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.cb CAUSALITY IN SYSTEMS OF INTERACTING AUTOMATA

	We wish to treat statements of the form %2A is the cause
of B%1.  The method is first to treat them in systems of interacting
finite automata and then in common sense statements about the
world.  It seems likely that causality will turn out to be
an %2approximate concept%1, i.e. a common sense statement of the
form %2A is the cause of B%1 is defined in terms of an approximate
theory that models some aspect of the world by an automaton system
or some generalization of an automaton system.  Attempts to define
truth conditions for the causality statement outside of the automaton
model will usually fail.  All this should become clearer after we
have discussed causality in automaton systems.  However, we will note
now that our concept of causality is stronger than a mere association
or even a law, i.e. there will be systems that satisfy laws of the
form %2A is always followed by B%1 for which we will not want to
say that %2A is the cause of B%1.

	We will use finite systems of finite automata, but our
concept of causality won't rely essentially on the discreteness
of time or the automata or on the finiteness of the systems or of
the automata themselves.

Definition: A simple event is a particular automaton of the system
being in a particular state at a particular time.  Thus we may
set ⊗E1 to be %2S3(5)_=_7%1.

Definition: A complex event is that the state of a particular automaton
at a particular time satisfies a particular predicate.  Thus
⊗E2 may be %2even_S3(5)%1.

Definition: A mixed event involves a relation between the states of
more than one automaton at a given time.  Thus ⊗E3 may be 
%2S3(5)_≤_S2(5)%1.

	We won't consider relations involving states at different times
although common sense allows "caused ⊗A to arrive before ⊗B".

	There seem to be many notions of causality in automaton
systems.  Here are a few of them.

	#. The simplest case of automaton causality involves a fixed
initial state of the system and a simple or complex event ⊗E1.  We say
Let ⊗E1 involve the state of automaton ⊗S1 at time ⊗t1. 
Suppose that in the particular initial configuration the event
⊗E2 occurs.  Suppose further that when we change the state of ⊗S1 at
time ⊗t1 to some value to so that ⊗E1 doesn't occur, then ⊗E2 doesn't
occur either.  Then we say that ⊗E1 is the cause of ⊗E2. 

	We would like to show that this definition is different from
the notion of there being a law that says that whenever ⊗E1 occurs,
⊗E2 will also occur.