Quotient of a formal language

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

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

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

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

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

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

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

Example

Consider

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

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

Now, if we insert a divider into an element of ${\displaystyle L_{1}}$, the part on the right is in ${\displaystyle L_{2}}$ 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 ${\displaystyle a^{n}b^{n-i}}$ or ${\displaystyle a^{n}b^{n}c^{n-j}}$; and ${\displaystyle L_{1}/L_{2}}$ can be written as

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

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

References