Seminars

Title: Bornes inférieures superpolynomiales pour les circuits de profondeur constante

Abstract:
Tout polynôme multivarié P(X_1,...,X_n) peut être écrit comme une somme de
monômes, i.e., une somme de produits de variables et de constantes du corps.
La taille naturelle d'une telle expression est le nombre de monômes. Mais,
que se passe-t-il si on rajoute un nouveau niveau de complexité en
considérant les expressions de la forme : somme de produits de sommes (de
variables et de constantes) ? Maintenant, il devient moins clair comment
montrer qu'un polynôme donné n'a pas de petite expression. Dans cet exposé
nous résoudrons exactement ce problème. Plus précisément, nous prouvons que
certains polynômes explicites n'ont pas de représentations "somme de
produits de sommes'' (SPS) de taille polynomiale. Nous pouvons aussi obtenir
des résultats similaires pour les SPSP, SPSPS, etc... pour toutes les
expressions de profondeur constante.
"

titre: Type-based complexity analysis in a parallel process calculus

Abstract:
Some type systems have been designed to analyse statically the time
coplexity of functional languages. A natural question is whether this approach
can be extended to parallel languages. We address this problem for the
Pi-calculus, a paradigmatic calculus for parallel and concurrent computation.
In Pi-calculus, processes communicate through channels that can carry values
and channel names. We will define notions of sequential and parallel complexity
for Pi-calculus, and present a type system that provides an upper bound on the
time complexity of processes.
This is based on joint work with Alexis Ghyselen (ESOP 2021).

Based on: link.springer.com/chap.....9-3_3

Zoom link: univ-lille-fr.zoom.us/j/95419000064



Titre: Lower bound for arithmetic circuits via the Hankel matrix

Abstract: We study the complexity of representing polynomials by arithmetic
circuits in both the commutative and the non-commutative settings. To
analyse circuits we count their number of parse trees, which describe the
non-associative computations realised by the circuit. In the non-commutative
setting a circuit computing a polynomial of degree d has at most 2^{O(d)}
parse trees. Previous superpolynomial lower bounds were known for circuits
with up to 2^{d^{1/3-ε}} parse trees, for any ε>0. Our main result is to
reduce the gap by showing a superpolynomial lower bound for circuits with
just a small defect in the exponent for the total number of parse trees,
that is 2^{d^{1-ε}}, for any ε>0. In the commutative setting a circuit
computing a polynomial of degree d has at most 2^{O(d \\log d)} parse trees.
We show a superpolynomial lower bound for circuits with up to 2^{d^{1/3-ε}}
parse trees, for any ε>0. When d is polylogarithmic in n, we push this
further to up to 2^{d^{1-ε}} parse trees. While these two main results hold
in the associative setting, our approach goes through a precise
understanding of the more restricted setting where multiplication is not
associative, meaning that we distinguish the polynomials (xy)z and yz).
Our first and main conceptual result is a characterization result: we show
that the size of the smallest circuit computing a given non-associative
polynomial is exactly the rank of a matrix constructed from the polynomial
and called the Hankel matrix. This result applies to the class of all
circuits in both commutative and non-commutative settings, and can be seen
as an extension of the seminal result of Nisan giving a similar
characterization for non-commutative algebraic branching programs. Our key
technical contribution is to provide generic lower bound theorems based on
analyzing and decomposing the Hankel matrix, from which we derive the
results mentioned above. The study of the Hankel matrix also provides a
unifying approach for proving lower bounds for polynomials in the
(classical) associative setting. We demonstrate this by giving alternative
proofs of recent lower bounds as corollaries of our generic lower bound
results.

Title : One-Tape Turing Machine and Branching Program Lower Bounds for MCSP
Abstract:
eccc.weizmann.ac.il/report/2020/103/

Speaker' webpage : dimyrisiotis.github.io/