13 mai 2019

Amaury Pouly (IRIF - CNRS)

In 1941, Claude Shannon introduced a continuous-time analog model of
computation, namely the General Purpose Analog Computer (GPAC). The
GPAC is a physically feasible model in the sense that it can be
implemented in practice through the use of analog electronics or
mechanical devices. It can be proved that the functions computed by a
GPAC are precisely the solutions of a special class of differential
equations where the right-hand side is a polynomial. Analog computers
have since been replaced by digital counterpart. Nevertheless, one can
wonder how the GPAC could be compared to Turing
machines.

A few years ago, it was shown that Turing-based paradigms and
the GPAC have the same computational power. However, this result did
not shed any light on what happens at a computational complexity
level. In other words, analog computers do not make a difference about
what can be computed; but maybe they could compute faster than a
digital computer. A fundamental difficulty of continuous-time model is
to define a proper notion of complexity. Indeed, a troubling problem is
that many models exhibit the so-called Zeno’s phenomenon, also known as
space-time contraction.

In this talk, I will present results from my thesis that give several
fundamental contributions to these questions. We show that the GPAC has
the same computational power as the Turing machine, at the complexity
level. We also provide as a side effect a purely analog, machine-
independent characterization of P and Computable Analysis.

I will present some recent work on the universality of polynomial
differential equations. We show that when we impose no restrictions at
all on the system, it is possible to build a fixed equation that
is universal in the sense it can approximate arbitrarily well any
continuous curve over R, simply by changing the initial condition of
the system.

If time allows, I will also mention some recent application of this
work to show that chemical reaction networks are strongly Turing
complete with the differential semantics.