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### Quotient Complexity of Regular Languages

Janusz Brzozowski
2009 Electronic Proceedings in Theoretical Computer Science
The problem 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  ...

### Quotient Complexities of Atoms of Regular Languages [article]

Janusz Brzozowski, Hellis Tamm
2012 arXiv   pre-print
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 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.  ...

### Quotient Complexities of Atoms of Regular Languages [chapter]

Janusz Brzozowski, Hellis Tamm
2012 Lecture Notes in Computer Science
We use one 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]

Janusz Brzozowski, Sylvie Davies
2015 arXiv   pre-print
The 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] .  ...

### Quotient Complexity of Bifix-, Factor-, and Subword-Free Regular Languages [article]

Janusz Brzozowski, Galina Jirásková, Baiyu Li, Joshua Smith
2011 arXiv   pre-print
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.  ...  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  ...

### Maximal Syntactic Complexity of Regular Languages Implies Maximal Quotient Complexities of Atoms [article]

Janusz Brzozowski, Gareth Davies
2013 arXiv   pre-print
We relate two measures 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

Janusz Brzozowski, Sylvie Davies
2015 Acta Cybernetica
The 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] .  ...

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

Janusz Brzozowski, Galina Jirásková, Li Baiyu, Joshua Smith
2014 Acta Cybernetica
The 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]

Janusz Brzozowski, Baiyu Li, Yuli Ye
2011 Lecture Notes in Computer Science
The syntactic 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]

Janusz Brzozowski, Galina Jirásková, Chenglong Zou
2010 Lecture Notes in Computer Science
We study the 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.  ...

### COMPLEXITY OF ATOMS OF REGULAR LANGUAGES

JANUSZ BRZOZOWSKI, HELLIS TAMM
2013 International Journal of Foundations of Computer Science
The 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  ...

### Complexity in Convex Languages [chapter]

Janusz Brzozowski
2010 Lecture Notes in Computer Science
The state 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]

Janusz A. Brzozowski
2017 arXiv   pre-print
the 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]

Cezar Câmpeanu, Nelma Moreira, Rogério Reis
2014 arXiv   pre-print
Given a 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.  ...

### Quotient Complexity of Closed Languages

Janusz Brzozowski, Galina Jirásková, Chenglong Zou
2013 Theory of Computing Systems
In 1994 the state 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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