COntinuous tiMe comPUTations. Computations on the Reals.

We propose to contribute to understand computation theories for continuous time systems. This is motivated by • understanding algorithmic complexity of automatic verification procedures for continuous and hybrid systems; • understanding some new models of computations. New models of computations under study include analog electronics models, and some recent sensor and telecommunication networks models. Hybrid systems include all systems that mix continuous dynamics with discrete transitions. We propose to do so to develop the model of R-recursive functions introduced by Moore in [49], using the recent framework of [24]. We expect by the end of this project to • Develop significantly computation theory for continuous time systems to noisy and robust systems. Expected implications are contributions to understand a famous conjecture in verification about decidability and termination of verification procedures for hybrid systems, and hence possibly new verification tools. • Revisit computations on the reals, to avoid references to Turing machines. Expected implications are lower and upper bounds on the algorithmic complexity of natural problems in verification and control , motivated by automatic verification procedures for continuous and hybrid systems. • Understand deeply some new computational models. Expected implications are better understanding of some recent models of sensor and telecommunication networks, that could be used to better program them. • Contribute to understand better the computational properties of models of natural inspiration, and in particular contribute to understand whether edge-of-chaos regimes may provide an appropriate setting for computational processes.