Kleene Star on Unary Regular Languages

Čevorová, Kristína

doi:10.1007/978-3-642-39310-5_26

Cited by 15 publications

(4 citation statements)

References 8 publications

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“…The state complexity of the star operation is 3/4 • 2 n with binary witness languages [10,15,23]. In the unary case, the tight bound on the state complexity of star is (n−1) 2 +1 [23,24]. The nondeterministic state complexity of star is n+1, with witnesses defined over a unary alphabet [7].…”

Section: Starmentioning

confidence: 99%

Self-Verifying Finite Automata and Descriptional Complexity

Jirásková

2016

Descriptional Complexity of Formal Systems

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Section: Starmentioning

confidence: 99%

Self-Verifying Finite Automata and Descriptional Complexity

Jirásková

2016

Descriptional Complexity of Formal Systems

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“…As shown in Table 2, the average state complexities of catenation and star on unary languages are bounded by a constant, and for intersection (and union) note that 3ζ (3) 2π 2 ≈ 0.1826907423. Magical numbers for the star operation on unary languages was studied by Čevorová [40]. Considering the gap between the worst-case upper bound, n 2 − 2n + 2, and the average case (less than a constant), it is not a surprise that for every n no more than 4 complexities are attainable between n 2 − 4n + 6 and the upper bound.…”

Section: Unary Regular Languagesmentioning

confidence: 99%

A Survey on Operational State Complexity

Gao,

Moreira,

Reis

et al. 2015

Preprint

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Descriptional complexity is the study of the conciseness of the various models representing formal languages. The state complexity of a regular language is the size, measured by the number of states of the smallest, either deterministic or nondeterministic, finite automaton that recognises it. Operational state complexity is the study of the state complexity of operations over languages. In this survey, we review the state complexities of individual regularity preserving language operations on regular and some subregular languages. Then we revisit the state complexities of the combination of individual operations. We also review methods of estimation and approximation of state complexity of more complex combined operations.

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“…Moreover, for every n, the numbers 1, n, and 2 n−1 + 2 n−1−k with 1 ≤ k ≤ n − 1 are attainable by the complexity of Kleene closure. The situation is completely different in the case of a unary alphabet, where two holes of length n exist for every n [2].…”

Section: Introductionmentioning

confidence: 99%

Kleene closure and state complexity

Palmovský

2016

RAIRO-Theor. Inf. Appl.

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We prove that the automaton presented by Maslov [Soviet Math. Doklady 11, 1373-1375(1970] meets the upper bound 3/4 · 2 n on the state complexity of Kleene closure. This fixes a small error in this paper that claimed the upper bound 3/4 · 2 n − 1. Our main result shows that the upper bounds 2 n−1 + 2 n−1−k on the state complexity of Kleene closure of a language accepted by an n-state DFA with k final states are tight for every k in the binary case. We also present some results of our calculations. We consider not only the worst case, but we study all possible values that can be obtained as the state complexity of Kleene closure of a regular language accepted by a minimal n-state DFA. Using the lists of pairwise non-isomorphic binary automata of 2,3,4, and 5 states, we compute the frequencies of the resulting complexities for Kleene closure, and show that every value in the range from 1 to 3/4 ·2 n occurs at least ones. In the case of n = 6, 7, 8, we change the strategy, and consider binary automata, in which the first symbol is a circular shift of the states, and the second symbol is generated randomly. We show that all values from 1 to 3/4 · 2 n are attainable, that is, for every m with 1 ≤ m ≤ 3/4 · 2 n , there exists an nstate binary DFA A such that the state complexity of L(A) * is exactly m.

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Kleene Star on Unary Regular Languages

Cited by 15 publications

References 8 publications

Self-Verifying Finite Automata and Descriptional Complexity

Self-Verifying Finite Automata and Descriptional Complexity

A Survey on Operational State Complexity

Kleene closure and state complexity

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