Lower Bounds on Words Separation: Are There Short Identities in Transformation Semigroups?

TL;DR

This paper investigates the separation of words by finite automata and the length of identities in transformation semigroups, providing new shorter identities and improved lower bounds for automata state complexity.

Contribution

It introduces the first series of shorter identities in transformation semigroups, improving lower bounds on words separation, and presents new short identities in symmetric groups.

Findings

Shorter identities in $T_k$ for infinitely many $k$

Improved lower bounds on $Sep(n)$

Computer search results for small $k$ identities

Abstract

The words separation problem, originally formulated by Goralcik and Koubek (1986), is stated as follows. Let $Sep(n)$ be the minimum number such that for any two words of length $\le n$ there is a deterministic finite automaton with $Sep(n)$ states, accepting exactly one of them. The problem is to find the asymptotics of the function $Sep$. This problem is inverse to finding the asymptotics of the length of the shortest identity in full transformation semigroups $T_k$. The known lower bound on $Sep$ stems from the unary identity in $T_k$. We find the first series of identities in $T_k$ which are shorter than the corresponding unary identity for infinitely many values of $k$, and thus slightly improve the lower bound on $Sep(n)$. Then we present some short positive identities in symmetric groups, improving the lower bound on separating words by permutational automata by a multiplicative constant. Finally, we present the results of computer search for short identities for small $k$.