Section: Scientific Foundations

Continuous Computation Theory

Today's classical computability and complexity theory deals with discrete time and space models of computation. However, discrete time models of machines working on a continuous space have been considered: see e.g. Blum, Shub and Smale machines [51] , or recursive analysis [110] . Models of machines working with continuous time and space can also be considered: see e.g. the General Purpose Analog Computer of Claude Shannon [102] .

Continuous models of computation lead to particular continuous dynamical systems. More generally, continuous time dynamical systems arise as soon as one attempts to model systems that evolve over a continuous space with a continuous time. They can even emerge as natural descriptions of discrete time or space systems. Utilizing continuous time systems is a common approach in fields such as biology, physics or chemistry, when a huge population of agents (molecules, individuals, ...) is abstracted into real quantities such as proportions or thermodynamic data [82] , [97] .

Computation theory of continuous dynamical systems allows us to understand both the hardness of questions related to continuous dynamical systems and the computational power of continuous analog models of computations.

A survey on continuous-time computation theory, with discussions of relations between both approaches, co-authored by Olivier Bournez and Manuel Campagnolo, can be found in [54] .