A crude sketch of a computational model of Interaction.

Stephen Paul King

Might it be possible to model the content of 1st person experience as a
computationally generated "simulation"? We can point to the body of work by
David Deutsch, such as that found in his book The Fabric of Reality, as
providing some excellent reasoning to at least consider that the answer to
our question might be: Yes. OK, given that, how might we model interactions
between such "simulations" in a way that would give us something that covers
many situations including those where we have events that cannot occur
simultaneously? I think there is.
Let us first point out some features of computations and the simulations
that they could generate. We know that computers can generate simulations of
other computational systems. We see this when we consider how one computer
can run software that emulates of some other computer. 
What about a computer generating a simulation of itself? What about a
computer X generating a simulation of some other computer Y that is running
a simulation of X? It seems that if we allow for unlimited computational
resources, we could have a computer generating a simulation of a computer
generating a simulation of a computer generating a simulation . What about a
computer X generating a simulation of computer Y that is generating a
simulation of X as it generates a simulation of Y . As so forth.
We can see that if there is a finite upper bound on the resources available
to the simulation generating computers then such expressions of infinite
regress cannot obtain, but the idea that one computational system can
generate simulations of other computational systems is not problematic and
maybe even useful to model interactions between computational system. 
Now we need to ask how it is that we distinguish a simulation of a
computational system from a "real" computational system in most discussions
of this idea? Given that we have the notion of Universal Computers and even
Universal Virtual Reality Machines (1) we find that the idea that we can
distinguish a simulation from the real thing to require some kind of notion
of a physical reality that is distinct from simulations of parts of it. In
other words that there is something about "reality" that is not capable of
being simulated by a computational system in principle. 
In the work of Bruno Marchal (2), building on prior work in modal logic, we
find some very good arguments that there does not exist a computational
means to decide which computation might be the one that exactly matches the
world of experience that I have as a 1-scape. We can conjecture that that
something has to do with the Hard Problem of Consciousness (3), but we can
set that aside for now since we are only considering those aspects that are
Additionally there are some other reasoning as to why it makes sense to
suspect that some kind of Cartesian-like dualism is involved is implied.(4)
We could go further and borrow from the brilliant writer and thinker Greg
Egan (5) the notion of a 1-scape; the landscape of the world as seen by 1
person and communicated about in the 1st person sense. 3-scapes would then
be considered as emerging from the intercommunications between many
We now move to considerations of multiple separate computational
simulations. I suspect that we can use the notion of bisimulation to enable
us to figure out when and if separate systems can be said to communicate
with each other if in the course of a conversation back and forth their
successive simulations of each other match up with the internal simulations
that they might have of each other. 
In other words, if my simulation of you telling me that X occurred matched
up with your simulation of yourself telling me that X occurred *and* if your
simulation of me responding to the occurrence of X matches up with my
simulation of my response to the occurrence of X then we can say that X is
communicated to me by you.
I strongly suspect that this idea is consistent with Shannon's notion of
information as the coincidence of joint allowed states between a pair of
systems and if so it may help us to take a step beyond the usual account of
communication between systems that assumes some kind of substance exchange.
Some of the algebra (6) of this idea if bisimulation is as follows.

Let "A simulation of B" be denoted  A ~ B.

Further, let ( B ~ A ) be called the "conjugate" of ( A ~ B ); since these
are not equal, the simulation is not commutative in general.

We see that  A = A ~ A  is the "real identity bisimulation" since the
simulation is equal to its' conjugate.

Then we state the "Woolsey identity":

         A ~ A  =  A ~ B ~ A

That is:  real identity bisimulation = simulation of the conjugate

This is a law of identity for computational bisimulation that implies that a
"real identity" occurs only when the conjugate of the bisimulation is equal
to itself.

This "law of real identity bisimulation" would then imply:  
A ~ B ~ C ~ A not=  A ~ A, since A ~ B ~ C ~ A not= A ~ C ~ B ~ A ;
the conjugate is not equal to itself and so does not form the real identity
of A ~ A  =  A.

But the law of real identity bisimulation would validate the following
 A ~ B ~ B ~ A  =  A ~ A,

and this is seen to be consistent with 
A ~ B ~ B ~ A  =  A ~ B ~ A  =  A ~A

since B ~ B  =  B.

Also, due to the law of conjugate bisimulation identity:

         A ~ A  =  A ~ B ~ C ~ B ~ A  =  A ~ B ~ A

this is "retractable path independence": path independence only over
retrace-able paths.
This is consistent since  B  =  B ~ C ~ B, and is the closest I can come to
Note that retractable path independence does not necessarily imply closure:

A ~ C  not=  A ~ B ~ C, 
since closure is assuming something beyond the law of real identity
bisimulation.  It seems likely that bisimulation between
three observers (or more) is not in general closed.

1.      http://everything2.com/title/Turing+principle
2.      http://iridia.ulb.ac.be/~marchal/
3.      http://en.wikipedia.org/wiki/Hard_problem_of_consciousness
4.      See http://xxx.lanl.gov/abs/math.HO/9911150 and
http://boole.stanford.edu/pub/ratmech.pdf for more.
5) http://gregegan.customer.netspace.net.au/
6) As developed and communicated to me by Paul Hanna.

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