[Inquiry] Re: Differential Logic -- Series B

[Jon Awbrey]

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DLOG.  Note B12

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I happened on the graphical syntax for propositional calculus that
I now call the "cactus language" while exploring the confluence of
three streams of thought.  There was C.S. Peirce's use of operator
variables in logical forms and the operational representations of
logical concepts, there was George Spencer Brown's explanation of
a variable as the contemplated presence or absence of a constant,
and then there was the graph theory and group theory that I had
been picking up, bit by bit, since I first encountered them in
tandem in Frank Harary's foundations of math course, c. 1970.

More on that later, as the memories unthaw, but for the moment
I want very much to take care of some long-unfinished business,
and give a more detailed explanation of how I used this syntax
to represent a popular exercise from the PDP literature of the
late 1980's, McClelland's and Rumelhart's "Jets and Sharks".

The knowledge base of the case can be expressed as a single proposition.
The following display presents it in the corresponding text file format.

File "jas.log".  Jets and Sharks Example
o-----------------------------------------------------------o
|                                                           |
|   (( art    ),( al   ),( sam  ),( clyde ),( mike  ),      |
|    ( jim    ),( greg ),( john ),( doug  ),( lance ),      |
|    ( george ),( pete ),( fred ),( gene  ),( ralph ),      |
|    ( phil   ),( ike  ),( nick ),( don   ),( ned   ),      |
|    ( karl   ),( ken  ),( earl ),( rick  ),( ol    ),      |
|    ( neal   ),( dave ))                                   |
|                                                           |
|   ( jets , sharks )                                       |
|                                                           |
|   ( jets ,                                                |
|    ( art    ),( al   ),( sam  ),( clyde ),( mike  ),      |
|    ( jim    ),( greg ),( john ),( doug  ),( lance ),      |
|    ( george ),( pete ),( fred ),( gene  ),( ralph ))      |
|                                                           |
|   ( sharks ,                                              |
|    ( phil ),( ike  ),( nick ),( don ),( ned  ),( karl ),  |
|    ( ken  ),( earl ),( rick ),( ol  ),( neal ),( dave ))  |
|                                                           |
|   (( 20's ),( 30's ),( 40's ))                            |
|                                                           |
|   ( 20's ,                                                |
|    ( sam    ),( jim  ),( greg ),( john ),( lance ),       |
|    ( george ),( pete ),( fred ),( gene ),( ken   ))       |
|                                                           |
|   ( 30's ,                                                |
|    ( al  ),( mike ),( doug ),( ralph ),( phil ),          |
|    ( ike ),( nick ),( don  ),( ned   ),( rick ),          |
|    ( ol  ),( neal ),( dave ))                             |
|                                                           |
|   ( 40's ,                                                |
|    ( art ),( clyde ),( karl ),( earl ))                   |
|                                                           |
|   (( junior_high ),( high_school ),( college ))           |
|                                                           |
|   ( junior_high ,                                         |
|    ( art  ),( al    ),( clyde  ),( mike  ),( jim ),       |
|    ( john ),( lance ),( george ),( ralph ),( ike ))       |
|                                                           |
|   ( high_school ,                                         |
|    ( greg ),( doug ),( pete ),( fred ),                   |
|    ( nick ),( karl ),( ken  ),( earl ),                   |
|    ( rick ),( neal ),( dave ))                            |
|                                                           |
|   ( college ,                                             |
|    ( sam ),( gene ),( phil ),( don ),( ned ),( ol ))      |
|                                                           |
|   (( single ),( married ),( divorced ))                   |
|                                                           |
|   ( single ,                                              |
|    ( art  ),( sam  ),( clyde ),( mike  ),( doug ),        |
|    ( pete ),( fred ),( gene  ),( ralph ),( ike  ),        |
|    ( nick ),( ken  ),( neal  ))                           |
|                                                           |
|   ( married ,                                             |
|    ( al  ),( greg ),( john ),( lance ),( phil ),          |
|    ( don ),( ned  ),( karl ),( earl  ),( ol   ))          |
|                                                           |
|   ( divorced ,                                            |
|    ( jim ),( george ),( rick ),( dave ))                  |
|                                                           |
|   (( bookie ),( burglar ),( pusher ))                     |
|                                                           |
|   ( bookie ,                                              |
|    ( sam  ),( clyde ),( mike ),( doug ),                  |
|    ( pete ),( ike   ),( ned  ),( karl ),( neal ))         |
|                                                           |
|   ( burglar ,                                             |
|    ( al     ),( jim ),( john ),( lance ),                 |
|    ( george ),( don ),( ken  ),( earl  ),( rick ))        |
|                                                           |
|   ( pusher ,                                              |
|    ( art   ),( greg ),( fred ),( gene ),                  |
|    ( ralph ),( phil ),( nick ),( ol   ),( dave ))         |
|                                                           |
o-----------------------------------------------------------o

Let's start with the simplest clause of the conjoint proposition:

    ( jets , sharks )

Drawn as the correspnding cactus graph, we have:

      jets   sharks
        o-----o
         \   /
          \ /
           @

According to my earlier, if somewhat sketchy interpretive suggestions,
we are supposed to picture a quasi-neural pool that contains a couple
of quasi-neural agents or "units", that between the two of them stand
for the logical variables "jets" and "sharks", respectively.  Further,
we imagine these agents to be mutually inhibitory, so that settlement
of the dynamic between them achieves equilibrium when just one of the
two is "active" or "changing" and the other is "stable" or "enduring".

To be continued ...

Jon Awbrey

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