# Re: Bisimulation Algebra

```On 8/25/2012 1:53 AM, Stephen P. King wrote:
```
``` On 8/25/2012 2:41 AM, meekerdb wrote:
> On 8/24/2012 11:19 PM, Stephen P. King wrote:
>> On 8/24/2012 11:33 PM, meekerdb wrote:
>>> On 8/24/2012 7:05 PM, Stephen P. King wrote:
>>>>
>>>> "...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 retractable paths.
>>>
>>> I don't understand this. You write A~(B~A) which implies that
>>> B~A is a "system" (in this case one being simulated by A).
>>
>> Dear Brent,
>>
>> The symbol "~" represent simulate, so the symbols A~(B~A) would
>> be read as "A simulating B while it is simulating A". A and B and
>> C and D ... are universal simulators ala David Deutsch.
```

```
```
But then A~B is a relation between simulators, not simulations of a system on two different simulators.
```
The can
```
```>> run on any physical system capable of universality.
>>
>>> But then you write
>>>
>>> A~B~A=A~A
>>
>> These would read as: "A simulating B simulating A", which is
>> different from "A simulating B while it is simulating A", a
>> subtle difference.
```
```
```
So subtle I fail to grasp it. What does A "while" add? Is A~B~A = A~(B~A)? You didn't answer my question about why you dropped the parentheses, even though you treat ~ as non-associative.
```
Brent

The former is simultaneous while the latter is
```
```>> not.
>
> The idea of simultaneity seems out of place in simulation. A
> simulation simulates the event relations that define time. Your
> distinction implies some external time that makes an essential
> difference within the simulation??

Dear Brent,

Good question! It matters at the interface - the input location vs.
the output location, but not for the internals of the computation
itself. You have to stop thinking of a computer as an isolated
system. Bruno does this and he wonders why I complain that he does
not understand implications of the body problem when it is reduced to
arithmetic. We have a "reality" full of separate minds that needs to
be explained. Explaining a single mind is easy; why we can construct
beautiful Peano arithmetic and Robinson Arithmetic models of it, but
a plurality of separate minds; that's hard! We have diary entries and
discussions of being at Washington or Helsinki or Moscow, but that do
these names mean to an isolated computation? Locating a place is not
the same as locating a number.

>>>
>>> and also
>>>
>>> A~B~C~A =/= A~C~B~A =/= A~A
>>>
>>> This seems inconsistent, since A~B~C~A = A~D~A where D=B~C,
>>
>> How do you get D=B~C from? That is inconsistent with the Woolsey
>> identity rule .
>
> It's just defining a symbol "D" to denote the system B~C.

B~C is not a system, B~C is system B simulating C. If D is a system
simulating B simulating C then it is its own self with its own
identity D which includes the ability to simulate B simulating C.
This does not make D into a system B~C. Sorry. Stop thinking off
things as isolated from each other, the entire idea of interaction
becomes mute when you do that!

>> For example C could be capable of simulating B in the process of
>> it simulating A, which is different in content from C simulating
>> A while A is simulating B. Simulators do not commute the way
>> numbers do.
>
> I didn't assume commutation. I denoted B~C by D and C~B by E,
> making no assumption that D=E.

But you did assume that D was a particular computation and not a
simulator capable of many simulations, not just B~C. I didn't define
that possibility, so where did it come from?

>> BTW, a simulation relation is not necessarily an identity like
>> "=".
>>
>>
>>> but then A~D~A=A~A. And A~C~B~A = A~E~A where E=C~B, and then
>>> A~E~A=A~A. But then A~B~C~A = A~C~B~A.
>>
>> I seem to be assuming a natural ordering on the symbols A, B, C,
>> D, etc.
>
> No I just followed the arbitrary convention of picking the next
> letter when I needed a new name. Put X for C and S for E if you
> like, they are just names of systems.

It helps to check to see if one's conjectures about a idea are
consistent with all of the idea, not just pieces of it. Naming
conventions are very tricky and lead us into all sorts of
temptations. ;-)

> Of course for real computers running simulations it is not
> necessarily the case that A~B~A=A~A, which would equal A, although
> that's the most efficient way for A to simulate B simulating A.

But there is a difference! A simulating B simulating A is the
internal map of a single program, A. A simulating B while it is
simulating A is a internal map (in A) of another program's (B)
simulation. A slight difference. Can we untangle computations from
each other such that they can have seperate identities or
localizations? There is a good point to your critique here and it is
that the two versions are equivalent to a separate computer that has
A, B and C as subroutines such that the input and outputs are the
same. But this equivalence is strictly internal to that seperate
system that might be, in words like Bruno's, evaluating the
difference. What I am trying to set up here is the map-territory
difference and where it vanishes. When does my mental image of you
and your mental image of yourself differ? When might it be the same?

> I don't find your notion of system and simulation very clear.

Good point, I am an amateur at this and I am learning. I do

> I suppose by "system" you mean a some definite set of things which
> are evolving by a defined process, some set of states which can be
> computed by an algorithm (or possibly including randomness?). Then
> a simulation is a different set of things evolving through states
> that are isomorphic to the system simulated?

I am just following Deutsch's idea of a universal simulator. I am not
sure if this ties to functional equivalence yet, I'm exploring and
reading lots of Chalmer's stuff (and Bruno's -again).

>
> Brent
>
>> and a notion of being at the same level in the ordering with the
>> "(..)" symbols. I should have made this clear. My apologies! Does
>> the comment about telescope property not make sense?
>>
>>>
>>> You drop the parentheses, implying the relation is associative,
>>> but then you treat it as though it isn't??
>>
>> Not having pointed out the ordering caused a confusion. My
>> apologies. Thank you for pointing this out! This idea still needs
>> a lot of work, that I do admit!
>>
>>>
>>> Brent
>>>
-- -- Onward!

Stephen

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