A Combinatorial Approach to Collapsing Words

Given a word w over a finite alphabet Σ and a finite deterministic automaton \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathcal A} = {\langle} Q,\Sigma,\delta {\rangle}$\end{document}, the inequality |δ(Q,w)| ≤|Q|–n means that under the natural action of the word w the image of the state set Q is reduced by at least n states. The word w is n-collapsing if this inequality holds for any deterministic finite automaton that satisfies such an inequality for at least one word. In this paper we present a new approach to the topic of collapsing words, and announce a few results we have obtained using this new approach. In particular, we present a direct proof of the fact that the language of n-collapsing words is recursive.