-up from f to F and top-down from F to f.