# Quotient Complexity of Regular Languages

@inproceedings{Brzozowski2010QuotientCO, title={Quotient Complexity of Regular Languages}, author={Janusz A. Brzozowski}, booktitle={J. Autom. Lang. Comb.}, year={2010} }

The past research on the state complexity of operations on regular languages is examined, and a new approach based on an old method (derivatives of regular expressions) is presented. 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". The problem of finding the quotient complexity of a language f(K,L) is considered, where K and L are regular…

## 82 Citations

### COMPLEXITY OF ATOMS OF REGULAR LANGUAGES

- Mathematics
- 2014

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…

### A Review on State Complexity of Individual Operations

- Computer Science
- 2011

This report reviews some of the results of state complexity of individual operations for regular and some subregular languages on the basis of subset construction and nondeterministic state complexity.

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

- Linguistics, Computer ScienceActa Cybern.
- 2014

Tight upper bounds are found on the quotient complexity of intersection, union, difference, symmetric difference, concatenation, star, and reversal in these three classes of languages.

### F L ] 2 3 M ay 2 01 5 Quotient Complexities of Atoms in Regular Ideal Languages ⋆

- Mathematics, Linguistics
- 2018

A (left) quotient of a language L by a word w is the language wL = {x | wx ∈ L}. The quotient complexity of a regular language L is the number of quotients of L; it is equal to the state complexity…

### Symmetric Groups and Quotient Complexity of Boolean Operations

- MathematicsICALP
- 2014

The notion of uniform minimality to direct products of automata is generalized and the non-trivial connection between complexity of boolean operations and group theory is established.

### Maximal Syntactic Complexity of Regular Languages Implies Maximal Quotient Complexities of Atoms

- MathematicsArXiv
- 2013

It is proved that if a language has maximal syntactic complexity, then it has 2^n atoms and each atom has maximal quotient complexity, but the converse is false.

### Syntactic Complexity of Regular Ideals

- LinguisticsTheory of Computing Systems
- 2017

It is proved that nn−1, nn −1 + n − 1, and nn+2 + (n − 2)2n−2 + 1 are tight upper bounds on the syntactic complexities of right ideals and prefix- closed languages, left ideals and suffix-closed languages, and two-sided ideals and factor-closed Languages, respectively.

### Descriptional Complexity of the Languages KaL: Automata, Monoids and Varieties

- MathematicsDCFS
- 2010

The first step when forming the polynomial hierarchies of languages is to consider languages of the form KaL where K and L are over a finite alphabet A and from a given variety V of languages, a…

### Advanced Topics on State Complexity of Combined Operations

- Computer Science
- 2010

This thesis discusses the state complexities of individual operations on regular languages, including union, intersection, star, catenation, reversal and so on, and introduces the concept of estimation and approximation of state complexity.

### Quotient Complexities of Atoms of Regular Languages

- MathematicsDevelopments in Language Theory
- 2012

It is proved that, for any language L with quotients complexity n, the quotient complexity of any atom of L with r complemented quotients has an upper bound of 2n−1 if r=0 or r=n, and $1+\sum_{k=1}^{r} \sum_{h=k+1]^{k+n-r} C_{h}^{n} \cdot C_{k}^{h}$ otherwise.

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