Directory:Jon Awbrey/Papers/Differential Analytic Turing Automata - Revision history

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Jon Awbrey: lose a few mathematical cliches

2009-03-22T21:39:13Z

lose a few mathematical cliches

@@ Line 117: / Line 117: @@
 & 1 & 0 & 0 \\
 & 1 & 0 & 0 \\
-& {}^{\shortparallel} & {}^{\shortparallel} & {}^{\shortparallel}
+& {}^\shortparallel & {}^\shortparallel & {}^\shortparallel
 \end{array}</math>
 |}
@@ Line 134: / Line 134: @@
 & 1 & 0 & 0 \\
 & 1 & 0 & 0 \\
-& {}^{\shortparallel} & {}^{\shortparallel} & {}^{\shortparallel}
+& {}^\shortparallel & {}^\shortparallel & {}^\shortparallel
 \end{array}</math>
 |}
@@ Line 1,920: / Line 1,920: @@
 ==Visualization==
-In my work on [[Differential Logic and Dynamic Systems 2.0|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 <math>F : [u, v] \to [u, v],\!</math> we've been making use of what I call the ''areal view'' of the extended universe of discourse, <math>[u, v, du, dv],\!</math> 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 <math>F\!</math> from several different angles, and see if it can be fitted into a coherent picture of the transformation <math>F : (u, v) \mapsto ( ~\texttt{((u)(v))}~, ~\texttt{((u, v))}~ ).</math>
+In my work on [[Differential Logic and Dynamic Systems 2.0|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 <math>F : [u, v] \to [u, v],\!</math> we've been making use of what I call the ''areal view'' of the extended universe of discourse, <math>[u, v, \mathrm{d}u, \mathrm{d}v],\!</math> 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 <math>F\!</math> from several different angles, and see if it can be fitted into a coherent picture of the transformation <math>F : (u, v) \mapsto ( ~ \texttt{((} u \texttt{)(} v \texttt{))} ~,~ \texttt{((} u \texttt{,~} v \texttt{))} ~ ).\!</math>
 In our first crack at the transformation <math>F,\!</math> we simply plotted the state transitions and applied the utterly stock technique of calculating the finite differences.
@@ Line 1,929: / Line 1,929: @@
+|
 <math>\begin{array}{c|cc|cc|}
-t &  u &  v & du & dv \\[8pt]
+t & u & v & \mathrm{d}u & \mathrm{d}v \\[8pt]
-&  1 &  1 &  0 &  0 \\
+& 1 & 1 & 0 & 0 \\
-& '' & '' & '' & '' \\
+& {}^\shortparallel & {}^\shortparallel & {}^\shortparallel & {}^\shortparallel
 \end{array}</math>
 |}
-A quick inspection of the first Table suggests a rule to cover the case when <math>\texttt{u~=~v~=~1},</math> namely, <math>\texttt{du~=~dv~=~0}.</math>  To put it another way, the Table characterizes Orbit&nbsp;1 by means of the data:  <math>(u, v, du, dv) = (1, 1, 0, 0).\!</math>  Another way to convey the same information is by means of the extended proposition:  <math>\texttt{u~v~(du)(dv)}.</math>
+A quick inspection of the first Table suggests a rule to cover the case when <math>u = v = 1,\!</math> namely, <math>\mathrm{d}u  = \mathrm{d}v = 0.\!</math>  To put it another way, the Table characterizes Orbit&nbsp;1 by means of the data:  <math>(u, v, \mathrm{d}u, \mathrm{d}v) = (1, 1, 0, 0).\!</math>  Another way to convey the same information is by means of the extended proposition:  <math>u v \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}.\!</math>
 {| align="center" cellpadding="8" style="text-align:center"
@@ Line 1,942: / Line 1,942: @@
+|
 <math>\begin{array}{c|cc|cc|cc|}
-t &  u &  v & du & dv & d^2 u & d^2 v \\[8pt]
+t & u & v & \mathrm{d}u & \mathrm{d}v & \mathrm{d}^2 u & \mathrm{d}^2 v \\[8pt]
-&  0 &  0 &  0 &  1 &     1 &     0 \\
+& 0 & 0 & 0 & 1 & 1 & 0 \\
-&  0 &  1 &  1 &  1 &     1 &     1 \\
+& 0 & 1 & 1 & 1 & 1 & 1 \\
-&  1 &  0 &  0 &  0 &     0 &     0 \\
+& 1 & 0 & 0 & 0 & 0 & 0 \\
-& '' & '' & '' & '' &    '' &    '' \\
+& {}^\shortparallel & {}^\shortparallel & {}^\shortparallel & {}^\shortparallel & {}^\shortparallel & {}^\shortparallel
 \end{array}</math>
 |}
-A more fine combing of the second Table brings to mind a rule that partly covers the remaining cases, that is, <math>\texttt{du~=~v}, ~\texttt{dv~=~(u)}.</math>  This much information about Orbit&nbsp;2 is also encapsulated by the extended proposition, <math>\texttt{(uv)((du, v))(dv, u)},</math> which says that <math>u\!</math> and <math>v\!</math> are not both true at the same time, while <math>du\!</math> is equal in value to <math>v\!</math> and <math>dv\!</math> is opposite in value to <math>u.\!</math>
+A more fine combing of the second Table brings to mind a rule that partly covers the remaining cases, that is, <math>\mathrm{d}u = v, ~ \mathrm{d}v = \texttt{(} u \texttt{)}.\!</math>  This much information about Orbit&nbsp;2 is also encapsulated by the extended proposition <math>\texttt{(} uv \texttt{)((} \mathrm{d}u \texttt{,} v \texttt{))(} \mathrm{d}v, u \texttt{)},\!</math> which says that <math>u\!</math> and <math>v\!</math> are not both true at the same time, while <math>\mathrm{d}u\!</math> is equal in value to <math>v\!</math> and <math>\mathrm{d}v\!</math> is opposite in value to <math>u.\!</math>
 ==Turing Machine Example==

