## Theme One Program • Jets and Sharks 1

It is easy to spend a long time on the rudiments of learning and logic before getting down to practical applications — but I think we’ve circled square one long enough to expand our scope and see what the category of programs envisioned in Theme One can do with more substantial examples and exercises.

During the development of the Theme One program I tested successive implementations of its Reasoning Module or Logical Modeler on appropriate examples of logical problems current in the literature of the day.  The PDP Handbook of McClelland and Rumelhart set one of the wittiest gems ever to whet one’s app‑titude so I could hardly help but take it on.  The following text is a light revision of the way I set it up in the program’s User Guide.

### Example 5. Jets and Sharks

The propositional calculus based on the minimal negation operator can be interpreted in a way resembling the logic of activation states and competition constraints in one class of neural network models.  One way to do this is to interpret the blank or unmarked state as the resting state of a neural pool, the bound or marked state as its activated state, and to represent a mutually inhibitory pool of neurons $A, B, C$ by the proposition $\texttt{(} A \texttt{,} B \texttt{,} C \texttt{)}.$  The manner of representation may be illustrated by transcribing a well-known example from the parallel distributed processing literature (McClelland and Rumelhart 1988) and working through a couple of the associated exercises as translated into logical graphs.

Displayed below is the text expression of a traversal string which Theme One parses into a cactus graph data structure in computer memory.  The cactus graph represents a single logical formula in propositional calculus and this proposition embodies all the logical constraints defining the Jets and Sharks data base.

$\text{Jets and Sharks} \stackrel{_\bullet}{} \text{Log File}$

To be continued …

### References

• McClelland, J.L. (2015), Explorations in Parallel Distributed Processing : A Handbook of Models, Programs, and Exercises, 2nd ed. (draft), Stanford Parallel Distributed Processing LabOnline, Section 2.3, Figure 2.1.
• McClelland, J.L., and Rumelhart, D.E. (1988), Explorations in Parallel Distributed Processing : A Handbook of Models, Programs, and Exercises, MIT Press, Cambridge, MA.  “Figure 1. Characteristics of a number of individuals belonging to two gangs, the Jets and the Sharks”, p. 39, from McClelland (1981).
• McClelland, J.L. (1981), “Retrieving General and Specific Knowledge From Stored Knowledge of Specifics”, Proceedings of the Third Annual Conference of the Cognitive Science Society, Berkeley, CA.

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