This note ponders definitions of what it means to defend some portion of space in a digital system. This is tangent to some comments on chaos but restricted here to strictly digital domains.

By digital domain I want to include digital computers, including those that are commercially available, and Conway’s Life game. Both of these systems can support complex patterns of behavior. I include the life game because it is a partly like our universe and also partly like digital computers. Also there are many discoveries about its nature.

The question is about what sorts of subsystems of the total system it might be possible to reason about, just as we understand our analog universe and reason about it by analyzing pieces of it.

Alan Turing described a class of mechanical mechanisms, called Turing machines now, that kept state on an infinite mutable tape. One result was that you could design a universal Turing machine which could emulate any particular Turing machine, which in turn means that for any Turing machine, it was possible to find a program which was a state of the tape, which the universal Turing machine would obey and thus emulate the particular machine.

Another fundamental result was the possibility of a Turing machine running two programs at once without either interfering with the other. This was accomplished by the Turing program alternating its attention between the two tasks that it emulated. This was perhaps the first instance of protection between tasks that were ignorant of each other.

It has been shown that a general purpose computer can be built within the confines of Conway’s game of life. Such constructions are very brittle and a single error may bring down the entire edifice. For that matter a single bit error in a multiprogramming operating system can likewise bring down the entire system. In each case there is some single authority granting existence to the other independent entities within the particular cyberspace.

Are there other possibilities? It is not clear yet what the question means. In the metaphor of the operating system we can imagine hardware that supports distinct regimes and keeps them apart, indeed such hardware has been built. But this is no victory for the central hardware has merely replaced the central operating system.

Life Regimes

Here is a proposition, but I do not expect ever to learn if it is true. Either it or its negation would be an interesting theorem.

There exists an integer n, two n by n Life patterns P0 and P1, and a boolean expression E(p) involving the n2 cells, such that E(P0) = 0 and E(P1) = 1 and such that E, applied to that particular array of cells, is constant as Life goes on. All of this in the presence of arbitrary initial conditions outside the n by n pattern. In short is it possible to build a mechanism to guard a single bit of storage?

My intuition is strongly that this is false. I cannot imagine the nature of a proof however. On the other hand I did not imagine a general purpose computer built on Life. I think that the known general purpose computers implemented within life assume a benign environment and are unlikely to survive most incoming gliders.

If this proposition were true it seems likely that it would further possible to guard some one bit of the pattern from change and then E could involve just that one bit. If this is so then perhaps we can build a computer that survives hostile environments. If that is possible it seems likely that it could sense its environment to a degree. If this is possible it might be able to expand its regime of control for most initial environments. But not for all environments, for there might be a competing regime nearby with similar plans. It is not entirely clear that it can even sense whether it has gained control.

My life conjectures can be summed up imprecisely by either of the following:

I think that integrity is possible in most randomly selected and thus benign life environments, after the gliders have died out. Bootstrapping might require waiting several million cycles and then invoking magic to overwrite your initial pattern. Getting you pattern into the universe takes some sort of magic in any case. I thus jump to the conclusion that if evolution were possible, predators would arise to eliminate all integrity. Our universe seems to provide for an imperfect degree of property protection and thus a degree of integrity. Single cellular life required this and so, I imagine, does evolution.

I reiterate that a general purpose computer built in life can provide an environment in which integrity is possible. But such secure systems are built on the computer platform—not life! I think there might be evolution in the computer but it seems clear to me that there would be no way for it to detect the life underpinnings. It is remarkable that we can learn anything about our universe, or as Einstein said of the universe: The most remarkable thing is that it is comprehensible.