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

Let L be a sparse context-free language over a finite alphabet A = {a1, . . . , at} and let fL : N t → N be its Parikh (counting) function. We prove that the map fL is a box spline, that is there exists a finite set Π of hyperplanes of R, through the origin, such that fL is a quasi-polynomial on every polyhedral cone determined by Π. Moreover we prove that the Parikh function of such a language is rational. This work was partially supported by MIUR project ‘‘Linguaggi formali e automi: teoria e applicazioni’’. The first author acknowledges the partial support of fundings ‘‘Facoltà di Scienze MM. FF. NN. 2006’’ of the University of Rome ‘‘La Sapienza’’ and fundings CRUI DAAD Program ‘‘Vigoni 2006’’.