## Monday, June 10, 2024

### Computation

I’ve been imagining a very slow embodiment of computation. You have some abstract computer program designed for a finite-time finite-space subset of a Turing machine. And now you have a big tank of black and white paint that is constantly being stirred in a deterministic way, but one that is some ways into the ergodic hierarchy: it’s weakly mixing. If you leave the tank for eternity, every so often the paint will make some seemingly meaningful patterns. In particular, on very rare occasions in the tank one finds an artistic drawing of the next step of the Turing machine’s functioning while executing that program—it will be the drawing of a tape, a head, and various symbols on the tape. Of course, in between these steps will be a millenia of garbage.

In fact, it turns out that (with probability one) there will be some specific number n of years such that the correct first step of the Turing machine’s functioning will be drawn in exactly n years, the correct second step in exactly 2n years, the correct third one in exactly 3n years, and so on (remembering that there is only a finite number of steps, since we have working with a finite-space subset). (Technically, this is because weak mixing implies multiple weak mixing.) Moreover, each step causally depends on the preceding one. Will this be computation? Will the tank of paint be running the program in this process?

Intuitively, no. For although we do have causal connections between the state in n years and the next state in 2n years and so on, those connections are too counterfactually fragile. Let’s say you took the artistic drawing of the Turing machine in the tank at the first step (namely in n years) and you perturbed some of the paint particles in a way that makes no visible difference to the visual representation. Then probably by 2n years things would be totally different from what they should be. And if you changed the drawing to a drawing of a different Turing machine state, the every-n-years evolution would also change.

So it seems that for computation we need some counterfactual robustness. In a real computer, physical states define logical states in a infinity-to-one way (infinitely many “small” physical voltages count as a logical zero, and infinitely many “larger” physical voltages count as a logical one). We want to make sure that if the physical states were different but not sufficiently different to change the logical states, this would not be likely to affect the logical states in the future. And if the physical states were different enough to change the logical states, then the subsequent evolution would likely change in an orderly way. Not so in the paint system.

But the counterfactual robustness is tricky. Imagine a Frankfurt-style counterfactual intervener who is watching your computer while your computer is computing ten thousand digits of π. The observer has a very precise plan for all the analog physical states of your computer during the computation, and if there is the least deviation, the observer will blow up the computer. Fortunately, there is no deviation. But now with the intervener in place, there is no counterfactual robustness. So it seems the computation has been destroyed.

Maybe it’s fine to say it has been destroyed. The question of whether a particular physical system is actually running a particular program seems like a purely verbal question.

Unless consciousness is defined by computation. For whether a system is conscious, or at least conscious in a particular specific way, is not a purely verbal question. If consciousness is defined by computation, we need a mapping between physical states and logical computational states, and what that mapping is had better not be a purely verbal question.