A formal scheme for avoiding undecidable problems. Applications to chaotic behavior characterization and parallel computation

Abstract

In this paper 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 and polinomially uneomputable problems. Our hypothesis is that these limitations depend essentially on the supposition of the existence of a fixed universal alphabet. The suggestion is thus of justifying a contextual redefinition of the alphabet symbols 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 defined on different axiomatizations of numbers obtained by a modification of classical Peano's axiom of successor. In such a way, from an experimental viewpoint., 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 this technique applied to classical chaotic attractors are discussed. Finally we discuss some specific topic about pattern recognition and parallel computation. In this context we show how the same class of new recursive functions allows us to avoid some classical limitations demonstrated by Minsky and Papert in Perceptrons concerning the possibility of true parallel computations. These results are applied to the problem of reM time automatic recognition via software of particle tracks in high energy physics experiments.

Intelligence and Reference

“…Two classes of logical symbols with their relations constituted via a common causal relation from the denoted object! It is evident that such a solution of the reference problem can be implemented only in a NC model where <v,w,z> can be interpreted as cognitive agents, particularly as (pseudo-)cycles of the same chaotic attractor, as we and others demonstrated elsewhere (See [30,18,31,32]; [33,34,35]). …”

“…a finitistic computational procedure always convergent in polynomial time, even in chaotic dynamics [30].…”

In search of the person. Towards a real revolution

“…In the naturalistic approach the mathematical laws of physics and their evolution are based on the evolution of the universe, because there is mutual determination between the mathematical laws and the physical processes. The principle of this approach is named computational coarse graining (Perrone 1995;Basti & Perrone & et al 1995;1996). Secondly, the studies on entropy and on the inflation of the universe revealed that the gravitational force can be interpreted as the entropic force that generates information.…”

Intentionality and Foundations of Logic: a New Approach to Neurocomputation

“…This complexity in-deed makes inapplicable to deterministic chaos classical statistical methods of signal analysis. In the last four years, however, one of us, developed a new effective technique of pseudo-cycle extraction of any length, with a computation time growing only linearly with the cycle length [42][43][44][45]. We have the definitive experimental evidence that this method, based on the new foundational ideas discussed in the next Section, can extract practically all the pseudo-cycles of a chaotic dynamics.…”