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Quotient Complexities of Atoms in Regular Ideal Languages

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Date

2015

Authors

Brzozowski, Janusz
Davies, Sylvie

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Institute of Informatics: University of Szeged

Abstract

A (left) quotient of a language L by a word w is the language w(-1) L = {x vertical bar wx is an element of L}. The quotient complexity of a regular language L is the number of quotients of L; it is equal to the state complexity of L, which is the number of states in a minimal deterministic finite automaton accepting L. An atom of L is an equivalence class of the relation in which two words are equivalent if for each quotient, they either are both in the quotient or both not in it; hence it is a non-empty intersection of complemented and uncomplemented quotients of L. A right (respectively, left and two-sided) ideal is a language L over an alphabet Sigma that satisfies L = L Sigma* (respectively, L = Sigma*L and L = Sigma*L Sigma*). We compute the maximal number of atoms and the maximal quotient complexities of atoms of right, left and two-sided regular ideals.

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Keywords

atom, left ideal, quotient, quotient complexity, regular language, right ideal, state complexity, syntactic semigroup, two-sided ideal

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