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

arXiv:0807.0718v1 [cs.DM] 4 Jul 2008 The Parikh functions of sparse context-free languages are quasi-polynomials ∗ Flavio D’Alessandro Dipartimento di Matematica, Universit`a di Roma “La Sapienza” Piazzale Aldo Moro 2, 00185 Roma, Italy. e-mail: dalessan@mat.uniroma1.it, http://www.mat.uniroma1.it/people/dalessandro Benedetto Intrigila Dipartimento di Matematica, Universit`a di Roma “Tor Vergata”, via della Ricerca Scientifica, 00133 Roma, Italy. e-mail intrigil@mat.uniroma2.it, Stefano Varricchio Dipartimento di Matematica, Universit`a di Roma “Tor Vergata”, via della Ricerca Scientifica, 00133 Roma, Italy. e-mail varricch@mat.uniroma2.it, http://www.mat.uniroma2.it/~varricch Abstract Let L be a sparse context-free language over a finite alphabet A = {a1, . . . , at} and let fL : Nt →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 Rt, 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 ‘‘Facolt`a di Scienze MM. FF. NN. 2006’’ of the University of Rome ‘‘La Sapienza’’ and fundings CRUI DAAD Program ‘‘Vigoni 2006’’. 1 1 Introduction Given a word w over an alphabet A = {a1, . . . , at}, the Parikh vector of w is the vector (n1, . . . , nt), where for every i, with 1 ≤i ≤t, ni is the number of occurrences of letter ai in w. Given a language L over A, the Parikh map fL is the map which associates with non negative integers n1, . . . , nt, the number fL(n1, . . . , nt) of all the words in L having Parikh vector equal to (n1, . . . , nt). If we impose restrictions on the growth rate of fL we obtain different classes of languages. In particular, a language is termed Parikh slender (see [13]) if there is a positive integer r such that for every n1, . . . , nt, fL(n1, . . . , nt) ≤r holds. In this paper we shall consider the Parikh map of bounded context-free lan- guages. Bounded context-free languages and their properties have been ex- tensively studied in [9], where, in particular, one proves that it is decidable whether a given context-free language is bounded. It is also known that bounded context-free languages are exactly the sparse context-free languages [21]. A for- mal language L is termed sparse if its counting function is upper-bounded by a polynomial or, equivalently, if its Parikh function is upper bounded by a mul- tivariate polynomial. It is clear that the class of sparse context-free languages include that of Parikh slender ones. A quasi-polynomial is a a map F : Nt →N defined by a finite family of multivariate polynomials, with rational coefficients, {p(d1,d2,···,dt) | d1, . . . , dt ∈ N, 0 ≤di < d}, where, for every (x1, . . . , xt) ∈N, if di is the reminder of the division of xi by d, one has: F(x1, . . . , xt) = p(d1,d2,···,dt)(x1, . . . , xt). In this paper we prove that the Parikh map fL of a sparse context free language L can be exactly calculated using a finite number of quasi-polynomials. More precisely, if L is a sparse context-free language, then there exist a partition of Nt into a finite number of polyhedral conic regions R1, . . . , Rs, determined by hyperplanes, through the origin, with rational equations, and a finite number of quasi-polynomials p1, . . . ps, such that for any (n1, . . . , nt) one has: fL(n1, . . . , nt) = pj(n1, . . . , nt) where j is such that (n1, . . . , nt) ∈Rj. This is obtained by reducing the computation of the Parikh map of L to the computation of the number of non-negative solutions of a system of diophantine linear equations of the form n X j=1 aijxj = ni, 1 ≤i ≤t, aij ∈N, (1) as a function F(n1, . . . , nt) of the constants n1, . . . , nt. The latter computation deserves a special mention. It was first considered in the context of Numerical Analysis where, in a celebrated paper by Dahmen and Micchelli [3], it has been proved that the counting function of a Diophantine system of linear equations can be described by a set of quasi-polynomials, under suitable conditions on the matrix of the system. Recently this result has been 2 object of further investigations in [4, 5, 24] where important theorems on the algebraic and combinatorial structure of partition functions have been obtained. In this paper, we present a combinatorial proof of the description of the counting map of the system (1). This proof, which appears to be new, is of ele- mentary character, effective and makes the paper self-contained completely. We remark that the regions R1, . . . , Rs as well as the quasi-polynomials p1, . . . , ps that gives a description of the Parikh map can be effectively computed from an effective presentation of the language L. The decidability of some problems on the Parikh map of context-free languages is an easy c

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