Changes

In my work on "[[Directory:Jon_Awbrey/Papers/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 "[[Directory:Jon_Awbrey/Papers/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 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 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

| <math>\text{Orbit 1}\!</math>

|     | d d |

|-

| u v | u v |

|

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

| 1 1 | 0 0 |

t &  u &  v & du & dv \\[8pt]

| " " | " " |

0 &  1 &  1 &  0 &  0 \\

|}

A quick inspection of the first Table suggests a rule

A quick inspection of the first Table suggests a rule

to cover the case when u = v = 1, namely, du = dv = 0.

to cover the case when u = v = 1, namely, du = dv = 0.

information is by means of the (first order) extended

information is by means of the (first order) extended

proposition:  u v (du)(dv).

proposition:  u v (du)(dv).

Orbit 2.  (u v)

o-----o-----o-----o

| <math>\text{Orbit 2}\!</math>

|    |    | d d |

|-

|     | d d | 2 2 |

|

| u v | u v | u v |

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

t &  u &  v & du & dv & d^2 u & d^2 v \\[8pt]

| 0 0 | 0 1 | 1 0 |

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

| 0 1 | 1 1 | 1 1 |

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

| 1 0 | 0 0 | 0 0 |

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

| " " | " " | " " |

|}

A more fine combing of the second Table brings to mind

A more fine combing of the second Table brings to mind

a rule that partly covers the remaining cases, that is,

a rule that partly covers the remaining cases, that is,