CHAOTIC NEURAL NETS, COMPUTABILITY, AND UNDECIDABILITY - TOWARD A COMPUTATIONAL DYNAMICS

In this article we intend to analyze a chaotic system from the standpo int of its computation capability. To pursue this aim, we refer to a c omplex chaotic dynamics that we characterize via its symbolic dynamics . We show that these dynamic systems are subjected to some typical und ecidable problems. Particularly, we stress the impossibility of decidi ng on a unique invariant measure. This depends essentially on the supp osition of the existence of a fixed universal grammar. The suggestion is thus of justifying a centextual redefinition of the grammar as a fu nction of the same evolution of the system. We propose on this basis a general theorem for avoiding undecidable problems in computability th eory by introducing a new class of recursive functions on different ax iomatizations of numbers. From it a series expansion on n algebraic fi elds can be defined. In such a way, we are able to obtain a very fast extraction procedure of unstable periodic orbits from a generic chaoti c dynamics. The computational efficiency of this algorithm allows us t o characterize a chaotic system by the complete statistics of its unst able cycles. Some examples of these two techniques are discussed. Fina lly, we introduce the possibility of an application of this same class of recursive functions to the calculus of the absolute minimum of ene rgy in neural nets, as far as it is equivalent to a well-formed formul a of a first-order predicate calculus. (C) 1995 John Wiley and Sons, I nc.