Right quotient

If $L 1$ 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.