dies out. But already in example (b) it is not so easy. One can go for 1000 steps and still not know what is going to happen. And only after 1017 steps does it finally become clear that the pattern in fact dies out.

So what about examples (c) and (d)? What happens to these? After a million steps neither has died out; in fact they are respectively 31,000 and 39,718 cells wide. And after 10 million steps both are still going, now 339,028 and 390,023 cells wide. But even having traced the evolution this far, one still has no idea what its final outcome will be.

And in any system the only way to be able to guarantee to know this in general is to have some way to shortcut the evolution of the system, and to be able to reduce to a finite computation what takes the system an infinite number of steps to do.

But if the behavior of the system is computationally irreducible—as I suspect is so for the cellular automaton on the facing page and for many other systems with simple underlying rules—then the point is that ultimately no such shortcut is possible. And this means that the general question of what the system will ultimately do can be considered formally undecidable, in the sense there can be no finite computation that will guarantee to decide it.

For any particular initial condition it may be that if one just runs the system for a certain number of steps then one will be able to tell what it will do. But the crucial point is that there is no guarantee that this will work: indeed there is no finite amount of computation that one can always be certain will be enough to answer the question of what the system does after an infinite number of steps.

That this is the case has been known since the 1930s. But it has normally been assumed that the effects of such undecidability will rarely be seen except in special and complicated circumstances. Yet what the picture on the facing page illustrates is that in fact undecidability can have quite obvious effects even with a very simple underlying rule and very simple initial conditions.

And what I suspect is that for almost any system whose behavior seems to us complex almost any non-trivial question about what the system does after an infinite number of steps will be undecidable. So, for example, it will typically be undecidable whether the evolution of

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From Stephen Wolfram: A New Kind of Science [citation]