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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 such that is in for some string in , where and are defined on the same alphabet . Formally:[1][2]

where is the Kleene star on , and is the empty set.

In other words, for all strings in that have a suffix in , the suffix (right part of the string) is removed.

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

In other words, for all strings in that have a prefix in , the prefix (left part of the string) is removed.

Note that the operands of are in reverse order, so that preceeds .

The right and left quotients of with respect to may also be written as and respectively.[1][3]

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Example

Consider and

If an element of is split into two parts, then the right part will be in if and only if the split occurs somewhere after the final . Assuming this is the case, if the split occurs before the first then and , otherwise and . For instance:

where is the empty string.

Thus, the left part will be either or ), and can be written as:

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Basic properties

If are languages over the same alphabet then:[3]

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Relationship between right and left quotients

Summarize
Perspective

The right and left quotients of languages and are related through the language reversals and by the equalities:[3]

Proof

To prove the first equality, let . Then there exists such that . Therefore, there must exist such that , hence by definition . It follows that , and so .

The second equality is proved in a similar manner.

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Other 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.

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See also

References

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