Does not Compute 3:

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1 Δεκ 2013 (πριν από 3 χρόνια και 7 μήνες)

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Does not Compute 3:

Awesomer

Cellular Automata

In which we consider how to upgrade our
cellular automata

A few key choices are considered, but
eventually we decide there is no real choice
but to buy the
Utimate

Superdeluxe

MegaAutomaton
, which has a killer feature
that puts them all to shame


After This Class…

1.
You will be able to give an example of a cellular
automata that really does some computation,
and have an intuition for how it works

2.
We will have considered the question of color
and size, and know the definitive answer for
which provides more computational power

3.
The
Utimate

Superdeluxe

MegaAutomata

arrives. But I won’t say more!

4.
If we have time, we briefly mention the Game of
Life, without which no discussion of cellular
automata would be complete


You will be able to give an example of a cellular
automata that really does some computation,
and have an intuition for how it works:

The Squaring Cellular Automaton

Questions


Questions…Part A
-

Colors


Your job is to prove that cellular automata
with additional colors can be used to simulate
cellular automata with greater amounts
“looking”.


Figure out a way to make a 8
-
color cellular
automata that looks only at 3 cells simulate a
2
-
color cellular automata that looks at 5 cells.


Hints


Realize that, because your cells have more
states, you can represent the state of more
than one cell of the 2
-
color automata within a
single cell of your 8
-
color automata.


Also note that representing the state of 2 cells
of a cellular automata makes things annoying
because it’s not odd length. Go ahead and
represented 3 cells
.




Where We Are…

1.
We used the Squaring Automaton as an example
of a CA that really does some computation, and
hopefully you have an intuition for how it works

2.
We have considered the question of color and
size, and know that (strangely enough) these
things are just two sides of the same coin. This
gives us hints of a new method of proving…

NEXT

The
Utimate

Superdeluxe

MegaAutomaton

arrives.

The
Utimate

Superdeluxe

MegaAutomaton


What Does This Mean?

1.
Cellular Automata can do a lot more than we
might initially think

2.
While we might imagine that computational
power increases as new features do…at least in
the case of cellular automata, there is a
“plateau” after which more features do not gain
you more power

3.
One of the primary ways we will want to prove
things about computational power is by getting
machines to simulate each other. Proof by
simulation!

The “Plateau” of CA Power

More Colors

More Computational Power

?

Conway’s Game of Life


For a space that is 'populated':

Each cell with one
or no neighbors dies, as if by loneliness. Each cell
with four or more neighbors dies, as if by
overpopulation. Each cell with two or three
neighbors survives.


For a space that is 'empty' or 'unpopulated'

Each
cell with three neighbors becomes populated.


This (should you doubt it even for a second) is
just a ‘ordinary’ 2D cellular automata

What Happened in Class

1.
We used the Squaring Automaton as an example
of a CA that really does computation

2.
We have considered the question of color and
size, and talked about the universal 1
-
D Cellular
automata. What you ought to remember:

1.
The method of proving by simulation, which we have
used in two different ways (how?)

2.
The surprising discovery of a computational plateau
for 1
-
D cellular automata

What is to Come


A new machine! Very different from the old
machine!


And with this machine we will actually be able
to find problems that it really cannot compute
the answers to

PS…although we will not actually use it in class (at least at the moment) you
owe it to yourself to check out the Game of Life.
NetLogo

has a simulation
(just called “Life”). You can also download a simulator here:

http://www.bitstorm.org/gameoflife/

Even if you don’t download a simulator yourself, the Life Lexicon will blow
your mind:

http://www.bitstorm.org/gameoflife/lexicon/