Basic Operations on Binary Suffix-Free Languages

Cmorik, Roland; Jir, Galina; skov,

doi:10.1007/978-3-642-25929-6_9

Cited by 22 publications

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“…Our results are summarized in Tables 1 and 2, where "B-, F-free" stands for bifixfree and factor-free, and "S-free" for subword-free. The bounds for operations on prefix-free languages are from [11,13], for operations on suffix-free languages from [9,12,14], and those for regular languages, from [16,17,27]. For languages over a unary alphabet Σ = {a}, the concepts prefix-, suffix-, factor-, and subwordfree coincide, and L is free with κ(L) = n if and only if L = {a n−2 }.…”

Section: Discussionmentioning

confidence: 99%

“…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.…”

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confidence: 99%

“…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.…”

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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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Section: Discussionmentioning

confidence: 99%

mentioning

confidence: 99%

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

Brzozowski¹,

Jirásková²,

Li³

et al. 2014

Acta Cybern

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“…By the restricted case, the states of Q ′ m × Q n are reachable and distinguishable using words in {a, b, d, e} * . Let R ∅ ′ = {(∅ ′ , q) | q ∈ Q n } and The complexity of suffix-free languages was studied in detail in [11,15,16,21,22,26]. For completeness we present a short summary of some of those results.…”

Section: Semigroupmentioning

confidence: 99%

Complexity of Left-Ideal, Suffix-Closed and Suffix-Free Regular Languages

Brzozowski

Sinnamon

2017

Language and Automata Theory and Applications

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A language L over an alphabet Σ is suffix-convex if, for any words x, y, z ∈ Σ * , whenever z and xyz are in L, then so is yz. Suffixconvex languages include three special cases: left-ideal, suffix-closed, and suffix-free languages. We examine complexity properties of these three special classes of suffix-convex regular languages. In particular, we study the quotient/state complexity of boolean operations, product (concatenation), star, and reversal on these languages, as well as the size of their syntactic semigroups, and the quotient complexity of their atoms.

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“…Researchers also considered the state complexity of multiple operations such as several concatenations or several intersections [5,7,22]. Jirásková and her co-authors studied the state complexity of some operations for binary languages [3,17,18]. Binary languages allow us to prove the tightness of the upper bound also in the case of reversal of deterministic unionfree languages, that is, languages represented by one-cycle-free-path deterministic automata, in which from each state there exists exactly one cycle-free accepting path [16].…”

Section: Introductionmentioning

confidence: 98%

State Complexity of k-Union and k-Intersection for Prefix-Free Regular Languages

Eom

Han

Salomaa

2015

Int. J. Found. Comput. Sci.

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We investigate the state complexity of multiple unions and of multiple intersections for prefix-free regular languages. Prefix-free deterministic finite automata have their own unique structural properties that are crucial for obtaining state complexity upper bounds that are improved from those for general regular languages. We present a tight lower bound construction for k-union using an alphabet of size k + 1 and for k-intersection using a binary alphabet. We prove that the state complexity upper bound for k-union cannot be reached by languages over an alphabet with less than k symbols. We also give a lower bound construction for k-union using a binary alphabet that is within a constant factor of the upper bound.

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Basic Operations on Binary Suffix-Free Languages

Cited by 22 publications

References 9 publications

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

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

Complexity of Left-Ideal, Suffix-Closed and Suffix-Free Regular Languages

State Complexity of k-Union and k-Intersection for Prefix-Free Regular Languages

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