Right quotient

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The right quotient (or simply quotient) of a formal language L_1 with a formal language L_2 is the language consisting of strings w such that wx is in L_1 for some string x in L_2.[1] 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 L_1 / L_2 is the prefix of a string wx in L_1, with the remainder of the word being a string in L_2.

Example[edit]

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 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 \ \ | \ \ p = q \ge r \ \ \or \ \ p \ge q \and r = 0 \}.

Properties[edit]

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[edit]

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.

References[edit]

  1. ^ Linz, Peter (2011). An Introduction to Formal Languages and Automata. Jones & Bartlett Publishers. pp. 104–108. ISBN 9781449615529. Retrieved 7 July 2014.