Growth and ergodicity of context-free languages

Ceccherini-Silberstein, Tullio; Woess, Wolfgang

doi:10.1090/S0002-9947-02-03048-9

Growth and ergodicity of context-free languages
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Trans. Amer. Math. Soc. 354 (2002), 4597-4625 Request permission

Abstract:

A language $L$ over a finite alphabet $\boldsymbol \Sigma$ is called growth-sensitive if forbidding any set of subwords $F$ yields a sublanguage $L^{F}$ whose exponential growth rate is smaller than that of $L$. It is shown that every ergodic unambiguous, nonlinear context-free language is growth-sensitive. “Ergodic” means for a context-free grammar and language that its dependency di-graph is strongly connected. The same result as above holds for the larger class of essentially ergodic context-free languages, and if growth is considered with respect to the ambiguity degrees, then the assumption of unambiguity may be dropped. The methods combine a construction of grammars for $2$-block languages with a generating function technique regarding systems of algebraic equations.

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