[Inquiry] Re: Differential Logic A -- Discussion

[Jon Awbrey]

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DLOG A.  Discussion Note 20

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| Yea, from the table of my memory
| I'll wipe away all trivial fond records,
| All saws of books, all forms, all pressures past,
| That youth and observation copied there,
| And thy commandment all alone shall live
| Within the book and volume of my brain
| Unmixed with baser matter.
|
| Hamlet, 1.5.98-104

Let me go back to one of the places where I came in,
where I woke up in one of my math psych courses and
began to pay attention to something inside that box
of our classroom one buzzy b(l)oomy midsommer's day.

As adapted from Ashby's 'Cybernetics' 4/1, here 'tis:

Table 1.  Finite Transducer, or Machine with Input
o---------o---------o---------o---------o---------o
|    |    |         |         |         |         |
|    |    |    a    |    b    |    c    |    d    |
|    V    |         |         |         |         |
o=========o=========o=========o=========o=========o
|         |         |         |         |         |
|   T_1   |    c    |    d    |    d    |    b    |
|         |         |         |         |         |
|   T_2   |    b    |    a    |    d    |    c    |
|         |         |         |         |         |
|   T_3   |    d    |    c    |    d    |    b    |
|         |         |         |         |         |
o---------o---------o---------o---------o---------o

As spied from the perspective of my 3rd eye,
it struck me as a table of 3-tuples, to wit:

   <1, a, c>
   <1, b, d>
   <1, c, d>
   <1, d, b>

   <2, a, b>
   <2, b, a>
   <2, c, d>
   <2, d, c>

   <3, a, d>
   <3, b, c>
   <3, c, d>
   <3, d, b>

At first sight, one usually sees the "input parameters", in this example
the identifiers in the set {1, 2, 3}, as signals or signlike data, while
the states are seen as objective conditions in which the machine resides
for moments at a time and transits between on cue from the input signals.

That view of the 3-adic relation M under survey casts it with a type
M c S x O x O, where S is a sign domain and O is an object domain.
Initially, then, the machine M appears to have some sort of dual
type in relation to the typical sign relation L c O x S x S.

It is obviously worth considering 3-adic relations of that type,
along with many others, but the apparent differences here tend
to disappear from more general points of view.  For example,
one may be thinking of the input parameters as relatively
direct data about the state of an external object system,
with the purpose of the machine being nothing more than
to store information about this object system in the
record of its states.

>From a still more general point of view, all three domains may be
the state spaces of systems, and here it is possible to think of
cases where there would be 1, 2, or 3 different systems related.

Consider the possibilities ...

Jon Awbrey

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