Abstract
Automata, Logic and Semantics Given a word w over a finite alphabet Sigma and a finite deterministic automaton A = < Q,Sigma,delta >, the inequality vertical bar delta(Q,w)vertical bar <= vertical bar Q vertical bar - k means that under the natural action of the word w the image of the state set Q is reduced by at least k states. The word w is k-collapsing (k-synchronizing) if this inequality holds for any deterministic finite automaton ( with k + 1 states) that satisfies such an inequality for at least one word. We prove that for each alphabet Sigma there is a 2-collapsing word whose length is vertical bar Sigma vertical bar(3)+6 vertical bar Sigma vertical bar(2)+5 vertical bar Sigma vertical bar/2. Then we produce shorter 2-collapsing and 2-synchronizing words over alphabets of 4 and 5 letters.
Highlights
Introduction and preliminariesIn this paper by an automaton A = Q, Σ, δ we mean a finite deterministic automaton with state set Q, input alphabet Σ, and transition function δ : Q × Σ → Q
The word w is k-collapsing (k-synchronizing) if this inequality holds for any deterministic finite automaton that satisfies such an inequality for at least one word
Otherwise if (Π, Υ) is a STEREO-role assignment (Π, {E1, E2}) we denote by Sk, k = 1, 2, the sets of the inner factors pi of w such that the last letter of ui−1 belongs to Ek
Summary
Given a word w over a finite alphabet Σ and a finite deterministic automaton A = Q, Σ, δ , the inequality |δ(Q, w)| ≤. |Q| − k means that under the natural action of the word w the image of the state set Q is reduced by at least k states. The word w is k-collapsing (k-synchronizing) if this inequality holds for any deterministic finite automaton (with k + 1 states) that satisfies such an inequality for at least one word. We prove that for each alphabet Σ there is a 2-
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