Monday, 9 October 2017

An Ashby Growth Machine

Imagine studying the dynamics of Ashby's homeostat where each unit produces a string of numbers which accord to the various values of each dial. The machine comes to its solution when the entropies of the dials are each 0 (redundancy 1). At this moment, the machine 'dies' - there's nothing else left to do.  

As the machine approaches its equilibrium, the constraint of each dial on every other can be explored by the relative entropies between the dials. If we wanted the machine to keep on searching (and living!) and not to settle, it's conceivable that we might add more dials into the mechanism as its relative entropy started to approach 0. What would this do? It would maintain a counterpoint in the relative entropies within the ensemble. 

So there's a kind of pattern: machine with n dials gradually approaches equilibrium. An observer measuring the relative entropy of the machine adds new dials when the relative entropy approaches 0. So, say there's n+1 dials, and the process is repeated. But growth also entails the death of parts of the machine. Maybe the same observer looks at the relative entropies between sub-sections of components. Maybe they decide that some subsections can be removed also as a way of increasing the relative entropy of the ensemble. 

But what about the entropy of the observer's actions in adding and taking away dials? Since this action is triggered by the relative entropy of the ensemble, the relative entropy between the relative entropy of the machine and the entropy of the observer should approach 0. What if we add observers? What would the relative entropy between the observers be?

Each observer might be called a "second-order" observer. Each dial in the homeostat is a first-order observer of the other dials. Each second-order observer sees the ensemble of first-order observers as if it was a single dial. A second second-order observer would also see this, and would see the other second-order observer. A third-order observer could add or remove a second-order observer. And so on. 

Does the growth of this Ashby organism display an emergent symmetry?

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