Inquiry Into Inquiry

Re: Theme One Program • Jets and Sharks • (1) • (2)

Example 5. Jets and Sharks (cont.)

Given a representation of the Jets and Sharks universe in computer memory, we naturally want to see if the memory serves to supply the facts a well-constructed data base should.

In their PDP Handbook presentation of the Jets and Sharks example, McClelland and Rumelhart suggest several exercises for the reader to explore the performance of their neural pool memory model on the tasks of retrieval and generalization (Exercise 2.1).

Using cactus graphs or minimal negations to implement pools of mutually inhibitory neurons lends itself to neural architectures on a substantially different foundation from the garden variety connectionist models.  At a high level of abstraction, however, there is enough homology between the two orders to compare their performance on many of the same tasks.  With that in mind, I tried Theme One on a number of examples like the ones suggested by McClelland and Rumelhart.

What follows is a brief discussion of two examples as given in the original User Guide.  Next time I’ll fill in more details about the examples and discuss their bearing on the larger issues at hand.

With a query on the name “ken” we obtain the following output, giving all the features associated with Ken.

With a query on the two features “college” and “sharks” we obtain the following outline of all features satisfying those constraints.

From this we discover all college Sharks are 30‑something and married.  Further, we have a complete listing of their names broken down by occupation.

References

• McClelland, J.L. (2015), Explorations in Parallel Distributed Processing : A Handbook of Models, Programs, and Exercises, 2nd ed. (draft), Stanford Parallel Distributed Processing Lab.  Online, 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.

Resources

• Theme One Program • Overview
• Theme One Program • Exposition
• Theme One Program • User Guide
• Survey of Theme One Program

cc: FB | Theme One Program • Laws of Form • Ma^thstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Example 5. Jets and Sharks (cont.)

As we saw last time, Theme One reads the text file shown below and constructs a cactus graph data structure in computer memory.  The cactus graph represents a single logical formula in propositional calculus and that proposition embodies the logical constraints defining the Jets and Sharks data base.

Our cactus graph incorporates a vocabulary of 41 logical terms, each of which represents a boolean variable, so the proposition in question, call it is a boolean function of the form   Given we know a truth table for takes over two trillion rows and a venn diagram for takes the same number of cells.  Topping it off, there are boolean functions of the form and is just one of them.

Measures of strategy are clearly needed to negotiate patches of cacti like those.

References

• McClelland, J.L. (2015), Explorations in Parallel Distributed Processing : A Handbook of Models, Programs, and Exercises, 2nd ed. (draft), Stanford Parallel Distributed Processing Lab.  Online, 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.

Resources

• Theme One Program • Overview
• Theme One Program • Exposition
• Theme One Program • User Guide
• Survey of Theme One Program

cc: FB | Theme One Program • Laws of Form • Ma^thstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

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 by the proposition   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.

References

• McClelland, J.L. (2015), Explorations in Parallel Distributed Processing : A Handbook of Models, Programs, and Exercises, 2nd ed. (draft), Stanford Parallel Distributed Processing Lab.  Online, 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.

Resources

• Theme One Program • Overview
• Theme One Program • Exposition
• Theme One Program • User Guide
• Survey of Theme One Program

cc: FB | Theme One Program • Laws of Form • Ma^thstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Transformation Rules and Equivalence Classes

The abstract character of the cactus language relative to its logical interpretations makes it possible to give abstract rules of equivalence for transforming cacti among themselves and partitioning the space of cacti into formal equivalence classes.  The transformation rules and equivalence classes are “purely formal” in the sense of being indifferent to the logical interpretation, entitative or existential, one happens to choose.

Two definitions are useful here:

• A reduction is a transformation which preserves equivalence classes and reduces the level of graphical complexity.
• A basic reduction is a reduction which applies to a basic connective, either a node connective or a lobe connective.

The two kinds of basic reductions are described as follows.

• A node reduction is permitted if and only if every component cactus joined to a node itself reduces to a node.

• A lobe reduction is permitted if and only if exactly one component cactus listed in a lobe reduces to an edge.

That is roughly the gist of the rules.  More formal definitions can wait for the day when we need to explain their use to a computer.

Resources

• Theme One Program • Overview
• Theme One Program • Exposition
• Theme One Program • User Guide
• Survey of Theme One Program

cc: FB | Theme One Program • Laws of Form • Ma^thstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Mathematical Structure and Logical Interpretation

The main things to take away from the previous post are the following two ideas, one syntactic and one semantic.

• Syntax.  The compositional structures of cactus graphs and cactus expressions are constructed from two kinds of connective operations.
• Semantics.  There are two ways of mapping the compositional structures of syntax into the compositional structures of propositional sentences.

The two kinds of connective operations are described as follows.

• The node connective joins a number of component cacti to a node, as shown below.

• The lobe connective joins a number of component cacti to a lobe, as shown below.

The two ways of mapping cactus structures to logical meanings are summarized in Table 3, which compares the entitative and existential interpretations of the basic cactus structures, in effect, the graphical constants and connectives.

Resources

• Theme One Program • Overview
• Theme One Program • Exposition
• Theme One Program • User Guide
• Survey of Theme One Program

cc: FB | Theme One Program • Laws of Form • Ma^thstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Logical Cacti

Up till now we’ve been working to hammer out a two‑edged sword of syntax, honing the syntax of cactus graphs and cactus expressions and turning it to use in taming the syntax of two‑level formal languages.

But the purpose of a logical syntax is to support a logical semantics, which means, for starters, to bear interpretation as sentential signs capable of denoting objective propositions about a universe of objects.

