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*To*: Ontology <ontology@ieee.org>*Subject*: ONT Re: Differential Analytic Turing Automata*From*: Jon Awbrey <jawbrey@att.net>*Date*: Thu, 11 Mar 2004 23:36:34 -0500*References*: <403F546C.D69B6918@att.net> <40476DB3.BC029F0@att.net>*Sender*: owner-ontology@majordomo.ieee.org

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DATA. Note 20 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o In my work on "Differential Logic and Dynamic Systems", I found it useful to develop several different ways of visualizing logical transformations, indeed, I devised four distinct styles of picture for the job. Thus far in our work on the mapping F : [u, v] -> [u, v], we've been making use of what I call the "areal view" of the extended universe of discourse, [u, v, du, dv], but as the number of dimensions climbs beyond four, it's time to bid this genre adieu, and look for a style that can scale a little better. At any rate, before we proceed any further, let's first assemble the information that we have gathered about F from several different angles, and see if it can be fitted into a coherent picture of the transformation F : <u, v> ~> <((u)(v)), ((u, v))>. In our first crack at the transformation F, we simply plotted the state transitions and applied the utterly stock technique of calculating the finite differences. Orbit 1. u v o-----o-----o | ` ` | d d | | u v | u v | o=====o=====o | 1 1 | 0 0 | | " " | " " | o-----o-----o A quick inspection of the first Table suggests a rule to cover the case when u = v = 1, namely, du = dv = 0. To put it another way, the Table characterizes Orbit 1 by means of the data: <u, v, du, dv> = <1, 1, 0, 0>. Last but not least, yet another way to convey the same information is by means of the (first order) extended proposition: u v (du)(dv). Orbit 2. (u v) o-----o-----o-----o | ` ` | ` ` | d d | | ` ` | d d | 2 2 | | u v | u v | u v | o=====o=====o=====o | 0 0 | 0 1 | 1 0 | | 0 1 | 1 1 | 1 1 | | 1 0 | 0 0 | 0 0 | | " " | " " | " " | o-----o-----o-----o A more fine combing of the second Table brings to mind a rule that partly covers the remaining cases, that is, du = v, dv = (u). To vary the formulation, this Table characterizes Orbit 2 by means of the following vector equation: <du, dv> = <v, (u)>. This much information about Orbit 2 is also encapsulated by the (first order) extended proposition, (uv)((du, v))(dv, u), which says that u and v are not both true at the same time, while du is equal in value to v, and dv is the opposite of u. To be continued ... Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o inquiry e-lab: http://stderr.org/pipermail/inquiry/ o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

**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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