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*To*: Ontology <ontology@ieee.org>*Subject*: ONT Re: Differential Analytic Turing Automata*From*: Jon Awbrey <jawbrey@att.net>*Date*: Mon, 08 Mar 2004 00:52:26 -0500*References*: <403F546C.D69B6918@att.net> <40476DB3.BC029F0@att.net> <404BB2D9.B81EF7A2@att.net>*Sender*: owner-ontology@majordomo.ieee.org

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DATA. Note 13 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o I think that it ought to be clear at this point that we need a more systematic symbolic method for computing the differentials of logical transformations, using the term "differential" in a loose way at present for all sorts of finite differences and derivatives, leaving it to another discussion to sharpen up its more exact technical senses. For convenience of reference, let's recast our current example in the form F = <f, g> = <((u)(v)), ((u, v))>. In their application to this logical transformation the operators E and D respectively produce the "enlarged map" EF = <Ef, Eg> and the "difference map" DF = <Df, Dg>, whose components can be given as follows, if the reader, in the absence of a special format for logical parentheses, can forgive syntactically 2-lingual phrases: Ef = ((u + du)(v + dv)) Eg = ((u + du, v + dv)) Df = ((u)(v)) + ((u + du)(v + dv)) Dg = ((u, v)) + ((u + du, v + dv)) But these initial formulas are purely definitional, and help us little to understand either the purpose of the operators or the significance of the results. Working symbolically, let's apply a more systematic method to the separate components of the mapping F. A sketch of this work is presented in the following series of Figures, where each logical proposition is expanded over the basic cells uv, u(v), (u)v, (u)(v) of the 2-dimensional universe of discourse U% = [u, v]. Computation Summary for f<u, v> = ((u)(v)) Figure 1 shows how f = ((u)(v)) expands over [u, v] to give the exclusive disjunction uv + u(v) + (u)v. o---------------------------------------o |```````````````````````````````````````| |```````````````````o```````````````````| |``````````````````/%\``````````````````| |`````````````````/%%%\`````````````````| |````````````````/%%%%%\````````````````| |```````````````o%%%%%%%o```````````````| |``````````````/%\%%%%%/%\``````````````| |`````````````/%%%\%%%/%%%\`````````````| |````````````/%%%%%\%/%%%%%\````````````| |```````````o%%%%%%%o%%%%%%%o```````````| |``````````/%\%%%%%/%\%%%%%/%\``````````| |`````````/%%%\%%%/%%%\%%%/%%%\`````````| |````````/%%%%%\%/%%%%%\%/%%%%%\````````| |```````o%%%%%%%o%%%%%%%o%%%%%%%o```````| |``````/%\%%%%%/%\%%%%%/%\%%%%%/%\``````| |`````/%%%\%%%/%%%\%%%/%%%\%%%/%%%\`````| |````/%%%%%\%/%%%%%\%/%%%%%\%/%%%%%\````| |```o%%%%%%%o%%%%%%%o%%%%%%%o%%%%%%%o```| |```|\%%%%%/%\%%%%%/`\%%%%%/%\%%%%%/|```| |```|`\%%%/%%%\%%%/```\%%%/%%%\%%%/`|```| |```|``\%/%%%%%\%/`````\%/%%%%%\%/``|```| |```|```o%%%%%%%o```````o%%%%%%%o```|```| |```|```|\%%%%%/`\`````/`\%%%%%/|```|```| |```|```|`\%%%/```\```/```\%%%/`|```|```| |```|`u`|``\%/`````\`/`````\%/``|`v`|```| |```o---+---o```````o```````o---+---o```| |```````|````\`````/`\`````/````|```````| |```````|`````\```/```\```/`````|```````| |```````|`du```\`/`````\`/```dv`|```````| |```````o-------o```````o-------o```````| |````````````````\`````/````````````````| |`````````````````\```/`````````````````| |``````````````````\`/``````````````````| |```````````````````o```````````````````| |```````````````````````````````````````| o---------------------------------------o Figure 1. f = ((u)(v)) Figure 2 shows how Ef = ((u + du)(v + dv)) expands over [u, v] as: uv.