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Quotient Complexity of Regular Languages

2009
*
Electronic Proceedings in Theoretical Computer Science
*

The problem

doi:10.4204/eptcs.3.2
fatcat:ytyoerzyifa6bmzbpev23tev7q
*of*finding the*quotient**complexity**of*a*language*f(K,L) is considered, where K and L are*regular**languages*and f is a*regular*operation, for example, union or concatenation. ... Since state*complexity*is a property*of*a*language*, it is appropriate to define it in formal-*language*terms as the number*of*distinct*quotients**of*the*language*, and to call it "*quotient**complexity*". ...*Quotient**Complexity**of**Regular**Languages**Regular**Languages**Quotient**Complexity*State*Complexity*Upper Bounds on*Complexity**of*Operations Results accepting or final*quotients*Why define the ...##
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Quotient Complexities of Atoms of Regular Languages
[article]

2012
*
arXiv
*
pre-print

An atom

arXiv:1201.0295v2
fatcat:tzfmcp2gxjehfmmgcku7kh5wmu
*of*a*regular**language*L with n (left)*quotients*is a non-empty intersection*of*uncomplemented or complemented*quotients**of*L, where each*of*the n*quotients*appears in a term*of*the intersection ... We prove that, for any*language*L with*quotient**complexity*n, the*quotient**complexity**of*any atom*of*L with r complemented*quotients*has an upper bound*of*2^n-1 if r=0 or r=n, and 1+∑_k=1^r∑_h=k+1^k+n-r ... Acknowledgments We are grateful to Baiyu Li for writing a program for evaluating the*quotient**complexity**of*atoms. ...##
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Quotient Complexities of Atoms of Regular Languages
[chapter]

2012
*
Lecture Notes in Computer Science
*

We use one

doi:10.1007/978-3-642-31653-1_6
fatcat:3anvc42r25fgpmiwn4dfrk6obi
*of*these two points*of*view or the other, depending on convenience. We study the*quotient**complexity**of*atoms*of**regular**languages*. Our main result is the following: ... An atom*of*a*regular**language*L with n (left)*quotients*is a non-empty intersection*of*uncomplemented or complemented*quotients**of*L, where each*of*the n*quotients*appears in a term*of*the intersection ... We are grateful to Baiyu Li for writing a program for evaluating the*quotient**complexity**of*atoms. ...##
###
Quotient Complexities of Atoms in Regular Ideal Languages
[article]

2015
*
arXiv
*
pre-print

The

arXiv:1503.02208v2
fatcat:mjpt2oiw7fezrotr5uy2t4idb4
*quotient**complexity**of*a*regular**language*L is the number*of**quotients**of*L; it is equal to the state*complexity**of*L, which is the number*of*states in a minimal deterministic finite automaton accepting ... We compute the maximal number*of*atoms and the maximal*quotient**complexities**of*atoms*of*right, left and two-sided*regular*ideals. ... The*quotient**complexity**of*atoms*of**regular**languages*has been studied in [4, 8, 11] . ...##
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Quotient Complexity of Bifix-, Factor-, and Subword-Free Regular Languages
[article]

2011
*
arXiv
*
pre-print

We study the

arXiv:1006.4843v3
fatcat:qhdi4acy3zgqnfsvva4o6k3kru
*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*. ... The*quotient**complexity**of*an operation in a class C*of**regular**languages*is the worst-case*quotient**complexity**of*the*language*resulting from the operation, taken as a function*of*the*quotient**complexities*...##
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Maximal Syntactic Complexity of Regular Languages Implies Maximal Quotient Complexities of Atoms
[article]

2013
*
arXiv
*
pre-print

We relate two measures

arXiv:1302.3906v2
fatcat:u3qkonhgsrghdf632ldg7i2ab4
*of**complexity**of**regular**languages*. The first is syntactic*complexity*, that is, the cardinality*of*the syntactic semigroup*of*the*language*. ... The second measure consists*of*the*quotient*(state)*complexities**of*the atoms*of*the*language*, where atoms are non-empty intersections*of*complemented and uncomplemented*quotients*. ... Maximal*quotient**complexities**of*atoms*of*a*regular**language*imply maximal*quotient**complexity**of*its reversal. ...##
###
Quotient Complexities of Atoms in Regular Ideal Languages

2015
*
Acta Cybernetica
*

The

doi:10.14232/actacyb.22.2.2015.4
fatcat:qf3un4ry3zholkyvp5ru6wrmhy
*quotient**complexity**of*a*regular**language*L is the number*of**quotients**of*L; it is equal to the state*complexity**of*L, which is the number*of*states in a minimal deterministic finite automaton accepting ... We compute the maximal number*of*atoms and the maximal*quotient**complexities**of*atoms*of*right, left and two-sided*regular*ideals. ... The*quotient**complexity**of*atoms*of**regular**languages*has been studied in [4, 8, 11] . ...##
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Quotient Complexity of Bifix-, Factor-, and Subword-free Regular Language

