Complexity of Atoms of Regular Languages

Abstract

The quotient complexity of a regular language L, which is the same as its state complexity, is the number of left quotients of L. An atom of a non-empty regular language L with n quotients is a non-empty intersection of the n quotients, which can be uncomplemented or complemented. An NFA is atomic if the right language of every state is a union of atoms. We characterize all reduced atomic NFAs of a given language, i.e., those NFAs that have no equivalent states. We prove that, for any language L with quotient complexity n, the quotient complexity of any atom of L with r complemented quotients has an upper bound of 2 n − 1 if r = 0 or r = n; for 1 r n − 1 the bound isFor each n 2, we exhibit a language with 2 n atoms which meet these bounds.Keywords: atoms, finite automaton, atomic NFA, quotient complexity, regular language, state complexity, syntactic semigroup, witness 2000 Mathematics Subject Classification: 68Q45, 68Q19, 68Q70 Terminology and NotationIn this section we provide some background information, introduce atoms of regular languages, and state our reasons for studying them. For basic properties of regular languages and finite automata see [7,9]. If Σ is a non-empty finite alphabet, then Σ * is the free monoid generated by Σ. A word is any element of Σ * , and the empty word is ε. A language over Σ is any subset of Σ * . The reverse of a language L is denoted by L R and defined as L R = {w R | w ∈ L}, where w R is w spelled backwards. The (left) quotient of a regular language L over an alphabet Σ by a word w ∈ Σ * is the language w −1 L = {x ∈ Σ * | wx ∈ L}. It is well known that the quotient of

Theory of átomata

“…Also, with the negative atom, a language L and its complement language L have the same atoms. Finally, we have symmetry between the atoms with 0 and n complemented quotients, and the same upper bounds on quotient complexity for both, as was shown in [6].…”

“…For completeness we mention that the quotient complexity (equivalent to state complexity) of atoms of regular languages was studied in [6] and [4].…”

“…The following properties of reduced atomic NFAs were proved in [6]. A similar approach was used more informally by Sengoku [16].…”

Most Complex Non-returning Regular Languages

“…An equivalence class of this relation is an atom of L [7]. Thus an atom is a non-empty intersection of complemented and uncomplemented quotients of L. The number of atoms and their state complexities were suggested as possible measures of complexity of regular languages [2], because all the quotients of a language, and also the quotients of atoms, are always unions of atoms [6,7,11].…”

Asymptotic Approximation for the Quotient Complexities of Atoms

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AbstractIn a series of papers, Brzozowski together with Tamm, Davies, and Szyku la studied the quotient complexities of atoms of regular languages [6,7,3,4]. The authors obtained precise bounds in terms of binomial sums for the most complex situations in the following five cases: (G): general, (R): right ideals, (L): left ideals, (T ): two-sided ideals and (S): suffix-free languages.

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