Section: New Results
Computation over the Continuum
Participants : Walid Gomaa, Emmanuel Hainry, Mathieu Hoyrup.
While the notion of computable function over the natural numbers is universally accepted, its counterpart over continuous spaces, as the real line, is subject to discussion. The wide range of possible formalizations partly has its origin in the diversity of structures continuous spaces can be endowed with: e.g. the real line can be seen as a topological space, a measure space, a field, a vector space, a manifold, etc., depending on the particular problem one is concerned with. It happens that the topological structure of the set of real numbers is usually implicitly taken as a reference for the theory of computable functions.
On the other hand, we are interested in the analysis of dynamical systems from the computability point of view. It happens that the probabilistic framework is of much interest to understand the behavior of dynamical systems, as it enables one to distinguish physically relevant features of such systems, providing at the same time a way to understand robustness to noise. In [26] , Mathieu Hoyrup, together with Peter Gács and Cristóbal Rojas, fully characterize the algorithmic effectivity of a natural class of properties arising naturally in dynamical systems.
As a result, restricting to a topological approach is somewhat limitative and we are interested in a theory of computable functions that would fit well with probabilities. We carry out such a development in [32] . Here the algorithmic theory of randomness, initiated by Martin-Löf in 1966 [92] , come into play. This theory offers a way to distinguish, in a probability space, elements that are plausible w.r.t. the probability measure put onto the space, the random elements. This theory is already at the intersection between probability and computability. In [32] Mathieu Hoyrup and Cristóbal Rojas show that it gives a powerful and elegant way of handling computability in a probabilistic context. They present applications of this framework in [33] .
Olivier Bournez, Walid Gomaa, and Emmanuel Hainry presented in [39] a framework that uses approximation to characterize both computability and complexity classes of functions from recursive analysis. This work provides an algebraical characterization of polynomial-time computable functions in the sense of Ko [84] and also extends techniques introduced in [59] for comparing discrete models with continuous models.
Walid Gomaa in [29] compares between computation over the space of continuous rational functions and the corresponding space of real functions. This investigation provides deeper insights into the role that continuity and smoothness of a real function play in the computability and/or complexity of the computation of such function.
