Chomsky–Schützenberger Enumeration Theorem
The Chomsky-Schützenberger enumeration theorem connects formal language theory and abstract algebra, focusing on word counts in unambiguous context-free grammars. It has applications in combinatorics and highlights ambiguity in certain languages.
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The Chomsky-Schützenberger enumeration theorem, developed by Noam Chomsky and Marcel-Paul Schützenberger, establishes a connection between formal language theory and abstract algebra. It focuses on the number of words generated by an unambiguous context-free grammar of a given length. The theorem states that if a context-free language has an unambiguous grammar, the power series representing the number of words of different lengths is algebraic over rational numbers. This theorem has applications in analytic combinatorics for estimating word counts as the length grows. It can also be used to demonstrate the inherent ambiguity of certain context-free languages, like the Goldstine language, showing that they do not admit unambiguous context-free grammars. The theorem's proofs are detailed in works by Kuich & Salomaa (1985) and Panholzer (2005), while its applications are further explored in various academic publications.
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