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Chomsky–Schützenberger Enumeration Theorem
In formal language theory, the Chomsky–Schützenberger enumeration theorem is a theorem derived by Noam Chomsky and Marcel-Paul Schützenberger about the number of words of a given length generated by an unambiguous context-free grammar. The theorem provides an unexpected link between the theory of formal languages and abstract algebra.
The theorem can be used in analytic combinatorics to estimate the number of words of length n generated by a given unambiguous context-free grammar, as n grows large. By applying methods from complex analysis to this equation, the number an{\displaystyle a_{n}} of words of length n generated by G can be estimated, as n grows large. This can be proved as follows: If gk{\displaystyle g_{k}} denotes the number of words of length k{\displaystyle k} in LG{\displaystyle L_{G}}, then for the associated power series holds G(x)=∑k=0∞gkxk=1−x1−2x−1x∑k≥1xk(k+1)/2−1{\displaystyle G(x)=\sum _{k=0}^{\infty }g_{k}x^{k}={\frac {1-x}{1-2x}}-{\frac {1}{x}}\sum _{k\geq 1}x^{k(k+1)/2-1}}.
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