[Inquiry] Re: Differential Logic -- Series B

[Jon Awbrey]

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DLOG.  Note B15

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Before we tangle with the rest of the Jets and Sharks example,
let's look at a cactus expression that's next in the series
we just considered, this time a lobe with three variables.
For instance, let's analyze the cactus form whose graph
and string expressions are shown in the next display.

o-------------------------------------------------o
|                                                 |
|                     x  y  z                     |
|                     o--o--o                     |
|                      \   /                      |
|                       \ /                       |
|                        @                        |
|                                                 |
o-------------------------------------------------o
|                    (x, y, z)                    |
o-------------------------------------------------o

As always in this competitive paradigm, we assume that
the units x, y, z are mutually inhibitory, so that the
only states that are possible at equilibrium are those
with exactly one unit charged and all the rest at rest.
Table 8 gives the lobal dynamics of the form (x, y, z).

Table 8.  Lobal Dynamics of the Form (x, y, z)
o-----------o-----------o-----------o-----------o
|     x     |     y     |     z     | (x, y, z) |
o-----------o-----------o-----------o-----------o
|           |           |           |           |
|  charged  |  charged  |  charged  |  charged  |
|           |           |           |           |
|  charged  |  charged  |  resting  |  charged  |
|           |           |           |           |
|  charged  |  resting  |  charged  |  charged  |
|           |           |           |           |
|  charged  |  resting  |  resting  |  resting  |
|           |           |           |           |
|  resting  |  charged  |  charged  |  charged  |
|           |           |           |           |
|  resting  |  charged  |  resting  |  resting  |
|           |           |           |           |
|  resting  |  resting  |  charged  |  resting  |
|           |           |           |           |
|  resting  |  resting  |  resting  |  charged  |
|           |           |           |           |
o-----------o-----------o-----------o-----------o

Given B = {charged, resting} the Table presents the appearance
of a function f : B x B x B -> B, where f[x, y, z] = (x, y, z).

If we make the identifications, charged = false, resting = true,
in accord with the so-called "existential" interpretation, then
the physical function f : B^3 -> B is tantamount to the logical
function that is suggested by the phrase "just 1 of 3 is false".
Table 9 is the truth table for the logical function that we get,
this time using 0 for false and 1 for true in the customary way.

Table 9.  Existential Interpretation of (x, y, z)
o-----------o-----------o-----------o-----------o
|     x     |     y     |     z     | (x, y, z) |
o-----------o-----------o-----------o-----------o
|                                   |           |
|     0           0           0     |     0     |
|                                   |           |
|     0           0           1     |     0     |
|                                   |           |
|     0           1           0     |     0     |
|                                   |           |
|     0           1           1     |     1     |
|                                   |           |
|     1           0           0     |     0     |
|                                   |           |
|     1           0           1     |     1     |
|                                   |           |
|     1           1           0     |     1     |
|                                   |           |
|     1           1           1     |     0     |
|                                   |           |
o-----------------------------------o-----------o

Jon Awbrey

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http://www.cs.bsu.edu/homepages/mighty/history.html
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