Computable classifications of continuous, transducer, and regular functions

TL;DR

This paper introduces a unified algorithmic framework for classifying continuous, transducer, and regular functions, establishing complexity results and characterizations using computability, automata, and functional analysis.

Contribution

It provides the first systematic classification of these functions, proving $oldsymbol{ ext{Sigma}}^0_2$-completeness and characterizing transducer functions as continuous regular functions.

Findings

Classification problem for continuous regular functions is $oldsymbol{ ext{Sigma}}^0_2$-complete.

Transducer functions are exactly the continuous regular functions.

The space $C[0,1]$ admits an arithmetical classification among Banach spaces.

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.