The Copenhagen Interpretation of quantum mechanics is a metaphysics which makes claims about scientific measurement and how states of the physical world relate to each other over time. We can build up a recipe for a Copenhagen universe, by starting from traditional cellular automata, and then making consecutive modifications until we reach something closer to what quantum mechanics suggests.

For the sake of digestibility, we will pretend our "universe" is a 1D strip of cells which take on states. Time runs downwards in the usual fashion, and each row corresponds to a singe time slice ; a snapshot of the universe at time t.

Multiple Futures

Traditional CA metaphysics says that the state of the world is a precise configuration given by the cell values, and the present state uniquely determines the future state.

The first modification is to give up on a unique future, instead have our CA rule merely suggest a future state as a probability over a large set of states. The tallness of the green arrow is a probability that the future cell will take on a black or white value. It can literally be either one, and over many time cycles the relative sizes of the arrows will bear out.

Superposition

So far it is obvious that measurement will always produce an exactly white or exactly black cell value, as this is the only admissible values for a cell at any time. To step closer to Copenhagen, we will now allow the cells to take on partial grey values between black and white, and call those superpositions. The rule now maps exact neighbor states to some distribution over greyish values transitioning from black to white.

Now that grey values are permitted, a conceptual problem emerges. How do we apply a rule to say, a collection of 3 mostly grey cells? The answer is strange. We change the rules so that they only apply when some person is taking a measurement of the world. In the cycles of time in which the cells are not measured, we allow cells to cycle between black and white and back again. This splits the "rules" into two contradictory phases. One in which the cells are not being measured and they smoothly oscillate, and one in which they were measured at time cycle (=t), in which case they undergo the traditional rule. The oscillations can best be viewed as going from black to white, and passing through a reddish region in between.