## Section: Scientific Foundations

### Computation over continuous structures

Classical recursion theory deals with computability over discrete structures (natural numbers, finite symbolic words). There is growing community of researchers working on the extension of this theory to continuous structures arising in mathematics. One goal is to give foundations of numerical analysis, by studying the limitations, in terms of computability or complexity, of machines when computing with real numbers. Classical questions are : if a function $f:\mathbb{R}\to \mathbb{R}$ is computable in some sense, are its roots computable? in which time? Another goal is to investigate the possibility of designing new computation paradigms, transcending the usual discrete-time, discrete-space computer model initiated by the Turing machine and underlying the modern computers.

While the notion of a computable function over discrete data is captured, according to the Church-Turing thesis, by the model of Turing machines, the situation is more delicate when the data are continuous, and several non-equivalent models exist. We mention computable analysis, which relates computability to topology [46] , [78] ; the Blum-Shub-Smale model (BSS), where the real numbers are treated as elementary entities [37] ; the General Purpose Analog Computer (GPAC) introduced by Shannon [72] where the time is continuous.