Creating order in sequence spaces with simple machines

Abstract

We intend to open a new research field towards, say, a theory of “creating order” under various constraints. As a prototype of problems guiding our investigations we study models involving sequence spaces. By “creating order” or equivalently “organization” we mean reducing in size the range of outputs by an “organizer” via a permuting channel (a simple machine), when it is fed by a given domain of inputs. The “creation of order” is assumed to come only from the permutation operation in these channels. Four types of “order creation” are considered depending on the structure of the knowledge of the organizer (limitations on mind) about the future input and past output sequences and the kinds of admissible permutations inside the channel (limitations on matter). In any case the organizer's goal is to produce output spaces of minimal cardinality (optimal organization). We present some strategies of ordering and some first and seemingly basic optimality results. After this more technical part of the paper we present some ideas about a general theory of ordering.