The Average State Complexity of Rational Operations on Finite Languages

Bassino, Frédérique; Nicaud, Cyril

doi:10.1142/s0129054110007398

Cited by 7 publications

(5 citation statements)

References 12 publications

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“…For instance, it was proved in [1,8] that the average complexity of Moore algorithm, another minimization algorithm, is significantly better than its worst-case complexity, making this algorithm a reasonable solution in practice. The reader can find some results on the average state complexity of operations under different settings in [16,3]. Let us also mention the recent article [2], in the same area, which focus on quantifying the probability that a random deterministic automaton is minimal.…”

Section: Introductionmentioning

confidence: 99%

Brzozowski Algorithm Is Generically Super-Polynomial for Deterministic Automata

Felice¹,

Nicaud²

2013

Developments in Language Theory

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Abstract. We study the number of states of the minimal automaton of the mirror of a rational language recognized by a random deterministic automaton with n states. We prove that, for any d > 0, the probability that this number of states is greater than n d tends to 1 as n tends to infinity. As a consequence, the generic and average complexities of Brzozowski minimization algorithm are super-polynomial for the uniform distribution on deterministic automata.

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Section: Introductionmentioning

confidence: 99%

Brzozowski Algorithm Is Generically Super-Polynomial for Deterministic Automata

Felice¹,

Nicaud²

2013

Developments in Language Theory

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“…The average state complexity on finite languages is addressed in two works. Gruber and Holzer [75] analysed the average state complexity of DFAs and NFAs based on a uniform distribution over finite languages whose longest word is of length at most n. Based on the size of finite languages as the summation of the lengths of all its words and a correspondent uniform distribution, Bassino et al [3] establish that the average state complexities of the basic regular operations are asymptotically linear.…”

Section: Some More Resultsmentioning

confidence: 99%

“…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].…”

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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“…As future work we plan to study the average transition complexity of these operations following the lines of Bassino et al [1].…”

Section: Final Remarksmentioning

confidence: 99%

“…Table 1 presents a comparison of the transition complexity on regular and finite languages, where the new results are highlighted. All the proofs not presented in this paper can be found in an extended version of this work 1 . Table 1.…”

Section: Introductionmentioning

confidence: 99%

Incomplete Transition Complexity of Basic Operations on Finite Languages

Maia

Moreira

Reis

2013

Implementation and Application of Automata

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Abstract. The state complexity of basic operations on finite languages (considering complete DFAs) has been in studied the literature. In this paper we study the incomplete (deterministic) state and transition complexity on finite languages of boolean operations, concatenation, star, and reversal. For all operations we give tight upper bounds for both descriptional measures. We correct the published state complexity of concatenation for complete DFAs and provide a tight upper bound for the case when the right automaton is larger than the left one. For all binary operations the tightness is proved using family languages with a variable alphabet size. In general the operational complexities depend not only on the complexities of the operands but also on other refined measures.

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The Average State Complexity of Rational Operations on Finite Languages

Abstract: Considering the uniform distribution on sets of m non-empty words whose sum of lengths is n, we establish that the average state complexities of the rational operations are asymptotically linear.

Cited by 7 publications

References 12 publications

Brzozowski Algorithm Is Generically Super-Polynomial for Deterministic Automata

Brzozowski Algorithm Is Generically Super-Polynomial for Deterministic Automata

A Survey on Operational State Complexity

Incomplete Transition Complexity of Basic Operations on Finite Languages

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