right quotient

L_1

and

L_2

are formal languages, then the right quotient of

L_1

with

L_2

is the language consisting of strings w such that wx is in

L_1

for some string x in

L_2

. In symbols, we write:

L_1 / L_2 = \{w \ | \ \ \exists x \in L_2 \ \ : \ \ wx \in L_1\}

Some common closure properties of quotient include:

The quotient of a regular language with any other language is regular.
The quotient of a context free language with a regular language is context free.
The quotient of two context free languages is context sensitive.
The quotient of a context sensitive language and any other language could be anything, recursively enumerable or nonrecursively enumerable.

There is a related notion of left quotient, which keeps the postfixes of

L_1

without the prefixes in

L_2

. Sometimes, though, "right quotient" is written simply as "quotient". The above closure properties hold for both left and right quotients.

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