quotient of languages

Let L1,L2 be two languagesPlanetmathPlanetmath over some alphabet Σ. The quotientPlanetmathPlanetmath of L1 by L2 is defined to be the set

L1/L2:={uΣ*uvL1 for some vL2}.

L1/L2 is sometimes written L1L2-1. The quotient so defined is also called the right quotient, for one can similarly define the left quotient of L1 by L2:

L1\L2:={uΣ*vuL1 for some vL2}.

L1\L2 is sometimes written L2-1L1.

Below are some examples of quotients:

  • If L1={anbncnn0} and L2={b,c}*, then

    • L1/L2={ambnmn0}

    • L2/L1=L2

    • L1\L2={λ}, the singleton consisting the empty word

    • L2\L1=L2

  • for any language L over Σ:

    • L/Σ* is the language of all prefixes of words of L

    • Σ*/L=Σ*

    • L\Σ* is the language of all suffixes of words of L

    • Σ*\L=Σ*

  • λL/LL\L, and if λL, then LL/LL\L.

Here are some basic properties of quotients:

  1. 1.


  2. 2.

    (L1/L2)L2(L1L2)/L2, and L2(L1\L2)(L2L1)\L2.

  3. 3.

    right and left quotients are related via reversal:

    (L1\L2)R = {uRvuL1 for some vL2}
    = {uR(vu)RL1R for some vRL2R}
    = {uRuRvRL1R for some vRL2R}
    = L1R/L2R.

A family of languages is said to be closed under quotient by a language L if for every language M, M/L. Furthermore, is said to be closed under quotient if M/L for any M,L. Closure under quotient is also termed closure under right quotient. Closure under left quotient is similarly defined.

It can be shown that the families of regular, context-free, and type-0 languages are closed under quotient (both left and right) by a regular language. The family of context-sensitive languages does not have this closure property.

Since all of the families mentioned above are closed under reversal, each of the families, except the context-sensitive family, is closed under left quotient by a regular language, according to the second property above.


Title quotient of languages
Canonical name QuotientOfLanguages
Date of creation 2013-03-22 18:56:06
Last modified on 2013-03-22 18:56:06
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 7
Author CWoo (3771)
Entry type Definition
Classification msc 68Q70
Classification msc 68Q45
Defines quotient
Defines left quotient
Defines right quotient