Theory of átomata

Brzozowski, Janusz; Tamm, Hellis

doi:10.1016/j.tcs.2014.04.016

Cited by 51 publications

(96 citation statements)

References 9 publications

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“…Atoms are defined by the following left congruence: two words x and y are equivalent if ux ∈ L if and only if uy ∈ L for all u ∈ Σ * . Thus x and y are equivalent if x ∈ u −1 L if and only if y ∈ u −1 L. An equivalence class of this relation is an atom of L [17,21]. Thus an atom is a non-empty intersection of complemented and uncomplemented quotients of L. If K 0 , .…”

Section: Motivationmentioning

confidence: 99%

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Complexity of Right-Ideal, Prefix-Closed, and Prefix-Free Regular Languages

Brzozowski¹,

Sinnamon²

2017

Acta Cybern

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A language L over an alphabet Σ is prefix-convex if, for any words x, y, z ∈ Σ * , whenever x and xyz are in L, then so is xy. Prefix-convex languages include right-ideal, prefix-closed, and prefix-free languages as special cases. We examine complexity properties of these special prefix-convex languages. In particular, we study the quotient/state complexity of boolean operations, product (concatenation), star, and reversal, the size of the syntactic semigroup, and the quotient complexity of atoms. For binary operations we use arguments with different alphabets when appropriate; this leads to higher tight upper bounds than those obtained with equal alphabets. We exhibit right-ideal, prefix-closed, and prefix-free languages that meet the complexity bounds for all the measures listed above.

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Section: Motivationmentioning

confidence: 99%

“…. , n − 1}, then atom A S is the intersection of quotients with subscripts in S and complemented quotients with subscripts in Q n \ S. For more information about atoms see [16,17,21].…”

Section: Motivationmentioning

confidence: 99%

Complexity of Right-Ideal, Prefix-Closed, and Prefix-Free Regular Languages

Brzozowski¹,

Sinnamon²

2017

Acta Cybern

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“…From [7] we know that the number of atoms of a regular language is equal to the state complexity of the reversal of the language. Hence this also proves a conjecture from [9], that a ternary witness is necessary to meet the bound for reversal of non-returning languages.…”

Section: Number Of Atomsmentioning

confidence: 99%

Most Complex Non-returning Regular Languages

Brzozowski

Davies

2017

Descriptional Complexity of Formal Systems

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Abstract. A regular language L is non-returning if in the minimal deterministic finite automaton accepting it there are no transitions into the initial state. Eom, Han and Jirásková derived upper bounds on the state complexity of boolean operations and Kleene star, and proved that these bounds are tight using two different binary witnesses. They derived upper bounds for concatenation and reversal using three different ternary witnesses. These five witnesses use a total of six different transformations. We show that for each n 4 there exists a ternary witness of state complexity n that meets the bound for reversal and that at least three letters are needed to meet this bound. Moreover, the restrictions of this witness to binary alphabets meet the bounds for product, star, and boolean operations. We also derive tight upper bounds on the state complexity of binary operations that take arguments with different alphabets. We prove that the maximal syntactic semigroup of a non-returning language has (n − 1) n elements and requires at least n 2 generators. We find the maximal state complexities of atoms of non-returning languages. Finally, we show that there exists a most complex non-returning language that meets the bounds for all these complexity measures.

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“…These are languages that meet all the upper bounds on the state complexities of Boolean operations, product, star, and reversal, have maximal syntactic semigroups and most complex atoms [13].…”

Section: Introductionmentioning

confidence: 99%

Syntactic Complexity of Regular Ideals

Brzozowski

Szykuła

2017

Theory Comput Syst

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The state complexity of a regular language is the number of states in a minimal deterministic finite automaton accepting the language. The syntactic complexity of a regular language is the cardinality of its syntactic semigroup. The syntactic complexity of a subclass of regular languages is the worst-case syntactic complexity taken as a function of the state complexity n of languages in that class. We prove that n n−1 , n n−1 + n − 1, and n n−2 + (n − 2)2 n−2 + 1 are tight upper bounds on the syntactic complexities of right ideals and prefix-closed languages, left ideals and suffix-closed languages, and two-sided ideals and factor-closed languages, respectively. Moreover, we show that the transition semigroups meeting the upper bounds for all three types of ideals are unique, and the numbers of generators (4, 5, and 6, respectively) cannot be reduced.

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Theory of átomata

Cited by 51 publications

References 9 publications

Complexity of Right-Ideal, Prefix-Closed, and Prefix-Free Regular Languages

Complexity of Right-Ideal, Prefix-Closed, and Prefix-Free Regular Languages

Most Complex Non-returning Regular Languages

Syntactic Complexity of Regular Ideals

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