Reversal on Regular Languages and Descriptional Complexity

Šebej, Juraj

doi:10.1007/978-3-642-39310-5_25

Cited by 12 publications

(8 citation statements)

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“…This improves the result from [8], where an alphabet of size growing exponentially with n is used to produce the whole range of complexities for the concatenation operation. Our result complements similar results from [10,14], where a linear alphabet is used to get the whole range of complexities for the reversal and Kleene closure operations.…”

Section: Discussionsupporting

confidence: 82%

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The Complexity of Languages Resulting from the Concatenation Operation

Jirásková

Szabari

Šebej

2016

Descriptional Complexity of Formal Systems

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Section: Discussionsupporting

confidence: 82%

“…The result for reversal and star was improved in [10,14] by showing that a linear alphabet is enough to produce the whole range of complexities.…”

Section: Introductionmentioning

confidence: 99%

The Complexity of Languages Resulting from the Concatenation Operation

Jirásková

Szabari

Šebej

2016

Descriptional Complexity of Formal Systems

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“…If n ≥ 5, then consider the language L = cKc, where K is a regular language over the alphabet {a, b} with κ(K) = n − 3 meeting the upper bound 2 n−3 for reversal [24]. The quotient automaton of L without the empty state is shown in Figure 11.…”

Section: Reversalmentioning

confidence: 99%

Quotient Complexity of Bifix-, Factor-, and Subword-free Regular Language

Brzozowski¹,

Jirásková²,

Li³

et al. 2014

Acta Cybern

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A language L is prefix-free if, whenever words u and v are in L and u is a prefix of v, then u = v. Suffix-, factor-, and subword-free languages are defined similarly, where "subword" means "subsequence". A language is bifix-free if it is both prefix-and suffix-free. We study the quotient complexity, more commonly known as state complexity, of operations in the classes of bifix-, factor-, and subword-free regular languages. We find tight upper bounds on the quotient complexity of intersection, union, difference, symmetric difference, concatenation, star, and reversal in these three classes of languages.One of the first results concerning the state complexity of an operation is the 1966 theorem by Mirkin [18], who showed that the bound 2 n for the reversal of an n-state dfa can be attained. In 1970 Maslov [17] stated without proof the bounds on the complexities of union, concatenation, star, and several other operations in the class of regular languages, and gave languages meeting these bounds. In 1994 these operations, along with intersection, reversal, and left and right quotients, were studied in detail by Yu, Zhuang and Salomaa [27].Results exist also for proper subclasses of the class of regular languages: unary [20,27], finite [8,10,26], cofinite [2], prefix-free [12,13], suffix-free [9,11,14], ideal [6], and closed [7]. The bounds can vary considerably.Free languages (with the exception of {ε}, where ε is the empty word) are codes, which constitute an important class of languages and have applications in such areas as cryptography, data compression, and information transmission. They have been studied extensively; see, for example, [3,15]. In particular, prefix and suffix codes [3] are prefix-free and suffix-free languages, respectively, infix codes [21,22] are factor-free, and hypercodes [21,22] are subword-free, where by subword we mean subsequence. Moreover, free languages are special cases of convex languages [1,23]. We are interested only in regular free languages.The state complexities of intersection, union, concatenation, star, and reversal were first studied by Han, K. Salomaa, and Wood [12] for prefix-free languages, and by Han and K. Salomaa [11] for suffix-free languages. In the present paper, these results are extended to bifix-, factor-and subword-free languages. In particular, we obtain tight upper bounds on the complexities of intersection, union, difference, symmetric difference, star, concatenation, and reversal in these three classes of free languages. PreliminariesIt is assumed that the reader is familiar with finite automata and regular languages as treated in [19,25], for example. If Σ is a finite non-empty alphabet, then Σ * is the set of all words over this alphabet, with ε as the empty word. For w ∈ Σ * , let |w| be the length of w. A language is any subset of Σ * .The following set operations are defined on languages: complement (L = Σ * \ L), union (K ∪ L), intersection (K ∩ L), difference (K \ L), and symmetric difference (K ⊕ L). A general boolean operation with two arguments ...

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“…Now we show that a binary alphabet, say Σ = {a, b}, is not enough to reach the upper bound. Let a ∈ Σ be a letter such that 0δ It is known that in the case of the class of all regular languages the resulted language of the reversal operation can have any state complexity in range of integers [log 2 n, 2 n ] [17,23], thus there are no gaps (magic numbers) in the interval of possible state complexities. The next theorem states that the situation is similar for the case of bifix-free languages.…”

Section: 2mentioning

confidence: 99%