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\C\F0Relativistic Systems of Automata\F1

\J	We are taking an idea from the theory relativity and applying it to systems of
interacting automata.  The idea is that instead of having a single time variable
of which the state at every position in space is a function, each body
has its own \F2local time\F1 and there is no notion of simultaneity for events
separated in space.  The following notion does not depend on understanding
anyting about relativity theory, and after we have explained it we will say what
it might be good for.

	We start with a collection of entities \f2A\f5i which are the automata of the
system.  Each \f2A\f5i has a set \F2states (A\F5i\F2)\F1 which is the set of its
possible states.  Another set associated with \f2\f5i is \F2times (A\F5i\F2)\F1
which has a partial ordering for which < is used.  A state history of an
automation A\f5i is a map.  Statehis: times(A\f5i)→states(A\f5i).  (In the
usual \F2non-relativistic\F1 automata theory, each \F2times (A\F5i\F2)\F1
is the set of integers and there is a distinguished isomorphism between \F2times
(A\F5i\F2)\F1 and a univeral set \F2times\F1 isomorphic to the integers.)
There are two main cases of interest for \F2times (A\F5i\F2)\F1, namely the
integers and the real numbers.  Some subautomata of a system may have integer
times while others may have real times.  Even if both have integer times they may
not correspond in a simple way - one clock may tick every second at WWV in
Boulder Colorado and the other may tick once a school day in Rio de Janeiro.

	In the examples we have given, the ordering is total, but you will see
that a system of automata can best be regarded as having partial ordered time
even though its subautomata have total ordered times.  Therefore, if we want
to take systems of automata as the constituent subautomata of yet bigger 
systems, the basic construction should use partial ordering.

	Besides the sets \F2states(A\F5i\F2)\F1 and \F2times(A\F5i\F2)\F1, we also
have the sets \F2inputs(A\F5i\F2) and \F2outputs(A\F5i\F2)\F1 which give the possible
values of inputs and outputs.  An input history of A\f5i is a map inhis:
times(A\f5i)→inputs(A\f5i) and an output history is a map outhis: times(A\f5i)
→outputs(A\f5i).  If we need it, we can use the concept of a \F2history\F1 of
\f2A\f5i which is a map\.

\←=375;\→.\F2hist: times(A\f5i)→inputs(A\f5i)⊗states(A\f5i)⊗outputs(A\f5i)\F1

\JLet \F2Inhis(A\f5i), Statehis(A\f5i), Outhis(A\f5i)\F1 and \F2His(A\f5i)\F1 (written
with initial caps) be the sets of all possible input histories, state histories,
output histories, and histories.

	The law of motion of an automaton gives the state at any local time as a 
function of the input history and the state history.  We impose the further
condition that the state at a given time depend only on the past of the state and
the input.  If the local time is taken as an integer, then the most important case
is when the state depends only on the immediately preceding state and input.
Moreover, the general case of integer time can be reduced to the case of 
depending only on the immediately preceding time by replacing the automaton A\f5i
by an automaton A\!jpb;\⊗\f5i\F1 

a differential equation\.

\!divi(w,(\F1d state),(\F1dt));\oa[1 = F(state,input,t)]\&a←L&w&a;\Oa;\;

(In order to be sure only the past is involved you could specify a left hand
derivative, but it won't matter if all the fnctions are smooth enough.)

	The output law of an automaton gives the output as a function of the
state.  (Something might be gained by letting the output depend on the pasts of
the state and input if it is computationally or mathematically convenient in
a particular case.)

	The interaction between automata is specified by a law which gives the
input of A\f5i as a function of the output histories of all the automata subject
to a causality condition.  The idea of the causality condition is again to allow 
the state to depend only on the past, but since we do not have a correspondence
between the times of the eifferent automata, we proceed as follows:

Form the disjoint union of all the local time spaces and take a partial ordering
< on that space consistent with the local orderings of local timer, i.e. 
if \F2t\f5iεtimes(A\f5i)\F1 and \F2t\!jpb;\⊗\f5iεtimes(A\f5i)\F1, then