On Thu, Jan 9, 2014 at 12:24 PM, John Clark <johnkcl...@gmail.com> wrote:
> On Wed, Jan 8, 2014 at 2:41 PM, Jesse Mazer <laserma...@gmail.com> wrote
> > For example, in Life one could define macrostates in terms of the ratio
>> of white to black cells [...]
> In the Game of Life the number of black cells is always infinite, so I
> don't see how you can do any ratios.
> John K Clark
Maybe that would be true for some ideal Platonic version of the Game of
Life on an infinite board, but any real-world implementation of a cellular
automaton involves a finite number of squares--usually this is done with a
periodic boundary condition, so squares on the left edge of the finite grid
are defined to be neighbors of squares on the right edge, and squares on
the top edge of the grid are defined to be neighbors of squares on the
bottom edge. Another alternative would be to imagine you do have an
infinite grid, but with a starting state where there are only a finite
pattern of black squares surrounded by an infinite number of white squares,
then you can expand the size of the simulated grid if the region of black
squares approaches its border, so that the grid always remains larger than
the region of black squares (you don't have to simulate regions beyond that
because any region that's all-white on a given time-step, and doesn't have
any black squares on its immediate border, will stay all-white on the next
In either of these cases (though it's easier to analyze the periodic
example since the grid size remains constant), the ratio of black squares
to white squares on the simulated grid region at any given time is
well-defined, so one can use this ratio to define the macrostate. And since
the rules of the Game of Life aren't reversible, and many different initial
states end up either in an all-white end state or an end-state with mostly
white and a few blinking black shapes, I'm pretty sure this would be a case
where an "entropy" defined in terms of these macrostates would tend to
decrease from a randomly-chosen initial finite pattern of black squares. Do
you disagree? (even if you're not as confident as I am that this would be
true for the Game of Life, one could easily define less "interesting"
transition rules where this is obviously the case, like a transition rule
that says that only if a black square has a single black neighbor will it
remain white, in every other case the square will turn white--hopefully
you'd at least agree that in this case, entropy would tend to decrease from
a random initial state on a periodic grid).
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