Non Recursive Functions Have Transcendental Generating Series

In this note we shaii prove that non primitive recursive functions have transcendental generating series. This resuit translates a certain measure of the complexity of a fonction, the fact of not being primitive recursive, into another measure of the complexity of the generating series associated to the function, the fact of being transcendental. Résumé. On démontre que les fonctions qui ne sont pas recursives primitives ont des séries génératrices transcendantes. Ce résultat traduit une certaine mesure de complexité d'une fonction, le fait de ne pas être recursive primitive, dans une autre mesure de la complexité de la série génératrice associée à cette fonction, le fait d'être transcendante. A relation between the generating series of a function and its algebraic character was established in [2], proving the following resuit: Let L be an unambiguous context-free language. Then the generating series of the language Ç (z) = S ÇM z" where ÇM = Card { œ e L/1 CO | = n }, is algebraic, This theorem was employed to prove that some languages are inherently ambiguous in [1] and [3]. Our main resuit is the following THEOREM 1: Let ƒ : N^Q be a total function and consider the generating series 00 eQ((z))*/ if r is the degree in Y of P we know that P has exactly r different roots in Q((z))*lg> cpA (z), . . ., (z) is one of them> To distinguish q>(z) between the others roots of P in Q[[z]]alg we can consider a natural number k and rational numbers c0, . . .,cfc such that (p(z) is the only root of P begining with co+ . . . +ckz . We will see now how to compute ƒ (n) for every n from the finite collection of data { P, k, cOï . . M q }. This is done using the indeterminate coefficients method ; i. e. once we know c0, . . ., ch such that cp (z) is the only root of P in Q [[]]aig begining with c0 + . . . + ch z > we consider the equality P (z, c0 + . . * + ch z H + A, z + . . . ) = 0. This translates into an infinité System of équations corresponding to the coefficients of the resulting series which must be all zero. But the équation corresponding to the term of least degree is an équation in X having only one solution because there is only one possible root of P beggining with co+ . . +chz . In fact, it is well known Informatique théorique et Applications/Theoretical Informaties and Applications TRANSCENDENTAL GENERATING SERIES 447 that when Computing the solutions of P in Q[l»]alg af ter this point the équations giving the coefficients of the solutions are linear in this coefficient, L e> the indeterminate X turns out to be a polynomial fonction of the previöusly computed numbers co> -.. •, ch (see [$]). We then compute ch+i, the only solution of this équation in h We have then seen that knowing the first ft + 1 coefficients co>. . .,ch we can compute the following ch+Ï9 i. e. we can recursively compute ƒ (n) for every », An alffiost immédiate conséquence of theorem 1 is CÔROLLARV: Let L a £* be a mcursively enumerable language, where £ is a jfînite alphabet, such that its generming séries mentiôned above ïs algebraic, Then L is recursîve* Prôôf: Let us consider ƒ : N -> M defined by ƒ («)—ÇB. Following theorem 1, we have that ƒ is recursim Then, we can compute the number of words in L having length n. Since there are only a finite number of words of length n in £* we compute in parallel their membership to L until we get Çn of them. At this moment we know we have all of them and we stop the computation. From this we conclude that every recursively enumerable but non recursive language in 2* has a transcendental generating series. In a classic paper about computability of real numbers (see [4]) Rice has shown the following result: the field of recursive real numbers ê is real closed (u e. ê [i] is algebraically closed). As a trivial conséquence of that we get that every real algebraic number is recursive. In some sense our theorem 1 is an analog of this last result. In the last part of this note we will push this analogy a bit further. DÉFINITION: Let Comp = {Q recursive such that cp = 2 ƒ (n) z }. We will say that Comp is the ring of computable power series over Q. Our last result is the following THEOREM 2: Comp is algebraically closed in Q [[z]]. Proof: Let cp e Q [[z]] and \|/0, . . ., \|/m e Comp such that 9 is a root of