2014
*
Acta Cybernetica
*

The

doi:10.14232/actacyb.21.4.2014.1
fatcat:bdqadrhzqnhvvaw5flb45cyuhe
*quotient**complexity**of*an operation on*regular**languages*is defined as the number*of*left*quotients**of*the result*of*the operation as a function*of*the numbers*of*left*quotients**of*the operands. ... The*quotient**complexity**of*a*regular**language*is the same as its state*complexity*, which is the number*of*states in the complete minimal deterministic finite automaton accepting the*language*. ... The*quotient**complexity**of*an operation in a class C*of**regular**languages*is the maximal*quotient**complexity**of*the*language*resulting from the operation, taken as a function*of*the*quotient**complexities*...##
###
Syntactic Complexity of Prefix-, Suffix-, and Bifix-Free Regular Languages
[chapter]

2011
*
Lecture Notes in Computer Science
*

The syntactic

doi:10.1007/978-3-642-22600-7_8
fatcat:kzxtjjpa4bhwfix3dh5jmdianu
*complexity**of*a subclass*of*the class*of**regular**languages*is the maximal syntactic*complexity**of**languages*in that class, taken as a function*of*the state*complexity*n*of*these*languages*... The syntactic*complexity**of*a*regular**language*is the cardinality*of*its syntactic semigroup. ... An equivalent notion is that*of**quotient**complexity*, which is the number*of*left*quotients**of*the*language*. ...##
###
Quotient Complexity of Closed Languages
[chapter]

2010
*
Lecture Notes in Computer Science
*

We study the

doi:10.1007/978-3-642-13182-0_8
fatcat:nuf7xkvhpzg6bkmpi35cokhv5a
*quotient**complexity*(usually called state*complexity*)*of*operations on prefix-, suffix-, factor-, and subword-closed*languages*. ... We find tight upper bounds on the*complexity**of*the prefix-, suffix-, factor-, and subword-closure*of*arbitrary*languages*, and on the*complexity**of*boolean operations, concatenation, star and reversal ... We consider here the case where the given*language*is closed and*regular*, and give upper bounds on the*quotient**complexity**of*the resulting*languages*. ...##
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COMPLEXITY OF ATOMS OF REGULAR LANGUAGES

2013
*
International Journal of Foundations of Computer Science
*

The

doi:10.1142/s0129054113400285
fatcat:tgtowdskdzdedhvm4c4isyil6m
*quotient**complexity**of*a*regular**language*L, which is the same as its state*complexity*, is the number*of*left*quotients**of*L. ... An atom*of*a non-empty*regular**language*L with n*quotients*is a non-empty intersection*of*the n*quotients*, which can be uncomplemented or complemented. ... OGP0000871, the ERDF funded Estonian Center*of*Excellence in Computer Science, EXCS, the Estonian Science Foundation grant 7520, and the Estonian Ministry*of*Education and Research target-financed research ...##
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Complexity in Convex Languages
[chapter]

2010
*
Lecture Notes in Computer Science
*

The state

doi:10.1007/978-3-642-13089-2_1
fatcat:zaikqi3zffas5hnirzepaw2khu
*complexities**of**regular*operations in subclasses*of*convex*languages*have been examined together, with considerable economies*of*effort. ... This paper surveys the recent results on convex*languages*with an emphasis on*complexity*issues. ... Acknowledgments This work was supported by the Natural Sciences and Engineering Research Council*of*Canada under Grant No. OGP000871. ...##
###
Towards a Theory of Complexity of Regular Languages
[article]

2017
*
arXiv
*
pre-print

the

arXiv:1702.05024v1
fatcat:gzkyubh2ireizdsuu6fpqy6ece
*quotient**complexity**of*its atoms -- basic components*of*every*regular**language*. ... We then turn to the study*of*the*quotient*/state*complexity**of*common operations on*regular**languages*: reversal, (Kleene) star, product (concatenation) and boolean operations. ... So it is natural to consider the number*of**quotients**of*a*regular**language*L as a*complexity*measure, which we call the*quotient**complexity**of*L and denote by κ(L). ...##
###
Distinguishability Operations and Closures on Regular Languages
[article]

2014
*
arXiv
*
pre-print

Given a

arXiv:1407.0160v3
fatcat:xbdk2mhrlnbv7mjwuegnxiibai
*regular**language*L, we study the*language**of*words D(L), that distinguish between pairs*of*different left-*quotients**of*L. ... We generalize the results for the*languages**of*words that distinguish between pairs*of*different right-*quotients*and two-sided*quotients**of*a*language*L. ... In Section 4, we study the state*complexity**of*D(L) as a function*of*the state*complexity**of*L, for any general*regular**language*L. ...##
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Quotient Complexity of Closed Languages

2013
*
Theory of Computing Systems
*

In 1994 the state

doi:10.1007/s00224-013-9515-7
fatcat:2jkix7i7bneurctxbjobtiszji
*complexities**of*concatenation, star, left and right*quotients*, reversal, intersection, and union*of**regular**languages*were examined in detail by Yu, Zhuang and K. Salomaa in [32]. ... We study the state*complexity*(which we prefer to call*quotient**complexity*)*of*operations on prefix-, suffix-, factor-, and subword-closed*languages*. ... The*quotient**complexity**of*L is the number*of*distinct*quotients**of*L and is denoted by κ(L).*Quotients**of**regular**languages*[4, 6] can be computed as follows. ...
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