Unary finite automata vs. arithmetic progressions

TL;DR

This paper identifies and corrects a subtle error in the proof of Chrobak's theorem regarding unary automata and arithmetic progressions, and discusses the computational complexity of Martinez's algorithm.

Contribution

The paper corrects the proof of Chrobak's theorem and adapts Martinez's polynomial-time algorithm to be correct, clarifying its space complexity limitations.

Findings

Corrected proof of Chrobak's theorem

Adjusted Martinez's algorithm for correctness

Established space complexity limitations of the algorithm

Abstract

We point out a subtle error in the proof of Chrobak's theorem that every unary NFA can be represented as a union of arithmetic progressions that is at most quadratically large. We propose a correction for this and show how Martinez's polynomial time algorithm, which realizes Chrobak's theorem, can be made correct accordingly. We also show that Martinez's algorithm cannot be improved to have logarithmic space, unless L = NL.