On the Gap Between Separating Words and Separating Their Reversals

TL;DR

This paper demonstrates that the difference in the minimal number of states needed for a DFA to separate two words and their reversals can be arbitrarily large over a binary alphabet, resolving an open problem in automata theory.

Contribution

It proves that the difference in separation complexity between words and their reversals is unbounded, answering an open question in the field.

Findings

The difference |sep(w,x)-sep(w^R,x^R)| can be arbitrarily large.

This unbounded difference holds over a binary alphabet.

The result addresses a previously open problem in automata theory.

Abstract

A deterministic finite automaton (DFA) separates two strings $w$ and $x$ if it accepts $w$ and rejects $x$. The minimum number of states required for a DFA to separate $w$ and $x$ is denoted by $sep(w,x)$. The present paper shows that the difference $|sep(w,x)-sep(w^R,x^R)|$ is unbounded for a binary alphabet; here $w^R$ stands for the mirror image of $w$. This solves an open problem stated in [Demaine, Eisenstat, Shallit, Wilson: Remarks on separating words. DCFS 2011. LNCS vol. 6808, pp. 147-157.]