John Conway giving a talk on knots

Originally uploaded by hzenilc.

It is well known that cellular automata (CA) can evolve in a few specific types of behavior, one of which is precisely the type discovered by John Conway in his Game of Life. I happen to think it is something of a misnomer to speak of John Conway as having “designed” the Game of Life. Working from an initial intuition (using it as a kind of filter) it is quite possible to find a CA of an equivalent type. The procedure is very simple. If you limit the space of all possible 2D CA to those that are clearly non-trivial (avoiding periodic and fractal evolutions) and apply a simple filter (based on compression, topology parameters or other known techniques), you end up with two basic classes identified and studied before by Stephen Wolfram as complex and random behavior (at the end, a single class, according to what he calls the Principle of Computational Equivalence or PCE). The probability that a complex 2D CA will behave like the Game of Life or another CA of equivalent sophistication is very high after the filtering process. As has been demonstrated, many of these interesting but not uncommon 1D, 2D or nD CAs share some fundamental properties, reaching a maximal degree of sophistication (namely complexity/universality).

All this notwithstanding, it is Conway’s great achievement to have drawn attention to this particular 2D CA and to have analyzed it in such detail. Another example of such a CA is the elementary cellular automaton Rule 110. What’s special about Rule 110 is not that it has been extensively studied by Wolfram and eventually proven to be universal by him and his assistant M. Cook, though this is no doubt significant. What is truly noteworthy is that Rule 110 is meant to prove that universality can be reached even by simple systems, leading one to expect that all manner of interesting/complex CA are likewise universal.

Almost every serious attempt to prove that a given arbitrary interesting CA is universal has been successful. Examples range from Conway’s Game of Life to Wolfram’s Rule 110 to Ed Fredkin’s Salt Model, and to cite the most recent, Maurice Margenstern and Yu Song’s universal cellular automaton on the ternary heptagrid (papers accepted for http://www.csc.liv.ac.uk/~rp2008/accepted.php, 2nd Workshop on Reachability Problems).

Rule 110, just like the Game of Life, was discovered, not built. These discoveries are the product of an exhaustive (‘random’) search (over the whole or a part of all possible rules)– which doesn’t mean that they are random discoveries. One can of course mistake such a discovery for an invention. For in low dimensional CA spaces it is very easy to explore the space of possible rules by changing the rules themselves before letting the system evolve through a few initial steps in order to observe said process of evolution. This may make it seem as if one has designed the system rather than simply explored a part of the space, but such is not the case. And even if it were, it is, strictly speaking, misguided to think in terms of designing rather than exploring for the reason given above—viz. that all complex systems have the potential to behave with a sophistication equivalent to that of the Game of Life.

Furthermore, by what Stephen Wolfram calls “irreducibility”, one cannot predict (and this applies particularly to complex CAs such as the Game of Life) how a CA rule will evolve without having run the said CA (and once proven to be universal most properties about this CA will be undecidable). Which explains why the study of CAs has been so successful lately, attracting many minds; because we now have the tools (computers) to let these systems run so we can observe how they develop. The idea of irreducible systems is very simple and powerful at the same time. Basically it holds that one cannot shortcut any of these interesting complex CAs, which means that there could have been no other way for Conway to test his CA apart from running it. Which in turn means that if it were not behaving in the right way, he would have had to change the rule of the CA and test it again, or even if he had the chance to find this particular CA at the first shot, he would have had to let it run for some steps before being sure about what he just made. This doesn’t mean that he wouldn’t have been able to figure out some of the basic properties of the rule he was working with beforehand, but it means that he wouldn’t ever have been certain of his conclusions save by actually running the CA itself. This has been the case with other rules as well, such as Rule 30, which no one has succeeded in cracking by coming up with a function or formula to shortcut the evolution of the rule able to provide the value at an arbitrary step without having to calculate the previous steps(i.e. without running the CA up to the step in question). Even if Rule 30 were crackable, based on computational complexity it would be expected that most of these systems are irreducible in several respects, from undecidability of the properties of the particular CA to incapability of our systems (including ourselves) to compress the information generated by these systems. Applying Wolfram’s PCE, a system with a maximal degree of computational sophistication would have the same computational power of another system of maximal degree, so one cannot expect any one of them to figure out what the other is doing without having to emulate it step by step.

This doesn’t mean that Conway does not deserve credit for the sheer cleverness of the Game of Life and for the discovery of this interesting evolution of a 2D CA. But the Game of Life has become so popular and interesting not because it is rare, but because it has been so extensively studied as a particular instance of an interesting complex 2D CA. Furthermore, there is the fact that it involves some basic game theoretical principles pertaining to how life actually develops.

I met John John Conway 3 weeks ago (and took the photo that goes along this text) and probably I should have brought up the subject with him. However, as far as I remember, Stephen Wolfram himself asked him how he came up with the Game of Life, but unfortunately he seemed to say that he was not certain anymore…

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