Valiant?s model and the cost of computing integers

Koiran, Pascal

doi:10.1007/s00037-004-0186-2

Cited by 25 publications

(27 citation statements)

References 16 publications

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“…The following result answers in the affirmative some questions posed by Koiran (2004). From the very general proof technique, it becomes obvious that this result actually holds for a large class of integer sequences, so the choice of the sequences below is for illustration and just motivated by Koiran's question.…”

Section: Connecting Valiant's Model To Integers and Univariate Polynomentioning

confidence: 64%

“…Koiran (2004) proved the following weaker version of the statement regarding the factorials: if (n!) is hard to compute, then VP 0 = VNP 0 or P = PSPACE.…”

Section: The Sequence Of Taylor Approximationsmentioning

confidence: 99%

“…This constant-free model has been systematically studied by Malod (2003). We briefly present the salient features following Koiran (2004).…”

Section: The Constant-free Valiant Modelmentioning

confidence: 99%

“…Lemma 4.4 in Koiran (2004) implies τ (a(n)) ≤ (2e(n) + 3)τ (2 e(n) a(n)). Altogether, we obtain τ (a(n)) = (log n) O(1) .…”

Section: Connecting Valiant's Model To Integers and Univariate Polynomentioning

confidence: 99%

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On Defining Integers And Proving Arithmetic Circuit Lower Bounds

Bürgisser

2009

comput. complex.

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Let τ (n) denote the minimum number of arithmetic operations sufficient to build the integer n from the constant 1. We prove that if there are arithmetic circuits of size polynomial in n for computing the permanent of n by n matrices, then τ (n!) is polynomially bounded in log n. Under the same assumption on the permanent, we conclude that the Pochhammer-Wilkinson polynomials n k=1 (X − k) and the Taylor approximations n k=0 1 k! X k and n k=1 1 k X k of exp and log, respectively, can be computed by arithmetic circuits of size polynomial in log n (allowing divisions). This connects several so far unrelated conjectures in algebraic complexity.

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Section: Connecting Valiant's Model To Integers and Univariate Polynomentioning

confidence: 64%

“…Koiran (2004) proved the following weaker version of the statement regarding the factorials: if (n!) is hard to compute, then VP 0 = VNP 0 or P = PSPACE.…”

Section: The Sequence Of Taylor Approximationsmentioning

confidence: 99%

“…This constant-free model has been systematically studied by Malod (2003). We briefly present the salient features following Koiran (2004).…”

Section: The Constant-free Valiant Modelmentioning

confidence: 99%

“…Lemma 4.4 in Koiran (2004) implies τ (a(n)) ≤ (2e(n) + 3)τ (2 e(n) a(n)). Altogether, we obtain τ (a(n)) = (log n) O(1) .…”

Section: Connecting Valiant's Model To Integers and Univariate Polynomentioning

confidence: 99%

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On Defining Integers And Proving Arithmetic Circuit Lower Bounds

Bürgisser

2009

comput. complex.

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“…Indeed, as explained below the study of VΠP 0 leads to meaningful results about the complexity of decision problems. This paper is therefore in the same spirit as [9], where it is shown that certain sequences of integers become easy to compute if certain classes of polynomial families coincide.…”

Section: Introductionmentioning

confidence: 97%

Valiant’s Model: From Exponential Sums to Exponential Products

Koiran

Perifel

2006

Lecture Notes in Computer Science

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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.

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