Automata - please comment; I'm over my head here - thanks, Alan

[Alan Sondheim]

(please comment if you can unpack the below - thanks, Alan)


Automata


I'm reading a book describing automata in practical terms - Automata
Theory: An Engineering Approach, Igor Aleksander and F. Keith Hanna, Crane
Russak, 1975. This is prior to Wolfram's main work of course on cellular
automata; it's classical theory. In any case, as far as I can tell, given
states and transitions, there is no indication _how long_ any particular
transition will take; i.e. this isn't queuing theory. So one might
consider the transitions occurring in present - as if digital-in-present,
collapsed across what might otherwise be temporal, analogic. Given
Wolfram's reformulation in A New Kind of Science, I wonder about the
ontology (mathematical, even within an engineering approach) and epistem-
ology (perhaps fundamentally discrete mathematics). Automata may at times
be partitioned into independent automata; the substitution property is
important here and seems to have implications for labeling. Again I'm
thinking in terms of temporal collapse, flattening. In any case:
"Formally, a partition of the set of states of an automaton has the
substitution property if and only if each of its boxes maps into a single
block under all possible inputs." And:

"The crucial significance of an automaton and an SP partition is that as
far as next-state mappings are concerned [...], the states within a block
could be considered as a _single state_ and an automaton could be found
which behaves in the same way as the original automaton, but with only one
state per block. This property is used directly in state minimization, as
will be seen later." Further: "Given an SP partition of the states of an
automaton, each block of which is completely output consistent, every such
block may be replaced by a single state, providing a state-minimal automa-
ton."

At this point, I'm well over my head (I've not reproduced the mathematical
apparatus of course). The two points I want to make are: the possibility
of state-minimal automatons as a rough model for language (think of the
emblematic), and the collapse of temporality in relation to presencing
present.

To push this analogy etc. into the realm of absurdity - consider a state-
minimal automaton network or sememe (holarchy); if we average across the
transitions as infinitesimals, can anything be located in relation to the
appearance of time (as well as subjective time); second, consider one of
the chaotic cellular automata described by Wolfram; if we average across
the 'landscape' as it develops, can we consider this the origin of the
analogic?

I realize neither my mathematics nor my grasp of automata concepts is
particularly strong, but I wanted to throw this out (not literally) for
comment -

Alan