Speaker : Sebastien Tavenas ( LAMA, Université de Savoie Mont-Blanc)
Title : SuperPolynomial lower bounds for circuits of constant depth
Any multivariate polynomial P(X_1,…,X_n) can be written as a sum of monomials, i.e., a sum of products of variables and constants of the field. The natural size of such an expression is the number of monomials. But, what happens if we add a new level of complexity by considering expressions of the form: sum of products of sums (of variables and constants)? Now it becomes less clear how to show that a given polynomial has no small expression. In this talk we will solve exactly this problem. More precisely, we prove that some explicit polynomials do not have polynomial-sized sum-of-products-of-sums (SPS) representations. We can also obtain similar results for SPSPs, SPSPSs, etc. for all expressions of constant depth. We will see also why this problem can be seen an important step for obtaining lower bounds on the size of algebraic circuits.
Speaker : Antonio E. Porreca (CANA, LIS)
Title : The algebra of alternation and synchronisation of finite dynamical systems
Many phenomena of scientific and engineering interest can be modelled as finite dynamical systems, that is, as deterministic transition functions iterated over an abstract set of states. Dynamical systems can be put together (or, from the opposite point of view, analysed) in terms of composition operations, among which two of the most basic ones are alternative and synchronous execution. These give rise to a commutative semiring (a “ring without subtraction”) over dynamical systems up to isomorphism, with some rather complex algebraic properties, notably the lack of unique factorisation into irreducibles. Several interesting decomposition properties can then be expressed in terms of polynomial equations. We discuss computability and complexity issues in solving these equations (most interesting cases are intractable, and the general problem is undecidable), as well as some properties of irreducible systems and the (possible) existence of prime ones.