Parikh's theorem

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 whether or not a string with a given number of some terminals is 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

Let ${\displaystyle \Sigma =\{a_{1},a_{2},\ldots ,a_{k}\}}$ be an alphabet. The Parikh vector of a word is defined as the function ${\displaystyle p:\Sigma ^{*}\to \mathbb {N} ^{k}}$, given by[1]

${\displaystyle p(w)=(|w|_{a_{1}},|w|_{a_{2}},\ldots ,|w|_{a_{k}})}$, where ${\displaystyle |w|_{a_{i}}}$ denotes the number of occurrences of the letter ${\displaystyle a_{i}}$ in the word ${\displaystyle w}$.

A subset of ${\displaystyle \mathbb {N} ^{k}}$ is said to be linear if it is of the form

${\displaystyle u_{0}+\langle u_{1},\ldots ,u_{m}\rangle =\{u_{0}+t_{1}u_{1}+\ldots +t_{m}u_{m}\mid t_{1},\ldots ,t_{m}\in \mathbb {N} \}}$ for some vectors ${\displaystyle u_{0},\ldots ,u_{m}}$.

A subset of ${\displaystyle \mathbb {N} ^{k}}$ is said to be semi-linear if it is a union of finitely many linear subsets.

The formal statement of Parikh's theorem is as follows. Let ${\displaystyle L}$ be a context-free language. Let ${\displaystyle P(L)}$ be the set of Parikh vectors of words in ${\displaystyle L}$, that is, ${\displaystyle P(L)=\{p(w)\mid w\in L\}}$. Then ${\displaystyle P(L)}$ is a semi-linear set.

Two languages are said to be commutatively equivalent if they have the same set of Parikh vectors. If ${\displaystyle S}$ is any semi-linear set, the language of words whose Parikh vectors are in ${\displaystyle S}$ is commutatively equivalent to some regular language. Thus, every context-free language is commutatively equivalent to some regular language.

Significance

Parikh's theorem proves 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.

References

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". Quartly Progress Report, Research Laboratory of Electronics, MIT.
4. ^ Parikh, Rohit (1966). "On Context-Free Languages". Journal of the Association for Computing Machinery. 13 (4).