The Parikh functions of sparse context-free languages are quasi-polynomials

We prove that the Parikh map of a bounded context-free language is a box spline. Moreover we prove that in this case, such a function is rational.

💡 Research Summary

**
The paper investigates the precise algebraic nature of Parikh maps for a special class of context‑free languages, namely bounded (or “sparse”) CFLs, and establishes that these maps are not only semilinear (as guaranteed by Parikh’s theorem) but enjoy a much richer structure: they are box splines, rational generating functions, and consequently quasi‑polynomials.

The authors begin by recalling that the Parikh map (\Psi) sends a word over an alphabet (\Sigma) to a vector of symbol frequencies in (\mathbb{N}^{|\Sigma|}). While Parikh’s theorem tells us that (\Psi(L)) is a finite union of linear sets for any CFL (L), it does not describe the exact functional relationship between the input word and its Parikh vector. The paper narrows the focus to bounded CFLs, i.e., languages that can be written as a subset of a product of finitely many Kleene stars (w_{1}^{}w_{2}^{}\dots w_{k}^{*}) for fixed words (w_{i}). This restriction enables a translation of the language’s derivations into a system of non‑negative integer variables (x_{1},\dots,x_{k}) that count how many times each block (w_{i}) is repeated. Consequently, the Parikh vector of any word in the language can be expressed as a linear map \