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*To*: Ontology <ontology@ieee.org>*Subject*: ONT Re: Differential Analytic Turing Automata*From*: Jon Awbrey <jawbrey@att.net>*Date*: Sun, 07 Mar 2004 18:40:09 -0500*References*: <403F546C.D69B6918@att.net> <40476DB3.BC029F0@att.net>*Sender*: owner-ontology@majordomo.ieee.org

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DATA. Note 12 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Way back in DATA Note 3, we guessed, or "abduced", as a line of logicians from Aristotle to Peirce and beyond would say, the form of a rule that adequately accounts for the finite protocol of states that we observed the system X pass through, as spied in the light of its boolean state variable x : X -> B, and that rule is well-formulated in any of these styles of notation: 1.1. f : B -> B such that f : x ~> (x) 1.2. x' = (x) 1.3. x := (x) 1.4. dx = 1 In the current example, having read the manual first, I guess, we already know in advance the program that generates the state transitions, and it is a rule of the following equivalent and easily derivable forms: 2.1. F : B^2 -> B^2 such that F : <u, v> ~> <((u)(v)), ((u, v))> 2.2. u' = ((u)(v)), v' = ((u, v)) 2.3. u := ((u)(v)), v := ((u, v)) 2.4. ??? Well, the last one is not such a fall off the log, but that is exactly the purpose for which we have been developing all of the foregoing machinations. Here is what I got when I just went ahead and calculated the finite differences willy-nilly: Incipit 1. <u, v> = <0, 0> d d | d d | d d | d d | d d | ... 0 0 | 1 1 | 2 2 | 3 3 | 4 4 | ... u v | u v | u v | u v | u v | ... ----+-----+-----+-----+-----+----- 0 0 | 0 1 | 1 0 | 0 1 | 1 0 | ... 0 1 | 1 1 | 1 1 | 1 1 | 1 1 | ... 1 0 | 0 0 | 0 0 | 0 0 | 0 0 | ... 1 0 | 0 0 | 0 0 | 0 0 | 0 0 | ... " " | " " | " " | " " | " " | Incipit 2. <u, v> = <1, 1> d d | d d | d d | d d | 0 0 | 1 1 | 2 2 | 3 3 | u v | u v | u v | u v | ----+-----+-----+-----+ 1 1 | 0 0 | 0 0 | = = | 1 1 | 0 0 | 0 0 | = = | " " | " " | " " | " " | To be honest, I have never thought of trying to hack the problem in such a brute-force way until just now, and so I know enough to expect a not unappreciable probability of error about all that I have taken the risk of writing here, but let me forge ahead and see what I can see. What we are looking for is, well, one rule to rule them all, that is, a rule that works at every state and at every time. What we see at first sight in the tables above are patterns of differential features that attach to the states in each orbit of the dynamics. Looked at locally to these orbits, the isolated fixed point at <1, 1> is no problem, as the rule du = dv = 0 describes it pithily enough. When it comes to the other orbit, the first thing that comes to mind is to write out the law du = v, dv = (u). I am going to take a nap to clear my head. Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o inquiry e-lab: http://stderr.org/pipermail/inquiry/ o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

**Follow-Ups**:**ONT Re: Differential Analytic Turing Automata***From:*Jon Awbrey <jawbrey@att.net>

**References**:**ONT Differential Analytic Turing Automata***From:*Jon Awbrey <jawbrey@att.net>

**ONT Re: Differential Analytic Turing Automata***From:*Jon Awbrey <jawbrey@att.net>

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