Parikh's theorem

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Parikh's theorem in theoretical computer science says that if one looks only at the number of occurrences of each terminal symbol in a context-free language, without regard to their order, then the language is indistinguishable from a regular language.[1] It is useful for deciding that strings with a given number of terminals are not accepted by a context-free grammar.[2] It was first proved by Rohit Parikh in 1961[3] and republished in 1966.[4]

Definitions and formal statement[edit]

Let be an alphabet. The Parikh vector of a word is defined as the function , given by[1]

where denotes the number of occurrences of the letter in the word .

A subset of is said to be linear if it is of the form

for some vectors . A subset of is said to be semi-linear if it is a union of finitely many linear subsets.

Statement 1: Let be a context-free language. Let be the set of Parikh vectors of words in , that is, . Then is a semi-linear set.

Two languages are said to be commutatively equivalent if they have the same set of Parikh vectors.

Statement 2: If is any semi-linear set, the language of words whose Parikh vectors are in is commutatively equivalent to some regular language. Thus, every context-free language is commutatively equivalent to some regular language.

These two equivalent statements can be summed up by saying that the image under of context-free languages and of regular languages is the same, and it is equal to the set of semilinear sets.

Strengthening for bounded languages[edit]

A language is bounded if for some fixed words . Ginsburg and Spanier [5] gave a necessary and sufficient condition, similar to Parikh's theorem, for bounded languages.

Call a linear set stratified, if in its definition for each the vector has the property that it has at most two non-zero coordinates, and for each if each of the vectors has two non-zero coordinates, and , respectively, then their order is not . A semi-linear set is stratified if it is a union of finitely many stratified linear subsets.

The Ginsburg-Spanier theorem says that a bounded language is context-free if and only if is a stratified semi-linear set.


The theorem has multiple interpretations. It shows that a context-free language over a singleton alphabet must be a regular language and that some context-free languages can only have ambiguous grammars[further explanation needed]. Such languages are called inherently ambiguous languages. From a formal grammar perspective, this means that some ambiguous context-free grammars cannot be converted to equivalent unambiguous context-free grammars.


  1. ^ a b Kozen, Dexter (1997). Automata and Computability. New York: Springer-Verlag. ISBN 3-540-78105-6.
  2. ^ Håkan Lindqvist. "Parikh's theorem" (PDF). Umeå Universitet.
  3. ^ Parikh, Rohit (1961). "Language Generating Devices". Quarterly Progress Report, Research Laboratory of Electronics, MIT.
  4. ^ Parikh, Rohit (1966). "On Context-Free Languages". Journal of the Association for Computing Machinery. 13 (4): 570–581. doi:10.1145/321356.321364. S2CID 12263468.
  5. ^ Ginsburg, Seymour; Spanier, Edwin H. (1966). "Presburger formulas, and languages". Pacific Journal of Mathematics. 16 (2): 285–296. doi:10.2140/pjm.1966.16.285.