Hi Giles,
Fractal dimensions are a fascinating topic. What do you
think, perhaps one can define a fractal dimension for certain
map-reduce patterns that characterizes certain flows,
streams, waves, whirls or vortices? That would be cool.
Has anyone considered the connection between Map-Reduce
and CA/NN/ABM in general? On the one hand Map-Reduce
algorithms can be used to calculate very large Cellular
Automata (CA), Neural Networks (NN) and perhaps Agent
Based Models (ABM). On the other hand, CA, NN and ABM
may shed some light on possible Map-Reduce patterns..
Map-Reduce is a basic pattern of distributed computation,
which can be found in CA and NN. Neurons in NN collect
informations through their synapses (reduce) and send
it to other neurons through their axon (map).
Cells in CA do the same, the collect information from
their neighbors in the previous timestep and process
the input (reduce), then they send it to to their
neighbors for the next timestep (map).
If we have only mapping, or if mapping prevails,
then we get a propagating outgoing wave, which ensures
activities. (This corresponds roughly to liveliness properties
in distributed algorithms, and Eppstein's CA classification
"Contraction impossible").
If we have only reducing, or if reducing prevails,
then we get an incoming wave, which ensures stability.
(This corresponds to safety property in distributed
algorithms, and Eppstein's CA classification
"Expansion impossible").
If we have both mapping and reducing, expansion and
contraction, we get all kinds of interesting complex
patterns. For instance if we embed a 2-dimension
manifold in a 3-dimensional space we get many
interesting patterns and strange attractors (Roessler,
Lorenz, etc). I wonder if it is possible to define a fractal
dimension for it in general, or if certain kinds of
"map-reduce" patterns may result in certain kinds
of streams, flows, vortices or whirls?
-J.
----- Original Message -----
From: Giles Bowkett
To: The Friday Morning Applied Complexity Coffee Group
Sent: Thursday, December 16, 2010 1:42 AM
Subject: Re: [FRIAM] hi friam - how do I calculate the fractal dimension of
repetitive text?
[...] So here you have a piece of code which contains another piece of code
highly similar to itself. The inner loop is like a version of the outer loop
copied and then slightly altered. You could imagine an infinite series of
for loops, each with a tiny tweak but otherwise highly similar to the
containing loop. Kind of like how the Koch curve contains an infinite number
of nested Koch curves, in a sense, but the difference is that the Koch curve
is a regular fractal, while this code, if it has fractal dimension,
represents an irregular fractal.
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