A Combinatorial Approach to Collapsing Words

Introduction New characterization Recognizing collapsing words Mono and stereo automata Questions Definitions We will consider finite deterministic automata with a total transition function: Remark Each element of Σ induces via δ a transformation of Q. We will view an automaton as a set of such transformations. Introduction New characterization Recognizing collapsing words Mono and stereo automata Questions Definitions We will consider finite deterministic automata with a total transition function: Remark Each element of Σ induces via δ a transformation of Q. We will view an automaton as a set of such transformations. Introduction New characterization Recognizing collapsing words Mono and stereo automata Questions Definitions Deficiency Given an automaton A and a word w ∈ Σ * we will be interested in the value of df A (w) = |Q| − |Qw| called the deficiency of w with respect to A. n-compressing words We say that w ∈ Σ * is n-compressing for A if df A (w) ≥ n (for n = |Q| − 1 such a word synchronizes A). Introduction New characterization Recognizing collapsing words Mono and stereo automata Questions Definitions Deficiency Given an automaton A and a word w ∈ Σ * we will be interested in the value of df A (w) = |Q| − |Qw| called the deficiency of w with respect to A. n-compressing words We say that w ∈ Σ * is n-compressing for A if df A (w) ≥ n (for n = |Q| − 1 such a word synchronizes A). Introduction New characterization Recognizing collapsing words Mono and stereo automata Questions Definitions Example w = baab 1 2 3 4 a a a b a b b b df A (w) = |{1, 2, 3, 4}| − |{1, 3}| = 2, w is 2-compressing for A. Introduction New characterization Recognizing collapsing words Mono and stereo automata Questions Definitions Example w = baab 1 2 3 4 a a a b a b b b df A (w) = |{1, 2, 3, 4}| − |{1, 3}| = 2, w is 2-compressing for A. Introduction New characterization Recognizing collapsing words Mono and stereo automata Questions Definitions Example w = baab 1 2 3 4 a a a b a b b b df A (w) = |{1, 2, 3, 4}| − |{1, 3}| = 2, w is 2-compressing for A. Introduction New characterization Recognizing collapsing words Mono and stereo automata Questions Definitions Example w …