A Formal Scheme for Avoiding Undecidable Problems - Applications to Chaotic Behaviour, Characterization and Parallel Computation

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 (incomputable 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.