One of the difficulties we face is that the words interpretation, meaning, semantics, and their ilk take on so many different meanings from one moment to the next of their use.  A dedicated neologician might be able to think up distinctive names for all the aspects of meaning and all the approaches to them that concern us, but I will do the best I can with the common lot of ambiguous terms, leaving it to context and intelligent interpreters to sort it out as much as possible.

The formal language of cacti is formed at such a high level of abstraction that its graphs bear at least two distinct interpretations as logical propositions.  The two interpretations concerning us here are descended from the ones C.S. Peirce called the entitative and the existential interpretations of his systems of graphical logics.

Existential Interpretation

Table 1 illustrates the existential interpretation of cactus graphs and cactus expressions by providing English translations for a few of the most basic and commonly occurring forms.

Entitative Interpretation

Table 2 illustrates the entitative interpretation of cactus graphs and cactus expressions by providing English translations for a few of the most basic and commonly occurring forms.

Resources

• Theme One Program • Overview
• Theme One Program • Exposition
• Theme One Program • User Guide
• Survey of Theme One Program

cc: FB | Theme One Program • Laws of Form • Ma^thstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Quickly recapping the discussion so far, we started with a data structure called an idea‑form flag and adopted it as a building block for constructing a species of graph-theoretic data structures called painted and rooted cacti.  We showed how to code the abstract forms of cacti into character strings called cactus expressions and how to parse the character strings into pointer structures in computer memory.

At this point we had to choose between two expository strategies.

A full account of Theme One’s operation would describe its use of cactus graphs in three distinct ways, called lexical, literal, and logical applications.  The more logical order would approach the lexical and literal tasks first.  That is because the program’s formal language learner must first acquire the vocabulary its propositional calculator interprets as logical variables.  The sequential learner operates at two levels, taking in sequences of characters it treats as strings or words plus sequences of words it treats as strands or sentences.

Finding ourselves more strongly attracted to the logical substance, however, we leave the matter of grammar to another time and turn to Theme One’s use of cactus graphs in its reasoning module to represent logical propositions on the order of Peirce’s alpha graphs and Spencer Brown’s calculus of indications.

Resources

• Theme One Program • Overview
• Theme One Program • Exposition
• Theme One Program • User Guide
• Survey of Theme One Program

cc: FB | Theme One Program • Laws of Form • Ma^thstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Lexical, Literal, Logical

Theme One puts cactus graphs to work in three distinct but related ways, called lexical, literal, and logical applications.  The three modes of operation employ three distinct but overlapping subsets of the broader species of cacti.  Accordingly we find ourselves working with graphs, expressions, and files of lexical, literal, and logical types, depending on the task at hand.

The logical class of cacti is the broadest, encompassing the whole species described above, of which we have already seen a typical example in its several avatars as abstract graph, pointer data structure, and string of characters suitable for storage in a text file.

Being a logical cactus is not just a matter of syntactic form — it means being subject to meaningful interpretations as a sign of a logical proposition.  To enter the logical arena cactus expressions must express something, a proposition true or false of something.

Fully addressing the logical, interpretive, semantic aspect of cactus graphs normally requires a mind‑boggling mass of preliminary work on the details of their syntactic structure.  Practical, pragmatic, and especially computational considerations will eventually make that unavoidable.  For the sake of the present discussion, however, let’s put that on hold and fast forward to the logical substance.

Resources

• Theme One Program • Overview
• Theme One Program • Exposition
• Theme One Program • User Guide
• Survey of Theme One Program

cc: FB | Theme One Program • Laws of Form • Ma^thstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Parsing Logical Graphs

It is possible to write a program that parses cactus expressions into reasonable facsimiles of cactus graphs as pointer structures in computer memory, making edges correspond to addresses and nodes correspond to records.  I did just that in the early forerunners of the present program, but it turned out to be a more robust strategy in the long run, despite the need for additional nodes at the outset, to implement a more articulate but more indirect parsing algorithm, one in which the punctuation marks are not just tacitly converted to addresses in passing, but instead recorded as nodes in roughly the same way as the ordinary identifiers, or paints.

Figure 3 illustrates the type of parsing paradigm used by the program, showing the pointer graph structure obtained by parsing the cactus expression in Figure 2.  A traversal of this graph naturally reconstructs the cactus string that parses into it.

The pointer graph in Figure 3, namely, the parse graph of a cactus expression, is the sort of thing we’ll probably not be able to resist calling a cactus graph, for all the looseness of that manner of speaking, but we should keep in mind its level of abstraction lies a step further in the direction of a concrete implementation than the last thing we called by that name.  While we have them before our mind’s eyes, there are several other distinctive features of cactus parse graphs we ought to notice before moving on.

In terms of idea‑form structures, a cactus parse graph begins with a root idea pointing into a by‑cycle of forms, each of whose sign fields bears either a paint, in other words, a direct or indirect identifier reference, or an opening left parenthesis indicating a link to a subtended lobe of the cactus.

A lobe springs from a form whose sign field bears a left parenthesis.  That stem form has an on idea pointing into a by‑cycle of forms, exactly one of which has a sign field bearing a right parenthesis.  That last form has an on idea pointing back to the form bearing the initial left parenthesis.

In the case of a lobe, aside from the single form bearing the closing right parenthesis, the by‑cycle of a lobe may list any number of forms, each of whose sign fields bears either a comma, a paint, or an opening left parenthesis signifying a link to a more deeply subtended lobe.

Just to draw out one of the implications of this characterization and to stress the point of it, the root node can be painted and bear many lobes, but it cannot be segmented, that is, the by‑cycle corresponding to the root node can bear no commas.

Resources

• Theme One Program • Overview
• Theme One Program • Exposition
• Theme One Program • User Guide
• Survey of Theme One Program

cc: FB | Theme One Program • Laws of Form • Ma^thstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science