Matrix Multinomial Systems with Finite Syntax

Abstract

Typically, describing complex processes and the sequences of events they generate requires both statistical and structural information. Statistical information alone does not suffice when intrinsic constraints allow a process to produce well-formed sequences of events but not others. Typically, processes become history dependent; the multiplicity of well-formed sequences with identical histogram and derived concepts, entropy for instance, start to depend on the structure, the grammar, of the underlying process. We demonstrate that for a sufficiently well behaved class of complex processes, it is possible to derive an exact criterion for deciding whether a sequence of arbitrary length is well formed or not. The approach is based on representing events by matrices and sequences of events by products of respective matrices. Formally such processes have a multinomial structure only that the elements are not numbers, but matrices. We demonstrate the approach by applying it to enumerate the well known Oslo sand-pile model, resulting in an elegant formula for the number of stable attractor states for Oslo sand-piles of arbitrary size.