Valiant’s Model: From Exponential Sums to Exponential Products

Abstract

We study the power of big products for computing multivariate polynomials in a Valiant-like framework. More precisely, we define a new class VΠP 0 as the set of families of polynomials that are exponential products of easily computable polynomials. We investigate the consequences of the hypothesis that these big products are themselves easily computable. For instance, this hypothesis would imply that the nonuniform versions of P and NP coincide. Our main result relates this hypothesis to Blum, Shub and Smale's algebraic version of P versus NP. Let K be a field of characteristic 0. Roughly speaking, we show that in order to separate P K from NP K using a problem from a fairly large class of "simple" problems, one should first be able to show that exponential products are not easily computable. The class of"simple"problems under consideration is the class of NP problems in the structure (K, +, −, =), in which multiplication is not allowed. Keywords:Algebraic complexity, Valiant's model, Blum-Shub-Smale's model, big products. RésuméCet articleétudie la puissance des gros produits pour le calcul de polynômesà plusieurs variables dans le cadre de la théorie de Valiant. Plus précisément, nous définissons pour cela une nouvelle classe VΠP 0 de familles de polynômes : il s'agit des produits de taille exponentielle de polynômes facilement calculables. Nousétudions les conséquences de l'hypothèse que ces gros produits sont euxmêmes facilement calculables. Par exemple, cela impliquerait que les versions non-uniformes de P et NP coïncident. Le résultat principal est un lien avec les classes algébriques P et NP du modèle BSS sur un corps K de caractéristique nulle. On pourrait l'énoncer ainsi : si nous voulons séparer P K de NP K grâceà des problèmes issus d'un ensemble important de problèmes « simples », il faut d'abordêtre capable de montrer que nos gros produits ne sont pas facilement calculables. L'ensemble des problèmes « simples » en question est NP sur la structure (K, +, −, =), dans laquelle la multiplication n'est pas autorisée.

VPSPACE and a Transfer Theorem over the Reals

“…This paper is therefore in the same spirit as Koiran & Perifel (2006). In that paper, we showed that if certain polynomials can be evaluated efficiently then certain decision problems become easy.…”

Interpolation in Valiant’s Theory

“…It contains in particular Twenty Questions and Subset Sum. We will need a result from [13]: At last, as a corollary of Theorem 3 again, we obtain the following result concerning question (*), suggesting that it will be hard to refute. As pointed out in the introduction, this result does not give any evidence concerning the answer to question (*) since the separation VP 0 = VNP 0 is very likely to be true.…”

“…One natural strategy for separating P C from NP C would be to exhibit a problem A in NP C \ P C . Drawing on results from [13], we show that this strategy is bound to fail for a fairly large class of "simple" problems A, unless one can prove that VP 0 = VNP 0 . The class of "simple" problems that we have in mind is NP (C,+,−,=) .…”

“…Unfortunately, in spite of all the work establishing close connections between the boolean model of computation and the algebraic models of Valiant and of Blum, Shub and Smale [2,4,5,8,9,[13][14][15] no such transfer theorem is known. In fact, we do not know of any hypothesis from boolean complexity theory that would imply the equality VP = VNP (but transfer theorems in the opposite direction were established in [4]).…”

VPSPACE and a Transfer Theorem over the Complex Field

“…This paper is therefore in the same spirit as [12] and [13]. In the present paper we work with the class of polynomial families VPSPACE introduced in [13].…”