Complexity of atoms, combinatorially

Iván, Szabolcs

doi:10.1016/j.ipl.2016.01.003

Cited by 22 publications

(19 citation statements)

References 7 publications

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“…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].…”

Section: Preliminariesmentioning

confidence: 99%

Most Complex Non-Returning Regular Languages

Brzozowski

Davies

2019

Int. J. Found. Comput. Sci.

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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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Section: Preliminariesmentioning

confidence: 99%

Most Complex Non-Returning Regular Languages

Brzozowski

Davies

2019

Int. J. Found. Comput. Sci.

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“…where |X| ≤ |S|, |Y | ≤ n − |S|, and X, Y ⊆ Q are disjoint. Using the approach from Iván [16] we define the DFA D S = (Q S , Σ, δ, (S, Q \ S), F S ) such that:…”

Section: Atom Complexitiesmentioning

confidence: 99%

Complexity of bifix-free regular languages

Ferens

Szykuła

2019

Theoretical Computer Science

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“…An equivalence class of this relation is an atom of L [10]. Atoms can be expressed as non-empty intersections of complemented and uncomplemented quotients of L. The number of atoms and their state complexities were suggested as measures of complexity of regular languages [4] because all quotients of a language and all quotients of its atoms are unions of atoms [9,10,14].…”

Section: Preliminariesmentioning

confidence: 99%

Most Complex Deterministic Union-Free Regular Languages

Brzozowski

Davies

2018

Descriptional Complexity of Formal Systems

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A regular language L is union-free if it can be represented by a regular expression without the union operation. A union-free language is deterministic if it can be accepted by a deterministic one-cycle-freepath finite automaton; this is an automaton which has one final state and exactly one cycle-free path from any state to the final state. Jirásková and Masopust proved that the state complexities of the basic operations reversal, star, product, and boolean operations in deterministic unionfree languages are exactly the same as those in the class of all regular languages. To prove that the bounds are met they used five types of automata, involving eight types of transformations of the set of states of the automata. We show that for each n 3 there exists one ternary witness of state complexity n that meets the bound for reversal and product. Moreover, the restrictions of this witness to binary alphabets meet the bounds for star and boolean operations. We also show that the tight upper bounds on the state complexity of binary operations that take arguments over different alphabets are the same as those for arbitrary regular languages. Furthermore, we prove that the maximal syntactic semigroup of a union-free language has n n elements, as in the case of regular languages, and that the maximal state complexities of atoms of union-free languages are the same as those for regular languages. Finally, we prove that there exists a most complex union-free language that meets the bounds for all these complexity measures. Altogether this proves that the complexity measures above cannot distinguish union-free languages from regular languages.

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Complexity of atoms, combinatorially

Cited by 22 publications

References 7 publications

Most Complex Non-Returning Regular Languages

Most Complex Non-Returning Regular Languages

Complexity of bifix-free regular languages

Most Complex Deterministic Union-Free Regular Languages

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