Maximally Atomic Languages

Brzozowski, Janusz; Davies, Gareth

doi:10.4204/eptcs.151.10

Cited by 4 publications

(12 citation statements)

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“…In [3,5], the authors achieved results on properties of atoms such as the number and state complexity of individual atoms, via studying the "átomaton" of L, which is a nondeterministic automaton, actually being isomorphic to the reversal of the determinized reversal of M . In this paper we suggest another way to study atoms and reprove the characterization of the so-called maximally atomic languages.…”

Section: Atoms Of a Regular Languagementioning

confidence: 99%

“…Employing the notion from [3], we introduce to each 0 ≤ k ≤ n the set of (n, k)-type states as follows:…”

Section: Atoms Of a Regular Languagementioning

confidence: 99%

“…In [3], an atom A S is said to have maximal complexity, or is simply called maximal, if its state complexity is Ψ(n, |S|). That is, if every (n, |S|)-state is indeed reachable from (S, S) and are pairwise distinguishable.…”

Section: Atoms Of a Regular Languagementioning

confidence: 99%

“…A language is defined to be maximally atomic in [3] if it has the maximal number of atoms possible and each of the individual atoms has the maximal possible state complexity. In [2], Brzozowski and Davies showed that maximal syntactic complexity implies maximal atomicity and in [3] they gave necessary and sufficient conditions for a language to be maximally atomic.…”

Section: Introductionmentioning

confidence: 99%

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Complexity of atoms, combinatorially

Iván

2016

Information Processing Letters

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Atoms of a (regular) language L were introduced by Brzozowski and Tamm in 2011 as intersections of complemented and uncomplemented quotients of L. They derived tight upper bounds on the complexity of atoms in 2013. In 2014, Brzozowski and Davies characterized the regular languages meeting these bounds. To achieve these results, they used the so-called "átomaton" of a language, introduced by Brzozowski and Tamm in 2011.In this note we give an alternative proof of their characterization, via a purely combinatorial approach.

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Section: Atoms Of a Regular Languagementioning

confidence: 99%

“…Employing the notion from [3], we introduce to each 0 ≤ k ≤ n the set of (n, k)-type states as follows:…”

Section: Atoms Of a Regular Languagementioning

confidence: 99%

Section: Atoms Of a Regular Languagementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Complexity of atoms, combinatorially

Iván

2016

Information Processing Letters

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“…Below we present an alternative characterization of atoms, which we use in our proofs. Earlier papers on atoms such as [3,8,9] take this as the definition of atoms, for it was not known until recently that atoms may be viewed as congruence classes (this fact was first noticed by Iván in [11]).…”

Section: Atomsmentioning

confidence: 99%

Quotient Complexities of Atoms in Regular Ideal Languages

Brzozowski¹,

Davies²

2015

Acta Cybern

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A (left) quotient of a language L by a word w is the language w −1 L = {x | wx ∈ 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.

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Symmetric Groups and Quotient Complexity of Boolean Operations

Bell

Brzozowski

Moreira

et al. 2014

Automata, Languages, and Programming

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The quotient complexity of a regular language L is the number of left quotients of L, which is the same as the state complexity of L. Suppose that L and L ′ are binary regular languages with quotient complexities m and n, and that the transition semigroups of the minimal deterministic automata accepting L and L ′ are the symmetric groups Sm and Sn of degrees m and n, respectively. Denote by • any binary boolean operation that is not a constant and not a function of one argument only. For m, n ≥ 2 with (m, n) ∈ {(2, 2), (3, 4), (4, 3), (4, 4)} we prove that the quotient complexity of L • L ′ is mn if and only either (a) m = n or (b) m = n and the bases (ordered pairs of generators) of Sm and Sn are not conjugate. For (m, n) ∈ {(2, 2), (3, 4), (4, 3), (4, 4)} we give examples to show that this need not hold. In proving these results we generalize the notion of uniform minimality to direct products of automata. We also establish a non-trivial connection between complexity of boolean operations and group theory. MotivationThe left quotient, or simply quotient, of a regular language L over an alphabet Σ by a word w ∈ Σ * is the regular language w −1 L = {x ∈ Σ * : wx ∈ L}. It is well known that a language is regular if and only if it has a finite number of ⋆

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Maximally Atomic Languages

Cited by 4 publications

References 5 publications

Complexity of atoms, combinatorially

Complexity of atoms, combinatorially

Quotient Complexities of Atoms in Regular Ideal Languages

Symmetric Groups and Quotient Complexity of Boolean Operations

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