On The Dynamical Nature Of Computation

Dynamical Systems theory generally deals with fixed point iterations of continuous functions. Computation by Turing machine although is a fixed point iteration but is not continuous. This specific category of fixed point iterations can only be studied using their orbits. Therefore the standard notion of chaos is not immediately applicable. However, when a suitable definition is used, it is found that the notion of chaos and fractal sets exists even in computation. It is found that a non terminating Computation will be almost surely chaotic, and autonomous learning will almost surely identify fractal only sets. 1. Computation As Dynamical System In general we associate the term dynamical system to a fixed point iteration of a continuous function ‘f ’ as such: xn = f(xn−1). Turing Machines can be treated as a fixed point iteration with suitable rationalisation of the tape symbols T with Tn denoting the state of the tape at iteration ‘n’: Tn = C(Tn−1) Due to rationalisation ρ(T ) ∈ XQ (definition A.16) of the tape T to XQ of definition (A.15) a general computation like fixed point iteration (FPI) is therefore an arbitrary function of the following form : f : XQ → XQ (1.1) But never the less, this pose a problem to analyse computation as a standard dynamical system, which in general is continuous nowhere. 2010 Mathematics Subject Classification. Primary 03D10; Secondary 65P20,68Q05,68Q87,68T05.