Analytic models and ambiguity of context-free languages
Theoretical Computer Science
The growth function of context-free languages
Theoretical Computer Science
Introduction to Computer Theory
Introduction to Computer Theory
Handbook of Formal Languages
Characterizing Regular Languages with Polynomial Densities
MFCS '92 Proceedings of the 17th International Symposium on Mathematical Foundations of Computer Science
Note: on universally easy classes for NP-complete problems
Theoretical Computer Science
Approximate recognition of non-regular languages by finite automata
ACSC '05 Proceedings of the Twenty-eighth Australasian conference on Computer Science - Volume 38
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The density of a language is defined as the function dL(n) = |L ∩ Σn| and counts the number of words of a certain length accepted by L. The study of the density of regular and context-free languages has attracted some attention culminating in the fact that such languages are either sparse, when the density can be bounded by a polynomial, or dense otherwise. We show that for all nonambiguous context-free languages the number of accepted words of a given length n can also be computed recursively using a finite combination of the number of accepted words smaller than n, or dL = Σj=1k ujdL(n - j). This extends an old result by Chomsky and provides us with a more expressive description and new insights into possible applications of the density function for such languages as well as possible characterizations of the density of higher languages.