# MacKay’s Mathematical Examples

Diakonova and MacKay exhibit mathematical models of strong emergence and a measure for it.

For deterministic or stochastic dynamics of networks of units … the space-time phases are the limit points of probability distributions for state … which can occur by evolving suitable initial probability distributions for state as a function of space from initial time going to minus infinity.

A system exhibits strong emergence if it has more than one space-time phase, yet is “indecomposable” … . The amount of strong emergence is the diameter of the set of space-time phases, using Dobrushin metric. …

Here, a variety of examples of complex dynamic systems are constructed that we prove to have interesting space-time phases.

These examples are based on the work of Toom. One has a network wrapped around a torus, with binary elements. The base example is to calculate a candidate next state that is the majority of the states of self and those to ‘north’ and ‘east’. The actual next state is then chosen with some characteristic probability, λ, to be either the candidate next state or its opposite. For extreme values of λ there are at least two very different space-time phases, and hence strong emergence.

Gielis & MacKay have previously constructed systems with period one or two ‘ferromagnetic’ or ‘antiferromagnetic’ phases. That is, where at a given time adjacent states are mostly similar, or mostly opposite. This paper shows how to ‘interlace’ examples to get arbitrarily many phases and hence more extreme emergence.

It is noted that an undesirable feature of the Toom method is that it relies on asymmetry in the update rule. (That is, one can’t take the majority across self, east, west, north, south.)

RC. Ball, M. Diakonova RS. Mackay Quantifying Emergence In Terms Of Persistent Mutual Information Advances in Complex Systems, Vol. 13, No. 3 (2010) 327–338

This gives some different examples of emergence, and some helpful descriptive prose.

Our sharpest measure of Strong Emergence is the Permanently Persistent Mutual Information (PPMI), that is the PMI I(∞) which persists to infinite time. This quantifies the degree of permanent choice spontaneously made by the system, which cannot be anticipated without observation but which persists for all time. A prominent class of example is spontaneous symmetry breaking by ordered phases of matter: here a physical system is destined to order in a state of lower symmetry than the probability distribution of initial conditions, and hence must make a choice (such as direction of magnetization) which (on the scale of microscopic times) endures forever. As a result, Strong Emergence can only be diagnosed by observing multiple independent realizations of the system, not just one long-time history.