Quotient of a formal language

In mathematics and computer science, the right quotient (or simply quotient) of a language $L_{1}$ with respect to language $L_{2}$ is the language consisting of strings w such that wx is in $L_{1}$ for some string x in $L_{2}$. Formally:

$$L_{1}/L_{2}=\{w\in \Sigma ^{*}\mid \exists x\in L_{2}\colon \ wx\in L_{1}\}$$

In other words, for all the strings in $L_{1}$ that have a suffix in $L_{2}$, the suffix is removed.

Similarly, the left quotient of $L_{1}$ with respect to $L_{2}$ is the language consisting of strings w such that xw is in $L_{1}$ for some string x in $L_{2}$. Formally:

$$L_{2}\backslash L_{1}=\{w\in \Sigma ^{*}\mid \exists x\in L_{2}\colon \ xw\in L_{1}\}$$

In other words, we take all the strings in $L_{1}$ that have a prefix in $L_{2}$, and remove this prefix.

Note that the operands of $\backslash$ are in reverse order: the first operand is $L_{2}$ and $L_{1}$ is second.

Example

Consider

$$L_{1}=\{a^{n}b^{n}c^{n}\mid n\geq 0\}$$

and

$$L_{2}=\{b^{i}c^{j}\mid i,j\geq 0\}.$$

Now, if we insert a divider into an element of $L_{1}$, the part on the right is in $L_{2}$ only if the divider is placed adjacent to a b (in which case i ≤ n and j = n) or adjacent to a c (in which case i = 0 and j ≤ n). The part on the left, therefore, will be either $a^{n}b^{n-i}$ or $a^{n}b^{n}c^{n-j}$; and $L_{1}/L_{2}$ can be written as

$$\{\ a^{p}b^{q}c^{r}\ \mid \ p=q\geq r\ \ {\text{or}}\ \ (p\geq q{\text{ and }}r=0)\ \}.$$

Properties

Some common closure properties of the quotient operation 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 can be any recursively enumerable language.
• The quotient of two recursively enumerable languages is recursively enumerable.

These closure properties hold for both left and right quotients.

See also

• Brzozowski derivative