# Re: Bisimulation algebra

```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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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 . 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. 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. 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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