[Causal networks for] 2D mobile automata As in 2D random walks, active cells in 2D mobile automata often do not return to positions they have visited before, with the result that no causal connections end up being created.

When GCD[k, n]  1 the dot can never visit position 0. … In general, the dot will visit position m = k^IntegerExponent[n, k] every MultiplicativeOrder[k, n/m] steps.

For from the outside, one can readily see each individual step in the evolution of a mobile automaton, and one can tell that there is just a single active cell that visits different parts of the system in sequence. … But if the observer itself just consists of a collection of cells inside a mobile automaton, then no such change can occur except on steps when the active cell in the mobile automaton visits this collection of cells.

And for any particular form of neighborhood, it is once again quite straightforward to find the best model for any given set of data. … Rather, the purpose is usually just A comparison between data generated by ordinary cellular automata and the probabilistic cellular automata that are considered the best fit to it.

An argument for the Second Law from around 1900, still reproduced in many textbooks, is that if a system is ergodic then it will visit all its possible states, and the vast majority of these will look random. But only very special kinds of systems are in fact ergodic, and even in such systems, the time necessary to visit a significant fraction of all possible states is astronomically long.

Another feature is that there is always a path of doubled connections (associated with the active cell) that visits every node in some order.

But other parts—often including sites important for function—seem more like random walks.

• Is there a path shorter than some given length that visits all of some set of points in the plane? … Examples include: • Finding a Hamiltonian circuit that visits once every node in a given network is NP-complete, but finding an Euler circuit that visits once every connection is in P. • Finding the longest path between two nodes in a network is NP-complete, but finding the shortest path is in P. • Determining satisfiability for a Boolean expression with 3 variables in each clause is NP-complete, but for one with 2 variables is in P.

In many ways these kinds of computer experiments thus manage to combine the best of both theoretical and experimental approaches to science. … Over and over again the single most important principle that I have learned is that the best computer experiments are ones that are as simple and straightforward as possible.

The behavior of a system will be repetitive in time whenever it effectively follows a closed curve—either literally in space, or in terms of states that it visits.