## Alexi Block Gorman ; Philipp Hieronymi ; Elliot Kaplan ; Ruoyu Meng ; Erik Walsberg et al. - Continuous Regular Functions

lmcs:5301 - Logical Methods in Computer Science, February 14, 2020, Volume 16, Issue 1 - https://doi.org/10.23638/LMCS-16(1:17)2020
Continuous Regular Functions

Authors: Alexi Block Gorman ; Philipp Hieronymi ; Elliot Kaplan ; Ruoyu Meng ; Erik Walsberg ; Zihe Wang ; Ziqin Xiong ; Hongru Yang

Following Chaudhuri, Sankaranarayanan, and Vardi, we say that a function $f:[0,1] \to [0,1]$ is $r$-regular if there is a Büchi automaton that accepts precisely the set of base $r \in \mathbb{N}$ representations of elements of the graph of $f$. We show that a continuous $r$-regular function $f$ is locally affine away from a nowhere dense, Lebesgue null, subset of $[0,1]$. As a corollary we establish that every differentiable $r$-regular function is affine. It follows that checking whether an $r$-regular function is differentiable is in $\operatorname{PSPACE}$. Our proofs rely crucially on connections between automata theory and metric geometry developed by Charlier, Leroy, and Rigo.

Volume: Volume 16, Issue 1
Published on: February 14, 2020
Accepted on: February 14, 2020
Submitted on: March 21, 2019
Keywords: Computer Science - Logic in Computer Science,Mathematics - Logic