In my complex systems paper I make extensive use of John Horton Conway's
little cellular automaton called Game of Life (GoL), but two people have
made objections to this on the grounds that GoL can be used to implement
a Turing Machine, and is therefore an example of me not knowing what I
am talking about.
In case anyone else wonders about the same question, I will explain why
the Turing machine equivalence has no relevance at all.
Game of Life consists of the rules that drive it, and the initial state
of its "universe" (the pattern of cells that are in the ON state at the
beginning).
There are an infinite number of possible initial states.
Now, the way in which I use GoL in my paper is as an example of a
complex system, but we have to be careful about what is meant by
"complex system". Just to make sure there was no confusion, I defined
the meaning of this term very carefully in the paper.
The entire point of my paper is to talk about how we build scientific
explanations for various different kinds of system. I cannot stress
this strongly enough: the issue is "How long will it take us to
construct a theory that 'explains' a given system, and how big will that
theory be when we find it?". Complex systems are purely defined by the
theory size. Nice and easy, really: people have conjectured that there
are certain kinds of system for which the smallest reasonable scientific
theory is so huge that for all practical purposes we may as well say
that it does not exist. The concept of a "scientific explanation" is
not precise (never has been and probably never will be), but that is
beside the point: this is a *practical* matter that affects how we go
about handling some sorts of system.
[One quick aside: it is not quite as simple as saying that some systems
have huge theories, because systems can be PARTIALLY complex, which
means that some aspects of their behavior are explicable, in an
approximate way, but some other aspects cannot be given an ordinary
explanation that is more compact than just a simulation of the system.]
So how does this relate to Conways Game of Life? Well, what we are
interested in is the entire collection of all possible implementations
of GoL that have different initial states. The question is, do we
notice any regularities in that collection of systems? You bet! We
notice that *some* patterns of ON cells can have cyclic behavior, or in
some other sense be 'regular'. Generally, there are some patterns that
are distinctively interesting because of their regularity and
non-randomness.
Now the crux of the matter: can we explain *which* patterns should have
those interesting properties? Out of the infinite possible set of
possible patterns that could exist, which of those have the property
that they come back to their start state and recur exactly afterwards
(barring interfence)?
If the answer is "No, we appear to have no idea how to build such a
theory" then the system is complex (= has a high degree of complexity).
As far as we can tell, GoL is an example of that class of system in
which we simply never will be able to produce a "theory" in which we
plug in the RULES of GoL, and get out a list of all the patterns in GoL
that are interesting. We cannot seem to be able to predict the set of
interesting patterns that are implicit in the rules.
[In my paper I also extend this idea to another global aspect of the GoL
system: the overall "generativeness" of the rules (whether the rules
themselves give rise to any interesting patterns, a few patterns, huge
numbers of patterns, and so on). This is a complex feature of the GoL
system, because *explaining* or *predicting* the generativeness of
various different possible cellular automaton rule-sets is for all
practical purposes impossible.]
Now, finally: if you choose the initial state of a GoL system very,
VERY carefully, it is possible to make a Turing machine. So, in the
infinite set of GoL systems, a very "small" fraction of that set can be
made to implement a Turing machine.
But what does this have to do with explaining the existence of patterns
in the set of ALL POSSIBLE GoL systems?? So what if a few of those GoL
instances have a peculiar property? bearing in mind the definition of
complexity I have stated above, how would it affect our attempts to
account for patterns that exist across the entire set?
It doesn't!
It has no bearing whatsoever on the questions I raised in my paper;
questions about whether certain kinds of systems must be treated
differently than others because of the size of the explanations that
relate their high-level and low-level properties
The Turing Completeness of GoL might be fascinating to some people, but
it is a complete and utter red herring.
What else can I say?
Richard Loosemore
P.S. Well, actually I *can* say something else.
I always welcome comments and criticisms of the form "You can't use GoL
as an example of a complex system because it is possible to make a
Turing machine out of it...". But it is quite another thing to be
attacked with words that go something along the lines of "You clearly
haven't a clue what you are talking about, because you can't use GoL as
an example ... etc etc etc."
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