A Model of the Process of Thinking Based on the Dynamics of Bundles of Branches and Sets of Bundles of p-ADIC Trees

Abstract

The system of p-adic numbers Q p , constructed by K. Hensel in the 1890s, was the first example of an infinite number field (i.e., a system of numbers where the operations of addition, subtraction, multiplication and division are well defined) which was different from a subfield of the fields of real and complex numbers. During much of the last 100 years p-adic numbers were considered only in pure mathematics, but in recent years they have been extensively used in theoretical physics (see, for example, the books [7] and [11]), the theory of probability [7] and investigations of chaos in dynamical systems [7]. In [1], [4] and [8] p-adic dynamical systems were applied to the simulation of functioning of complex information systems (in particular, cognitive systems). In this paper we continue these investigations. We study the collective dynamics of information states. We found that such a dynamics has some advantages compared to the dynamics of individual information states. First of all, the use of collections of sets (instead of single points) as primary information (in particular, cognitive) units extremely extends the ability of an information system to operate with large volumes of information. Another advantage is that (in the opposite to the dynamics of single states) the collective dynamics is essentially more regular. As we have seen [1] and [8] discrete dynamical systems over fields of p-adic numbers have the large spectrum of non-attracting behaviours. Starting with the initial point x 0 ∈ Q p iterations need not arrive to an attractor.