Parikhs theorem
Parikh's theorem A theorem in formal language theory that concerns the nature of contextfree languages when order of letters is disregarded.
Let the alphabet Σ be the set {a_{1},…,a_{n}}. The letter distribution, φ(w), of a Σword w is the ntuple 〈N_{1},…,N_{n}〉
with N_{i} the number of occurrences of a_{i} in w. The Parikh image, φ(L), of a Σlanguage L is {φ(w)  w ∈ L}
i.e. the set of all letterdistributions of words in L. L_{1} and L_{2} are letterequivalent if φ(L_{1}) = φ(L_{2})
Letter distributions may be added componentwise as vectors. This leads to the following: a set S of letter distributions is linear if, for some distributions d and d_{1},…,d_{k}, S is the set of all sums formed from d and multiples of d_{i}. S is semilinear if it is a finite union of linear sets.
Parikh's theorem now states that if L is contextfree φ(L) is semilinear. It can also be shown that φ(L) is semilinear if and only if L is letterequivalent to a regular language. Hence any contextfree language is letterequivalent to a regular language – although not all such languages are contextfree.
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