Composing short 3-compressing words on a 2 letter alphabet

TL;DR

This paper develops a method to find short words that compress the state set of automata by at least three states on a two-letter alphabet, providing bounds on the shortest such words.

Contribution

It introduces a systematic approach to compute short 3-compressing words and establishes bounds for the shortest 3-collapsing words on a two-letter alphabet.

Findings

Constructed a set of short words for 3-compressibility

Derived a 3-collapsing word of length 53

Established bounds 34 ≤ c(2,3) ≤ 53 for shortest 3-collapsing words

Abstract

A finite deterministic (semi)automaton $\mathcal{A} =(Q,\Sigma,\delta)$ is $k$-compressible if there is some word $w\in \Sigma^+$ such that the image of its state set $Q$ under the natural action of $w$ is reduced by at least $k$ states. Such word, if it exists, is called a $k$-compressing word for $\mathcal{A}$. A word is $k$-collapsing if it is $k$-compressing for each $k$-compressible automaton. We compute a set $W$ of short words such that each $3$-compressible automata on a two letter alphabet is $3$-compressed at least by a word in $W$. Then we construct a shortest common superstring of the words in $W$ and, with a further refinement, we obtain a $3$-collapsing word of length $53$. Moreover, as previously announced, we show that the shortest $3$-synchronizing word is not $3$-collapsing, illustrating the new bounds $34\leq c(2,3)\leq 53$ for the length $c(2,3)$ of the shortest $3$-collapsing word on a two letter alphabet.