Changes

3 & 1 & 0 & 0 \\

3 & 1 & 0 & 0 \\

4 & 1 & 0 & 0 \\

4 & 1 & 0 & 0 \\

5 & {}^{\shortparallel} & {}^{\shortparallel} & {}^{\shortparallel}

5 & {}^\shortparallel & {}^\shortparallel & {}^\shortparallel

\end{array}</math>

\end{array}</math>

|}

|}

3 & 1 & 0 & 0 \\

3 & 1 & 0 & 0 \\

4 & 1 & 0 & 0 \\

4 & 1 & 0 & 0 \\

5 & {}^{\shortparallel} & {}^{\shortparallel} & {}^{\shortparallel}

5 & {}^\shortparallel & {}^\shortparallel & {}^\shortparallel

\end{array}</math>

\end{array}</math>

|}

|}

==Visualization==

==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.

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.

|

|

<math>\begin{array}{c|cc|cc|}

<math>\begin{array}{c|cc|cc|}

t & u & v & du & dv \\[8pt]

t & u & v & \mathrm{d}u & \mathrm{d}v \\[8pt]

0 & 1 & 1 & 0 & 0 \\

0 & 1 & 1 & 0 & 0 \\

1 & '' & '' & '' & '' \\

1 & {}^\shortparallel & {}^\shortparallel & {}^\shortparallel & {}^\shortparallel

\end{array}</math>

\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"

{| align="center" cellpadding="8" style="text-align:center"

|

|

<math>\begin{array}{c|cc|cc|cc|}

<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 & 0 & 1 &     1 &     0 \\

0 & 0 & 0 & 0 & 1 & 1 & 0 \\

1 & 0 & 1 & 1 & 1 &     1 &     1 \\

1 & 0 & 1 & 1 & 1 & 1 & 1 \\

2 & 1 & 0 & 0 & 0 &     0 &     0 \\

2 & 1 & 0 & 0 & 0 & 0 & 0 \\

3 & '' & '' & '' & '' &   '' &   '' \\

3 & {}^\shortparallel & {}^\shortparallel & {}^\shortparallel & {}^\shortparallel & {}^\shortparallel & {}^\shortparallel

\end{array}</math>

\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==

==Turing Machine Example==