(du dv) + u(v).(du (dv)) + (u)v.((du) dv) + (u)(v).((du)(dv)) o---------------------------------------o |```````````````````````````````````````| |```````````````````o```````````````````| |``````````````````/%\``````````````````| |`````````````````/%%%\`````````````````| |````````````````/%%%%%\````````````````| |```````````````o%%%%%%%o```````````````| |``````````````/%\%%%%%/%\``````````````| |`````````````/%%%\%%%/%%%\`````````````| |````````````/%%%%%\%/%%%%%\````````````| |```````````o%%%%%%%o%%%%%%%o```````````| |``````````/%\%%%%%/`\%%%%%/%\``````````| |`````````/%%%\%%%/```\%%%/%%%\`````````| |````````/%%%%%\%/`````\%/%%%%%\````````| |```````o%%%%%%%o```````o%%%%%%%o```````| |``````/%\%%%%%/%\`````/%\%%%%%/%\``````| |`````/%%%\%%%/%%%\```/%%%\%%%/%%%\`````| |````/%%%%%\%/%%%%%\`/%%%%%\%/%%%%%\````| |```o%%%%%%%o%%%%%%%o%%%%%%%o%%%%%%%o```| |```|\%%%%%/`\%%%%%/%\%%%%%/`\%%%%%/|```| |```|`\%%%/```\%%%/%%%\%%%/```\%%%/`|```| |```|``\%/`````\%/%%%%%\%/`````\%/``|```| |```|```o```````o%%%%%%%o```````o```|```| |```|```|\`````/%\%%%%%/%\`````/|```|```| |```|```|`\```/%%%\%%%/%%%\```/`|```|```| |```|`u`|``\`/%%%%%\%/%%%%%\`/``|`v`|```| |```o---+---o%%%%%%%o%%%%%%%o---+---o```| |```````|````\%%%%%/`\%%%%%/````|```````| |```````|`````\%%%/```\%%%/`````|```````| |```````|`du```\%/`````\%/```dv`|```````| |```````o-------o```````o-------o```````| |````````````````\`````/````````````````| |`````````````````\```/`````````````````| |``````````````````\`/``````````````````| |```````````````````o```````````````````| |```````````````````````````````````````| o---------------------------------------o Figure 2. Ef = ((u + du)(v + dv)) Figure 3 shows how Df = f + Ef expands over [u, v] to give: uv.du dv + u(v).du(dv) + (u)v.(du)dv + (u)(v).((du)(dv)) o---------------------------------------o |```````````````````````````````````````| |```````````````````o```````````````````| |``````````````````/`\``````````````````| |`````````````````/```\`````````````````| |````````````````/`````\````````````````| |```````````````o```````o```````````````| |``````````````/`\`````/`\``````````````| |`````````````/```\```/```\`````````````| |````````````/`````\`/`````\````````````| |```````````o```````o```````o```````````| |``````````/`\`````/%\`````/`\``````````| |`````````/```\```/%%%\```/```\`````````| |````````/`````\`/%%%%%\`/`````\````````| |```````o```````o%%%%%%%o```````o```````| |``````/`\`````/`\%%%%%/`\`````/`\``````| |`````/```\```/```\%%%/```\```/```\`````| |````/`````\`/`````\%/`````\`/`````\````| |```o```````o```````o```````o```````o```| |```|\`````/%\`````/%\`````/%\`````/|```| |```|`\```/%%%\```/%%%\```/%%%\```/`|```| |```|``\`/%%%%%\`/%%%%%\`/%%%%%\`/``|```| |```|```o%%%%%%%o%%%%%%%o%%%%%%%o```|```| |```|```|\%%%%%/%\%%%%%/%\%%%%%/|```|```| |```|```|`\%%%/%%%\%%%/%%%\%%%/`|```|```| |```|`u`|``\%/%%%%%\%/%%%%%\%/``|`v`|```| |```o---+---o%%%%%%%o%%%%%%%o---+---o```| |```````|````\%%%%%/`\%%%%%/````|```````| |```````|`````\%%%/```\%%%/`````|```````| |```````|`du```\%/`````\%/```dv`|```````| |```````o-------o```````o-------o```````| |````````````````\`````/````````````````| |`````````````````\```/`````````````````| |``````````````````\`/``````````````````| |```````````````````o```````````````````| |```````````````````````````````````````| o---------------------------------------o Figure 3. Df = f + Ef I'll break this here in case anyone wants to try and do the work for g on their own. 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>

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

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