Re: The Ghost of Von Neumann: Automata and Physics

On Aug 30, 12:37 am, markw...@xxxxxxxxx wrote:
A distinguishing feature ofvonNeumann'sapproach was to algebraize everything at the foundational
level.

What's not widely appreciated is that the two pursuits were two
strands on a single thread...

The Untold Story of Formal Languages
http://federation.g3z.com/CompSci/index.htm#Untold
In [this synthesis], the notion of state plays a central role. The
central theme of vonNeumann's enterprise -- if ever there were
one -- was this very notion.

This redacted series is but a small part of the much larger synthesis...

As von Neumann wrote

"We are very far from possessing a theory of automata which deserves
that name, that is, a properly mathematical-logical theory",
-- The General and Logical Theory of Automata, von Neumann

he was referring just as much to the stochastic or dynamic aspect of
the enterprise, and not just to the combinatorial aspect whose
synthesis in the above spirit I had described in the previous article.

Stochastic Expressions and State Spaces
http://federation.g3z.com/CompSci/index.htm#Stochastic1

This further expands the von Neumann correspondence previously
outlined, linking the von Neumann formalism for state spaces to power
series representations for expressions in formal language theory.

Directly below it is the following:

Stochastic Finite Automata
This is a generalization both of Markov Models and Hidden Markov
Models. A stochastic finite automaton may be thought of as a finite
automaton whose transition arcs are labelled with probabilities, and
its states with "final state" probabilities, subject to the
normalization condition that the total of all the transition
probabilities for transitions emanating from a state, plus the state's
final state probability should be 1. A Hidden Markov Model is
conceived of as a limiting case where the final state probabilities
tend to 0 (and, thus, the mean time to termination, to infinity).

.