Quotient Complexities of Atoms of Regular Languages

Brzozowski, Janusz; Tamm, Hellis

doi:10.1007/978-3-642-31653-1_6

Cited by 16 publications

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“…It was proved in [7] that the complexity of the atoms with 0 or n complemented quotients is bounded from above by 2 n −1, and the complexity of any atom with r complemented quotients, where 1 r n − 1, by…”

Section: A4mentioning

confidence: 99%

“…The positive closure of a language L is L + = ∞ n=1 L n , and the Kleene closure or star of L is L * = ∞ n=0 L n = L + ∪ {ε}. An atom a [6,7] of a regular language L with quotients K 0 , . .…”

Section: Introductionmentioning

confidence: 99%

“…a Atoms of regular languages were introduced in 2011 by Brzozowski and Tamm [6], and the theory was slightly modified in 2012 [7]. The newer model, which admits up to 2 n atoms, is used here.…”

Section: Introductionmentioning

confidence: 99%

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In Search of Most Complex Regular Languages

Brzozowski

2012

Implementation and Application of Automata

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Received (Day Month Year) Accepted (Day Month Year) Communicated by (xxxxxxxxxx)Sequences (Ln | n k), called streams, of regular languages Ln are considered, where k is some small positive integer, n is the state complexity of Ln, and the languages in a stream differ only in the parameter n, but otherwise, have the same properties. The following measures of complexity are proposed for any stream: 1) the state complexity n of Ln, that is, the number of left quotients of Ln (used as a reference); 2) the state complexities of the left quotients of Ln; 3) the number of atoms of Ln; 4) the state complexities of the atoms of Ln; 5) the size of the syntactic semigroup of Ln; and the state complexities of the following operations: 6) the reverse of Ln; 7) the star of Ln; 8) union, intersection, difference and symmetric difference of Lm and Ln; and 9) the concatenation of Lm and Ln. A stream that has the highest possible complexity with respect to these measures is then viewed as a most complex stream. The language stream (Un(a, b, c) | n 3) is defined by the deterministic finite automaton with state set {0, 1, . . . , n−1}, initial state 0, set {n−1} of final states, and input alphabet {a, b, c}, where a performs a cyclic permutation of the n states, b transposes states 0 and 1, and c maps state n − 1 to state 0. This stream achieves the highest possible complexities with the exception of boolean operations where m = n. In the latter case, one can use Un(a, b, c) and Un(b, a, c), where the roles of a and b are interchanged in the second language. In this sense, Un(a, b, c) is a universal witness. This witness and its extensions also apply to a large number of combined regular operations.

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Section: A4mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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In Search of Most Complex Regular Languages

Brzozowski

2012

Implementation and Application of Automata

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“…• Since |{AS | S ⊆ QL}| ≤ 2 n and since every element in BQ(L) is a union of atoms, we have |BQ(L)| ≤ 2 is optimal: It is proved in [6] that for every n ≥ 2 there exists a language L of complexity n with 2 n atoms. As AS ∩ AT = ∅ for S = T , the atoms form a partition of Σ * .…”

Section: Introductionmentioning

confidence: 99%

“…

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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Asymptotic Approximation for the Quotient Complexities of Atoms

Diekert¹,

Walter²

2015

Acta Cybern

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In 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. In each case let κC(n) be the maximal complexity of an atom of a regular language L, where L has complexity n ≥ 2 and belongs to the class C ∈ {G, R, L, T , S}. It is known that κT (n) ≤ κL(n) = κR(n) ≤ κG(n) < 3 n and κS (n) = κL(n − 1). We show that the ratiotends exponentially fast to 3 in all five cases but it remains different from 3. This behaviour was suggested by experimental results of Brzozowski and Tamm; and the result for G was shown independently by Luke Schaeffer and the first author soon after the paper of Brzozowski and Tamm appeared in 2012. However, proofs for the asymptotic behavior ofwere never published; and the results here are valid for all five classes above. Moreover, there is an interesting oscillation for all C: for almost all n we have κ C (n) κ C (n−1) > 3 if and only if κ C (n+1) κ C (n) < 3.

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Most Complex Regular Right-Ideal Languages

Brzozowski

Davies

2014

Descriptional Complexity of Formal Systems

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Abstract.A right ideal is a language L over an alphabet Σ that satisfies the equation L = LΣ * . We show that there exists a sequence (Rn | n 3) of regular right-ideal languages, where Rn has n left quotients and is most complex among regular right ideals under the following measures of complexity: the state complexities of the left quotients, the number of atoms (intersections of complemented and uncomplemented left quotients), the state complexities of the atoms, the size of the syntactic semigroup, the state complexities of reversal, star, product, and all binary boolean operations that depend on both arguments. Thus (Rn | n 3) is a universal witness reaching the upper bounds for these measures.

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Quotient Complexities of Atoms of Regular Languages

Abstract: The definition in [3] does not consider the intersection of all the complemented quotients to be an atom. Our new definition adds symmetry to the theory.

Cited by 16 publications

References 9 publications

In Search of Most Complex Regular Languages

In Search of Most Complex Regular Languages

Asymptotic Approximation for the Quotient Complexities of Atoms

Most Complex Regular Right-Ideal Languages

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