Right quotient

The right quotient (or simply quotient) of a formal language L₁ with a formal language L₂ is the language consisting of strings w such that wx is in L₁ for some string x in L₂. In symbols, we write:

$L_1 / L_2 = \{w \ | \ \exists x ((x \in L_2) \land (wx \in L_1)) \}$

In other words, each string in L₁ / L₂ is the prefix of a string wx in L₁, with the remainder of the word being a string in L₂.

Examples

Consider

$L_1 = \{ a^n b^n c^n \ \ |\ \ n\ge 0 \}$

and

$L_2 = \{ b^i c^j \ \ | \ \ i,j\ge 0 \}$.

Now, if we insert a divider into the middle of an element of L₁, the part on the right is in L₂ 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 aⁿb^{n − i} or aⁿbⁿc^{n − j}; and L₁ / L₂ can be written as

$\{ a^p b^q c^r \ \ | \ \ p = q \ge r \ \ \or \ \ p \ge q \and r = 0 \}$.

Properties

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

Left and right quotients

There is a related notion of left quotient, which keeps the postfixes of L₁ without the prefixes in L₂. Sometimes, though, "right quotient" is written simply as "quotient". The above closure properties hold for both left and right quotients.