Computable Classifications of Continuous, Transducer, and Regular Functions

Abstract

We develop a systematic algorithmic framework that unites global and local classification problems for functional separable spaces and apply it to attack classification problems concerning the Banach space C[0,1] of real-valued continuous functions on the unit interval. We prove that the classification problem for continuous (binary) regular functions among almost everywhere linear, pointwise linear-time Lipshitz functions is $\Sigma^0_2$-complete. We show that a function $f\colon [0,1] \rightarrow \mathbb{R}$ is (binary) transducer if and only if it is continuous regular; interestingly, this peculiar and nontrivial fact was overlooked by experts in automata theory. As one of many consequences, our $\Sigma^0_2$-completeness result covers the class of transducer functions as well. Finally, we show that the Banach space $C[0,1]$ of real-valued continuous functions admits an arithmetical classification among separable Banach spaces. Our proofs combine methods of abstract computability theory, automata theory, and functional analysis.