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Chaotic neural nets, computability, and undecidability: Toward a computational dynamics
Author(s) -
Basti Gianfranco,
Perrone Antonio L.
Publication year - 1995
Publication title -
international journal of intelligent systems
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.291
H-Index - 87
eISSN - 1098-111X
pISSN - 0884-8173
DOI - 10.1002/int.4550100106
Subject(s) - undecidable problem , computable function , chaotic , computability , computer science , dynamical systems theory , invariant (physics) , mathematics , theoretical computer science , algorithm , decidability , artificial intelligence , physics , quantum mechanics , mathematical physics
In this article we intend to analyze a chaotic system from the standpoint of its computation capability. to pursue this aim, we refer to a complex chaotic dynamics that we characterize via its symbolic dynamics. We show that these dynamic systems are subjected to some typical undecidable problems. Particularly, we stress the impossibility of deciding on a unique invariant measure. This depends essentially on the supposition of the existence of a fixed universal grammar. the suggestion is thus of justifying a contextual redefinition of the grammar as a function of the same evolution of the system. We propose on this basis a general theorem for avoiding undecidable problems in computability theory by introducing a new class of recursive functions on different axiomatizations of numbers. From it a series expansion on n algebraic fields can be defined. In such a way, we are able to obtain a very fast extraction procedure of unstable periodic orbits from a generic chaotic dynamics. the computational efficiency of this algorithm allows us to characterize a chaotic system by the complete statistics of its unstable cycles. Some examples of these two techniques are discussed. Finally, we introduce the possibility of an application of this same class of recursive functions to the calculus of the absolute minimum of energy in neural nets, as far as it is equivalent to a well‐formed formula of a first‐order predicate calculus. © 1995 John Wiley & Sons, Inc.