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. … And so this implies that continuous systems must always in effect be able to operate on infinite sequences.

Nand and Nor are the only 2-input functions universal in this sense. ( {Equal} can for example reproduce only functions {9, 10, 12, 15} , {Implies} only functions {10, 11, 12, 13, 14, 15} , and {Equal, Implies} only functions {8, 9, 10, 11, 12, 13, 14, 15} .)

Time and gravity General relativity implies that time can be affected by gravitational fields—and that for example a process in a lower gravitational field will seem to be going faster if it is looked at by an observer in a higher gravitational field. … One might have thought that this would imply in reverse that to an observer moving with the object the whole infinite future of the outside universe would in effect seem to go by in a finite time. … But for the reasons mentioned above, this very fact presumably implies instability, and the whole effect disappears if there is any deviation from perfect spherical symmetry.

For if more than half the strings are generated, then somewhere both a string and its negation would have to appear, implying that the system must be inconsistent. And similarly, if less than half the strings are generated, there must be some string for which neither that string nor its negation ever appear, implying that the system is incomplete.

The phenomenon of computational irreducibility implies that to find out what some specific system with complex behavior will do can require explicit simulation that involves an irreducible amount of computational work. … But what the Principle of Computational Equivalence implies is that there are actually a vast range of very different kinds of rules that all lead to exactly the same computational capabilities—and so can all in principle be used as a basis for making computers.

Such uniformity in frequency is implied by standard quantum theory to exist in the idealized zero-point fluctuations of a free quantum field—with direct consequences for such semiclassical phenomena as the Casimir effect and Hawking radiation.

But in the previous chapter we saw that it is possible for quite different programs to yield essentially the same large-scale behavior, implying that with programs there can be many models that have the same consequences but different detailed underlying structure.

Two sine functions Sin[a x] + Sin[b x] can be rewritten as 2 Sin[1/2(a + b) x] Cos[1/2(a - b) x] (using TrigFactor ), implying that the function has two families of equally spaced zeros: 2 π n/(a + b) and 2 π (n + 1/2)/(b - a) .

However, particularly on timescales less than a day, it has in the past decade become clear that, as suggested by Benoit Mandelbrot in the early 1960s, large price fluctuations are significantly more common than a Gaussian distribution would imply. … And the Black–Scholes model from 1973 implies that prices of suitably constructed options should depend in a sense only on such variances.

But as soon as different particles are related by a non-Abelian symmetry group, then the discreteness of the representations of such a group immediately implies that all charges must be rational multiples of each other.