Nonnegative Rank Measures and Monotone Algebraic Branching Programs

Inspired by Nisan\u27s characterization of noncommutative complexity (Nisan 1991), we study different notions of nonnegative rank, associated complexity measures and their link with monotone computations. In particular we answer negatively an open question of Nisan asking whether nonnegative rank characterizes monotone noncommutative complexity for algebraic branching programs. We also prove a rather tight lower bound for the computation of elementary symmetric polynomials by algebraic branching programs in the monotone setting or, equivalently, in the homogeneous syntactically multilinear setting

Etude de l'extension de la notion d'abstraction du lambda-calcul.

Le programme Elody repose principalement sur les concepts du lambda-calcul, notamment les notions d'abstraction et d'application. L'abstraction classique a été étendue dans [leplatre] afin de permettre des manipulations plus puissantes de lambda-termes. Cette extension reposait sur une relation de généralité définie entre deux termes. Nous étudions ici plus précisément comment définir une relation de généralité entre deux termes, dans un but un peu différent du précédent: nous souhaitons pouvoir représenter des ensembles de termes, avec comme intuition qu'un terme représente lénsemble de tous les termes moins généraux que lui. On cherche donc d'abord à obtenir une définition précise à partir de cette idée intuitive, en envisageant différentes définitions possibles, et on étudie les conséquences du choix qui semble le plus logique. La définition adoptée permet de définir une relation d'ordre et une relation d'équivalence qu'il convient de caractériser. On montre ensuite qu'il est possible de définir la borne supérieure et la borne inférieure d'un ensemble fini de termes, ce qui correspond à l'union et à l'intersection pour les ensembles de termes. On obtient finalement un cadre théorique assez net autour de la notion de généralité, qui permet dénvisager d'autres développements en manipulant des ensembles de termes. A titre déxemple, on présente les algorithmes permettant déffectuer les opérations sur les termes discutées précédemment, et enfin le code source commenté d'une implémentation simple en Caml

Towards Optimal Depth-Reductions for Algebraic Formulas

Classical results of Brent, Kuck and Maruyama (IEEE Trans. Computers 1973) and Brent (JACM 1974) show that any algebraic formula of size s can be converted to one of depth O(log s) with only a polynomial blow-up in size. In this paper, we consider a fine-grained version of this result depending on the degree of the polynomial computed by the algebraic formula. Given a homogeneous algebraic formula of size s computing a polynomial P of degree d, we show that P can also be computed by an (unbounded fan-in) algebraic formula of depth O(log d) and size poly(s). Our proof shows that this result also holds in the highly restricted setting of monotone, non-commutative algebraic formulas. This improves on previous results in the regime when d is small (i.e., d = s^o(1)). In particular, for the setting of d = O(log s), along with a result of Raz (STOC 2010, JACM 2013), our result implies the same depth reduction even for inhomogeneous formulas. This is particularly interesting in light of recent algebraic formula lower bounds, which work precisely in this "low-degree" and "low-depth" setting. We also show that these results cannot be improved in the monotone setting, even for commutative formulas

Towards Optimal Depth-Reductions for Algebraic Formulas

Classical results of Brent, Kuck and Maruyama (IEEE Trans. Computers 1973) and Brent (JACM 1974) show that any algebraic formula of size s can be converted to one of depth O(log s) with only a polynomial blow-up in size. In this paper, we consider a fine-grained version of this result depending on the degree of the polynomial computed by the algebraic formula. Given a homogeneous algebraic formula of size s computing a polynomial P of degree d, we show that P can also be computed by an (unbounded fan-in) algebraic formula of depth O(log d) and size poly(s). Our proof shows that this result also holds in the highly restricted setting of monotone, non-commutative algebraic formulas. This improves on previous results in the regime when d is small (i.e., d<<s). In particular, for the setting of d=O(log s), along with a result of Raz (STOC 2010, JACM 2013), our result implies the same depth reduction even for inhomogeneous formulas. This is particularly interesting in light of recent algebraic formula lower bounds, which work precisely in this ``low-degree" and ``low-depth" setting. We also show that these results cannot be improved in the monotone setting, even for commutative formulas

Homomorphism Polynomials Complete for VP

The VP versus VNP question, introduced by Valiant, is probably the most important open question in algebraic complexity theory. Thanks to completeness results, a variant of this question, VBP versus VNP, can be succinctly restated as asking whether the permanent of a generic matrix can be written as a determinant of a matrix of polynomially bounded size. Strikingly, this restatement does not mention any notion of computational model. To get a similar restatement for the original and more fundamental question, and also to better understand the class itself, we need a complete polynomial for VP. Ad hoc constructions yielding complete polynomials were known, but not natural examples in the vein of the determinant. We give here several variants of natural complete polynomials for VP, based on the notion of graph homomorphism polynomials

Polynômes et coefficients

Valiant defines algebraic analogues of the classes P and NP. We characterize the classes VP and VQP, yielding a simplified proof of VNP = VNPe and of the VQP-completeness of the determinant, and a proof of a conjecture by Bürgisser. The classes VPo and VNPo, defined without arbitrary constants, yield a link between the complexity of a polynomial and that of its coefficient function: VNPo is stable for the operation of taking coefficient functions; claiming that this holds for VPo is equivalent to VPo = VNPo. For polynomials of unbounded degree, one needs efficient computations of binomial coefficients, which can be done in positive characteristic but are unlikely in characteristic 0. At last we study the related problem of the effect of derivation on complexity. After a new proof of a result by Kaltofen (the number of variables matters more than the derivation order) we show how to simultaneously compute partial derivatives.Valiant définit des analogues algébriques des classes P et NP. Nous caractérisons les classes VP et VQP, d'où une preuve simplifiée de VNP = VNPe et de la VQP-complétude du déterminant, et la preuve d'une conjecture de Bürgisser. Les classes VPo et VNPo, définies sans constantes arbitraires, donnent facilement un lien entre la complexité d'un polynôme et celle de sa fonction coefficient: VNPo est stable par passage à la fonction coefficient et réciproquement; supposer ce résultat pour VPo est équivalent à VPo = VNPo. Pour traiter le cas du degré non borné, il faut un calcul rapide des coefficients binomiaux, faisable en caractéristique positive et peu probable en caractéristique 0. Nous étudions enfin un problème lié: l'effet de la dérivation sur la complexité. Nous retrouvons le résultat de Kaltofen (le nombre de variables fait exploser la taille plus que l'ordre de dérivation) et donnons un calcul simultané des dérivées partielles

Universal Relations and #P-complteness

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