Given my previous definition of system simulation (aka modelling) it seems intuitive that a finite system cannot model itself except insofar as it is itself. Even more obviously, no proper subsystem of a system could simulate its “parent”. A proper subsystem by definition has a smaller size than the enclosing system, but needs to be at least as big in order to model it.

(An infinite subsystem of an infinite system is not a case I care to think too hard about, though in theory it could violate this rule? Although some infinities are bigger than others, so… ask a set theorist.)

However, an abstraction of a system can be substantially smaller (i.e. require fewer bits of information) than the underlying system. This means that a system can have subsystems which recursively model abstractions of their parents. Going back to our game-of-life/glider example, this means that you could have a section of a game of life which computationally modeled the behaviour of gliders in that very same system. The model cannot be perfect (that would require the subsystem to be larger than its “parent”) so the abstraction must of necessity be incomplete, but as we saw in that example being incomplete doesn’t make it useless.

## 3 thoughts on “Recursive Abstractions and Approximate Models”