Skip to main content

Parikhs theorem

Parikh's theorem A theorem in formal language theory that concerns the nature of context-free languages when order of letters is disregarded.

Let the alphabet Σ be the set {a1,…,an}. The letter distribution, φ(w), of a Σ-word w is the n-tuple 〈N1,…,Nn

with Ni the number of occurrences of ai in w. The Parikh image, φ(L), of a Σ-language L is {φ(w) | wL}

i.e. the set of all letter-distributions of words in L. L1 and L2 are letter-equivalent if φ(L1) = φ(L2)

Letter distributions may be added component-wise as vectors. This leads to the following: a set S of letter distributions is linear if, for some distributions d and d1,…,dk, S is the set of all sums formed from d and multiples of di. S is semilinear if it is a finite union of linear sets.

Parikh's theorem now states that if L is context-free φ(L) is semilinear. It can also be shown that φ(L) is semilinear if and only if L is letter-equivalent to a regular language. Hence any context-free language is letter-equivalent to a regular language – although not all such languages are context-free.

Cite this article
Pick a style below, and copy the text for your bibliography.

  • MLA
  • Chicago
  • APA

"Parikhs theorem." A Dictionary of Computing. . 18 Feb. 2019 <>.

"Parikhs theorem." A Dictionary of Computing. . (February 18, 2019).

"Parikhs theorem." A Dictionary of Computing. . Retrieved February 18, 2019 from

Learn more about citation styles

Citation styles gives you the ability to cite reference entries and articles according to common styles from the Modern Language Association (MLA), The Chicago Manual of Style, and the American Psychological Association (APA).

Within the “Cite this article” tool, pick a style to see how all available information looks when formatted according to that style. Then, copy and paste the text into your bibliography or works cited list.

Because each style has its own formatting nuances that evolve over time and not all information is available for every reference entry or article, cannot guarantee each citation it generates. Therefore, it’s best to use citations as a starting point before checking the style against your school or publication’s requirements and the most-recent information available at these sites:

Modern Language Association

The Chicago Manual of Style

American Psychological Association

  • Most online reference entries and articles do not have page numbers. Therefore, that information is unavailable for most content. However, the date of retrieval is often important. Refer to each style’s convention regarding the best way to format page numbers and retrieval dates.
  • In addition to the MLA, Chicago, and APA styles, your school, university, publication, or institution may have its own requirements for citations. Therefore, be sure to refer to those guidelines when editing your bibliography or works cited list.