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On the structure of the counting function of sparse context-free languages. (English) Zbl 1160.68407

Summary: We give an exact description of the counting function of a sparse context-free language. Let \(L\) be a sparse context-free language and let \(f_L\) be its counting function. Then there exist polynomials \(P_0, P_1,\dots,P_k-1\), with rational coefficients, and an integer constant \(k_0\), such that for any \(n\geq k_0\) one has \(f_L (n) = p_j (n)\) where \(j\) is such that \(j \equiv n \pmod k\). As a consequence one can easily show the decidability of some questions concerning sparse context-free languages. Finally, we show that for any sparse context-free language \(L\) there exists a regular language \(L'\) such that for any \(n\geq 0\) one has \(f_L (n) = f_L' (n)\) and, therefore, \(f_L\) is rational.

MSC:

68Q45 Formal languages and automata
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