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3. Innovative Aspects

The proposed use of operator algebras has several advantages. First, operator algebras constitute themselves as a natural link between algebraic and analytical models.

Second, C -algebras are well studied object. From the mathematical point of view, there exists a quite extensive knowledge. At the same time C -algebras are also a still developing branch of mathematics. To make use of this mathematical knowledge should help to gain a better understanding of various aspects of computation; it might prevent us from reinventing the wheel all over again.

Third, C -algebras are used within several fields closely related to neural networks and other models of dynamic systems. Some of these concern various ``dynamic'' and ``physical'' aspects computation, like Markov Chains [Cuntz and Krieger, 1980], Dynamical Systems [Connes, 1994], and -- last but not least -- Quantum Mechanics [Bratteli and Robinson, 1979]. And there are also several recent approaches in logic and computer science which are related to concepts based in C -algebras (Girard's Geometry of Interaction, Quantales, etc).

Common models in computer science usually are based on discrete mathematics. Analytical models like operator algebras might thus seem superficial. But incorporating elements and methods from mathematical disciplines like functional analysis together with logical frameworks will increase our knowledge about computational processes, e.g. by embedding discrete models (e.g. a proof tree) in a continuous model (e.g. operator algebra) -- after all, it is not obvious that discrete problems need discrete methods to solve them, neither are discrete problems easier to handle (e.g. diophantine equations are by no means simpler than those over the real numbers).

Herbert Wiklicky
Fri Jun 14 13:32:12 BST 1996