ultimate computational capabilities of fluids must be those of a system of discrete particles.
But while it is known that many systems in nature are made up of discrete elements, it is still almost universally believed that there are some things that are fundamentally continuous—notably positions in space and values of quantum mechanical probability amplitudes.
Yet as I discussed in Chapter 9 my strong suspicion is that at a fundamental level absolutely every aspect of our universe will in the end turn out to be discrete. And if this is so, then it immediately implies that there cannot ever ultimately be any form of continuity in our universe that violates the Principle of Computational Equivalence.
But what if one somehow restricts oneself to a domain where some particular system seems continuous? Can one even at this level perform more sophisticated computations than in a discrete system?
My guess is that for all practical purposes one cannot. Indeed, it is my suspicion that with almost any reasonable set of assumptions even idealized perfectly continuous systems will never in fact be able to perform fundamentally more sophisticated computations.
In a sense the most basic defining characteristic of continuous systems is that they operate on arbitrary continuous numbers. But just to represent every such number in general requires something like an infinite sequence of digits. And so this implies that continuous systems must always in effect be able to operate on infinite sequences.
But in itself this is not particularly remarkable. For even a one-dimensional cellular automaton can be viewed as updating an infinite sequence of cells at every step in its evolution. But one feature of this process is that it is fundamentally local: each cell behaves in a way that is determined purely by cells in a local neighborhood around it.
Yet even the most basic arithmetic operations on continuous numbers typically involve significant non-locality. Thus, for example, when one adds two numbers together there can be carries in the digit sequence that propagate arbitrarily far. And if one computes even a function like 1/x almost any digit in x will typically have an effect on almost any digit in the result, as the pictures on the facing page indicate.