@@ Line 1: / Line 1: @@
 {{DISPLAYTITLE:Differential Analytic Turing Automata}}
 The task ahead is to chart a course from general ideas about ''transformational equivalence classes of graphs'' to a notion of ''differential analytic turing automata'' (DATA).  It may be a while before we get within sight of that goal, but it will provide a better measure of motivation to name the thread after the envisioned end rather than the more homely starting place.
@@ Line 2,911: / Line 2,910: @@
 # http://stderr.org/pipermail/inquiry/2004-June/001641.html
 # http://stderr.org/pipermail/inquiry/2004-June/001642.html

@@ Line 3: / Line 3: @@
 The task ahead is to chart a course from general ideas about ''transformational equivalence classes of graphs'' to a notion of ''differential analytic turing automata'' (DATA).  It may be a while before we get within sight of that goal, but it will provide a better measure of motivation to name the thread after the envisioned end rather than the more homely starting place.
-The basic idea is as follows.  One is given a set <math>\mathcal{G}</math> of graphs and a set <math>\mathcal{T}</math> of transformation rules, where each rule <math>\operatorname{t} \in \mathcal{T}</math> has the effect of transforming graphs into graphs, <math>\operatorname{t} : \mathcal{G} \to \mathcal{G}.</math>  In the cases that we shall be studying, this set of transformation rules partitions the set of graphs into ''transformational equivalence classes'' (TECs).
+The basic idea is as follows.  One has a set <math>\mathcal{G}</math> of graphs and a set <math>\mathcal{T}</math> of transformation rules, and each rule <math>\operatorname{t} \in \mathcal{T}</math> has the effect of transforming graphs into graphs, <math>\operatorname{t} : \mathcal{G} \to \mathcal{G}.</math>  In the cases that we shall be studying, this set of transformation rules partitions the set of graphs into ''transformational equivalence classes'' (TECs).
 There are many interesting excursions to be had here, but I will focus mainly on logical applications, and and so the TECs I talk about will almost always have the character of ''logical equivalence classes'' (LECs).