Computable classifications of continuous, transducer, and regular functions
Johanna N. Y. Franklin, Rupert Hölzl, Alexander Melnikov, Keng Meng Ng, Daniel Turetsky
TL;DR
The paper develops an index-set framework to unify global and local classification problems for functions in $C[0,1]$, establishing that the problem of recognizing continuous binary regular functions among almost everywhere linear, pointwise linear-time Lipschitz functions is $\Sigma^0_2$-complete, and proving that continuous regular functions coincide with transducer computations under suitable representations. It further shows that the global problem of classifying $C[0,1]$ among separable Banach spaces is arithmetical, with an upper bound of $\Sigma^0_7$, using a notion of local independence to approximate a standard basis. The results bridge abstract computability, automata theory, and functional analysis, illustrating both the limits of simple characterizations and a surprising tractability for identifying $C[0,1]$ among separable spaces. Overall, the work provides a coherent framework that connects local computability (via automata) with global structural classification in analysis.
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 $Σ^0_2$-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 $Σ^0_2$-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.