Computable classifications of continuous, transducer, and regular functions

Abstract

We develop a systematic algorithmic framework that unites global and local classification problems using index sets. We prove that the classification problem for continuous (binary) regular functions among almost everywhere linear, pointwise linear-time Lipschitz functions is $\Sigma^0_2$-complete. (Every regular function is pointwise linear-time Lipschitz.) We show that a function $f\colon [0,1] \rightarrow \mathbb{R}$ is (binary) transducer if and only if it is continuous regular. 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.