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Quotient of a formal language

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

In other words, for all the strings in that have a suffix in , the suffix is removed.

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

In other words, we take all the strings in that have a prefix in , and remove this prefix.

Note that the operands of are in reverse order: the first operand is and is second.

Example

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Consider and

Now, if we insert a divider into an element of , the part on the right is in 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 or ; and can be written as

Properties

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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

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References

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