my machine definition attachment seemed to fail. if this is a
duplicate, I apologize
Don Mikulecky
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WHAT IS A MACHINE? By Don Mikulecky
Let us look at a category theory formulation of a machine. It is surprisingly
simple. It embodies everything a Turing machine can do in an abstract way in
terms of mappings. It also very nicely shows where dynamics comes into the
entire picture.
We describe a SEQUENTIAL MACHINE which can be in one of a finite number of
states, receive one of a finite number of inputs, and emit one of a finite
number of outputs.
DEFINITION: SEQUENTIAL MACHINE: SM = ( Xo, Q, Delta, qo,Y,Beta)
Where
Xo is the set of inputs
Q is the set of states
Delta: (Q X Xo) maps to Q is the dynamics
qo is in Q is the initial state
Y is the set of outputs
Beta is the output map
(From: Arrows, Structures, and Functors: The Categorical Imperative, Michael
Arbib and Ernest G. Manes, Acedemic Press, NY , 1975 pp 93-106)
It has been said that this algorithmic character is restricted to sequential
machines. I now will show that this is an illusion. A machine is a machine.
Now can we describe a "PARALLEL MACHINE" as anything different?
DEFINITION: PARALLEL MACHINE: PM = ( Xo, Q, Delta, qo,Y,Beta)
Where
Xo is the set of inputs
Q is the set of states
Delta: (Q X Xo) maps to Q is the dynamics
qo is in Q is the initial state
Y is the set of outputs
Beta is the output map
Is there anything missing from this formulation? No, there is not, really. The
reason that they are the same in actuality is quite simple, it is always
possible to simulate a parallel machine with a sequential machine. It is true
that the symbols X and Q stand for different sets and it is also true that if Q
is the set of states for the system it contains a subset, N which is the states
of all the nodes in the parallel system. (In an ANN these would be the
neurons). More importantly, the rules or algorithms behind the mapping ,
Delta, are very different in the two examples.
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