Nesting of shuffle closure is important

Jȩdrzejowicz, Joanna

doi:10.1016/0020-0190(87)90213-4

Cited by 8 publications

(6 citation statements)

References 4 publications

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“…The shu e operation, being in some sense a mathematical model of parallel computation, has been intensively studied in formal language theory. For example, some types of regular expressions of shu e operators are dealt with in [1,2,[13][14][15][16][17]. A related decidability problem for commutative regular languages is solved in [8].…”

Section: Introductionmentioning

confidence: 99%

Shuffle and scattered deletion closure of languages

Itō¹,

Kari²,

Thierrin³

2000

Theoretical Computer Science

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We introduce and study the notion of shu e residual of a language L: the set containing the words whose shu e with L is completely included in L. Several properties and a characterization of the shu e residual of a language are obtained. The shu e closure of a language L (the smallest language that is shu e closed and contains L) is investigated. Moreover, conditions for the existence of maximal languages whose shu e residual equals a given language are obtained. The paper also considers an operation dual to shu e, namely scattered deletion: the scattered deletion of a word w from u consists of the words obtained by sparsely deleting from u the letters of w, in the order in which they appear in w. The scattered deletion residual and scattered deletion closure of a language are deÿned and studied. Finally, relationships and interdependencies between shu e, scattered deletion, and other insertion and deletion operations are obtained.

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Section: Introductionmentioning

confidence: 99%

Shuffle and scattered deletion closure of languages

Itō¹,

Kari²,

Thierrin³

2000

Theoretical Computer Science

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“…Interesting sub-hierarchies can be found hiding even in the lower hierarchy, as shown in [19], [20].…”

Section: Discussionmentioning

confidence: 99%

“…In [19], a subclass SSE (simple shuffle expressions) of SE, differing from the latter by prohibiting the nesting of iterated shuffles in a term, is shown to be strictly included in SE. Lemma 3.3 of [19] states that (a ⊙ (bc) ⊙ de) ∈ SSE, hence (⊙, )(FIN ) ⊆ SSE, showing how the lower hierarchy can be refined.…”

Section: ⊓ ⊔mentioning

confidence: 99%

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A Hierarchy of Languages with Catenation and Shuffle

Kudlek

Flick

2013

Fundamenta Informaticae

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We present basic structures, definitions, normal forms, and a hierarchy of languages based on catenation, shuffle and their iterations, defined by algebraic closures or least fixed point solutions to systems of equations.In this section, we introduce the basic structures, and two ways of defining language families based on these. Formal language theory deals with subsets of Σ * , sets of finite words over a finite alphabet, using as basic binary operation catenation, denoted by ⊙ in the sequel. This gives a basic monoid with ⊙ and λ, the empty word. Other binary operations have also been considered, such as the shuffle, denoted by , defined below. In contrast to catenation, is commutative; like catenation, it can be used to define another monoid with the finite subsets of Σ * as domain. Another possibility is to combine ⊙ and . Basic StructuresHere we present, partially recalling, the definitions of such structures more precisely. For w ∈ Σ * let w denote the length of w and w a the number of occurrences of a ∈ Σ in w. In particular, λ = 0. For A ⊆ Σ * let A = max{ w | w ∈ A} the norm of A. If A ⊆ Σ * then |A| denotes the cardinality of A which also can be infinite. M ⊙ = (Σ * ; λ, ⊙) is a monoid, where Σ is a (finite) alphabet, ⊙ stands for the binary operation catenation, and λ is the neutral element. Elements (words) w ∈ Σ * or singletons {w} can be seen as basic elements (atoms). Let 2 Σ * 1 = {α ∈ 2 Σ * | |α| = 1}, the class of singletons. The finite sets of words over Σ, FIN = 2 Σ * f = {α ∈ 2 Σ * | |α| < ∞}, can also serve as basic elements. The shuffle operation : Σ * × Σ * → 2 Σ * can be defined recursively as follows: For all a, b ∈ Σ and v, w ∈ Σ * , λ w = w λ = {w}; aw bv = {a} ⊙ (w bv) ∪ {b} ⊙ (aw v). Both operations are extended to sets: for shuffle, : 2 Σ * × 2 Σ * → 2 Σ * , A B = w∈A,v∈B w v gives the monoid M = (2 Σ * f ; {λ}, ). A bi-monoid is a structure D = (D; 1, ⊙, ⊗) where ⊙ and ⊗ are associative binary operations on D, and 1 is the common neutral element. A bi-semiring is a structure B = (B; 0, 1, ⊕, ⊙, ⊗) where (B; 0, 1, ⊕, ⊙) and (B; 0, 1, ⊕, ⊗) are semirings, i.e. ⊕ is a commutative and associative binary operation on B, ⊙ and ⊗ are associative binary operations on B, 0 is the neutral element of ⊕, and 1 the common neutral element of ⊙ and ⊗. Furthermore, ⊕ is distributive with ⊙ and ⊗, i.e. x ⊕ (y ⊙ z) = (x ⊙ y) ⊕ (x ⊙ z) and x ⊕ (y ⊗ z) = (x ⊗ y) ⊕ (x ⊗ z) for all x, y, z ∈ B. A bi-semiring B is ω-complete

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“…The opérations shuffle and shuffle closure have been introduced to describe sequentialized exécution histories of concurrent processes [7,8]. Together with other opérations they describe various classes of languages which have been extensively studied (see [1,[3][4][5]10]). Here, we consider the class of shuffle languages which émerges from the class of finite languages through regular opérations (union, concaténation, Kleene star) and shuffle opérations (shuffle and shuffle closure).…”

Section: Introductionmentioning

confidence: 99%

Lower Space Bounds for Accepting Shuffle Languages

Szepietowski

1999

RAIRO-Theor. Inf. Appl.

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Nesting of shuffle closure is important

Cited by 8 publications

References 4 publications

Shuffle and scattered deletion closure of languages

Shuffle and scattered deletion closure of languages

A Hierarchy of Languages with Catenation and Shuffle

Lower Space Bounds for Accepting Shuffle Languages

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