Quotient Complexities of Atoms in Regular Ideal Languages

Abstract

A (left) quotient of a language L by a word w is the language w −1L = {x | wx ϵ 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 Σ that satisfies L = LΣ* (respectively, L = Σ*L and L = Σ*LΣ*). We compute the maximal number of atoms and the maximal quotient complexities of atoms of right, left and two-sided regular ideals.