ConTComp: Continuous time computation and complexity

1. There are several potencial applications of computation and complexity theory for continuous systems. Among these are verification and control of continuous time systems, analog computation and the analysis of continuous models of large scale discrete systems like distributed computing in sensor and telecommunication networks. Toward these goals, a variety of models of continuous time computation were compared and explored. Their dynamics, computational bounds and robustness to noise have been assessed. Replace with a representative image, composite image or figure. Image resolution must be compatible with high quality printing. • Research team • Manuel Campagnolo • Paula Gouveia • Daniel Graça • Carlos Lourenço • Kerry Ojakian Contract number if applicable • ConTComp: Continuous time computation • and complexity • Context • Continuous time computation has been considered under two approaches. One is related to continuous time analog machines and has his roots in models of natural or artificial analog machinery. The other, which is broader in scope, arises from system theory, and encompasses for instance the computational analysis of hybrid systems and timed automata. The problems addressed by the theory of continuous time computation come from, among others, the fields of verification, control theory, VLSI design and neural networks. • Unlike discrete computation which is unified under the Turing paradigm, continuous time computation covers a wide variety of models which are apparently distinct. To build a fruitful theory of computation for continuous systems one needs to unify the existent models under a common framework. Complexity of continous time systems is still a ill-defined notion. In general, continuous systems can suffer space and time contractions, which makes it difficult to compare them with respect to complexity. While the notion of complexity is easier to define for dissipative systems that converge to attractors, this still relies on the notion of “natural” time. Complexity of continuous time models can instead be studied with respect to the computational complexity of the functions that those models allow to compute. In that case, standard and continuous models match nicely for a wide range of complexity classes.

2. Replace with a representative image, or figure. Image resolution must be compatible with high quality printing. Replace with a representative image, or figure. Image resolution must be compatible with high quality printing. • Discrete computation by a continuous • dynamical system: the Poincaré map. • Neural network inspired by Nature with excitatory and inhibitory neurons for dynamical computation. • Results • Systems of polynomial differential equations have been thoroughly analized. These are equivalent to Shannon’s GPAC, a model of the analog computer known as the Differential Analyzer. Turing machines can be robustly simulated by such systems. Using a new and more natural notion of computability for continuous dynamical systems, polynomial differential equations compute precisely the real functions that type-2 Turing machines compute. This result establishes an equivalence between two major paradigms of computation over the reals and set the foundations for a unified theory. On the negative side, it was shown that the boundness of the domain of definition is undecidable which prevents in principle the verification of processes modeled by such dynamical systems. • New techniques were developed to provide machine-independent proofs of the equivalence between different notion of computability for functions over the reals. It was found that a correct notion of approximation, more flexible than exact computation, is at the core of the equivalence between models. • One other area of application of the project has been neural networks described by ordinary or partial differential equations, with rich dynamical features, and connectivity patterns that mimic the massively parallel nets of Biology. The role of chaos in the computation achieved by those systems has been elucidated. This has been applied to diverse tasks like Selenoprotein discovery and pattern processing. • Perspectives • Continuous time systems arise as soon as one attempts to model systems which evolve in continuous space and time. They also emerge as natural descriptions of discrete time or space systems when a huge population of agents (molecules, individuals, processors) is abstracted into real quantities such as proportions or thermodynamic data. Therefore, continuous time models may have a prominent role in analyzing massively parallel systems. The results about decidability, robustness to noise and computational abilities of continuous time models resulting from the project may be used to better understand such systems. In collaboration with INRIA (Nancy) these issues will be further explorer. • Contact information mlc@math.isa.utl.pt