Decidability of regular language genus computation

TL;DR

This paper investigates the genus of regular languages, introduces the concept of topological size, and proves the conjecture that the genus is computable for a broad class of languages, including planarity.

Contribution

It generalizes previous results by constructing high-genus regular languages and introduces the topological size to analyze automata complexity.

Findings

Minimal automata can differ greatly in genus and size from automata of minimal genus.

Topological size can grow exponentially relative to language size.

Conjecture that genus is computable is proven for languages without short cycles.

Abstract

The article continues the study of the genus of regular languages that the authors introduced in a 2012 paper. Generalizing a previous result, we produce a new family of regular languages on a two-letter alphabet having arbitrary high genus. Let $L$ be a regular language. In order to understand the genus $g(L)$ of $L$, we introduce the topological size of $|L|_{\rm{top}}$ to be the minimal size of all finite deterministic automata of genus $g(L)$ computing $L$. We show that the minimal finite deterministic automaton of a regular language can be arbitrary far away from a finite deterministic automaton realizing the minimal genus and computing the same language, both in terms of the difference of genera and in terms of the difference in size. In particular, we show that the topological size $|L|_{\rm{top}}$ can grow at least exponentially in size $|L|$. We conjecture however the genus of every regular language to be computable. This conjecture implies in particular that the planarity of a regular language is decidable, a question asked in 1976 by R.V. Book and A.K. Chandra. We prove here the conjecture for a fairly generic class of regular languages having no short cycles.