And even if all one knows is that the original message was in some definite language this is still typically good enough. … For one can pick out of the encrypted message all the squares that occur at a certain point in the repetition period of the A simple example of an encryption system in which the encrypting sequence is obtained by repetitively cycling through the elements of the key.

automaton all cells are always effectively treated as being exactly the same. And to emulate a mobile automaton with a cellular automaton it turns out that all one need do is to divide the possible colors of cells in the cellular automaton into two sets: lighter ones that correspond to ordinary cells in the mobile automaton, and darker ones that correspond to active cells.

And I have little doubt that most if not all of these are universal. … But it turns out that 14 of them—all essentially equivalent—produce considerable complexity, even when started from a blank tape.

Outline of the Principle Across all the vastly different processes that we see in nature and in systems that we construct one might at first think that there could be very little in common. … There are various ways to state the Principle of Computational Equivalence, but probably the most general is just to say that almost all

For while I believe that the basic science that I develop in this book provides a remarkably powerful new framework, coming up with an actual model requires all sorts of detailed work and analysis. Certainly it would be wonderful if one could just take the ideas and results in this book and somehow immediately use them to create models for all sorts of systems. … And instead—just as in any other framework—there will be no choice but first to learn all sorts of details of a system, and then to use judgement and creativity to see which of them are really essential to a model and which are not.

The idea in all such systems is to have an array of spins, each of which can be either up or down. … In the ordinary Ising model, this ground state is simply all spins up or all spins down.

all numbers.

In cases where the underlying rules have only rather simple behavior—as with rules 90 and 184—it turns out that it is never possible to emulate more than a Illustrations of how rule 30 can be set up to emulate a single step in the evolution of all elementary cellular automata.

Methods for resolving some of the equations in the first column were known in antiquity; all had been resolved by the 1800s.

The sine-Gordon equation represents nonlinear waves obtained for example as the limit of a very large number of pendulums all connected to a spring. … It is notable that unlike with ODEs, essentially all PDEs that have been widely studied come quite directly from physics.