## 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

### Keywords:

sparse language; contex-free language; counting function
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### References:

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