Quantitative Computation by Hilbert Machines

In order to extend classical models of computing with symbols we introduce a model for quantitative computation which is based on innnite-dimensional topological linear structures. In particular machines which operate on data taken from Hilbert spaces will be looked at. These Hilbert machines (and other topological linear machines) allow the adequate treatment of concepts like innniteness and similarity as they are based on a combination of a simple algebraic structure together with a topological one. We rst discuss the diierences between qualitative data representation using symbols and quantitative descriptions. Hilbert machines are then introduced on a concrete as well as on a purely abstract { C-algebraic { level. Furthermore, the relation to other computational models will be investigated. It will also be shown that various novel computational models like Quantum Computation, Girard's Geometry of Interaction, or Neural Networks are actually instances of Hilbert machines. Finally, we will discuss the problem of programming in the context of such linear machines.