# Re: Bisimulation algebra

```On 8/24/2012 11:19 PM, Stephen P. King wrote:
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```On 8/24/2012 11:33 PM, meekerdb wrote:
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```On 8/24/2012 7:05 PM, Stephen P. King wrote:
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"...due to the law of conjugate bisimulation identity:

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

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this is "retractable path independence": path independence only over retractable paths.
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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).
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Dear Brent,

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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. The can run on any physical system capable of universality.
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```  But then you write

A~B~A=A~A
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These would read as: "A simulating B simulating A", which is different from "A simulating B while it is simulating A", a subtle difference. The former is simultaneous while the latter is not.
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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??
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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,
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How do you get D=B~C from? That is inconsistent with the Woolsey identity rule .
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It's just defining a symbol "D" to denote the system B~C.

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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.
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I didn't assume commutation.  I denoted B~C by D and C~B by E, making no
assumption that D=E.

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```BTW, a simulation relation is not necessarily an identity like "=".

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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.
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I seem to be assuming a natural ordering on the symbols A, B, C, D, etc.
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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.
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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. I don't find your notion of system and simulation very clear. 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?
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Brent

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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?
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You drop the parentheses, implying the relation is associative, but then you treat it as though it isn't??
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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!
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Brent

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--
Onward!

Stephen

http://webpages.charter.net/stephenk1/Outlaw/Outlaw.html
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