## Theme One Discussion • 1

Warfield gets it right about the relationship between object languages and metalanguages.  Something about the prefix meta- has contributed to a not uncommon misconception that metalanguages are formalized to a higher degree than the languages they objectify whereas in fact the opposite is true.

As it happens, the relation of informal contexts to formal contexts and what I’ve elsewhere called the formalization arrows between them are themes of major importance in my study of Inquiry Driven Systems.  Being short on time at the moment, I’ll just give a pointer into one of many relevant discussions and hope to elaborate further at the next opportunity.

## Theme One Exposition • 3

### Coding Logical Graphs

My earliest experiments with coding logical graphs as pointer data structures taught me that conceptual and computational efficiencies of a critical sort could be achieved by generalizing their abstract graphs from trees to the variety graph theorists call cacti.  The genesis of that generalization is a tale worth telling another time, but for now it’s best to jump right in and proceed by way of generic examples.

Figure 1 shows a typical example of a painted and rooted cactus.

```       o
a   |       d
o---o       o
\ /  b c   |
o----o----o b e
\       /
\     /
\   /
\ /
@ a c e

Figure 1.  Painted And Rooted Cactus
```

Figure 2 shows a way to visualize the correspondence between cactus graphs and cactus strings, demonstrated on the cactus from Figure 1.  By way of convenient terminology, the polygons of a cactus graph are called its lobes.  An edge not a part of a larger polygon is called a 2-gon or a bi-gon.  A terminal bi-gon is called a spike.

```       o
a  (|)        d
o---o         o
(\ /)  b c   (|)
o--,--o--,--o b e
\         /
\       /
(  \     /  )
\   /
\ /
@ a c e

( ( a , ( ) ) , b c , ( d ) b e ) a c e

Figure 2.  Cactus Graph and Cactus Expression
```

The correspondence between a cactus graph and a cactus string is obtained by an operation called traversing the graph in question.

• One traverses a cactus graph by beginning at the left hand side of the root node, reading off the list of paints one encounters at that point.  Since the order of elements at any node is not significant, one may start the cactus string with that list of paints or save them for the end.  We have done the latter in this case.
• One continues by climbing up the left hand side of the leftmost lobe, marking the ascent by means of a left parenthesis, traversing whatever cactus one happens to reach at the first node above the root, that done, proceeding from left to right along the top side of the lobe, marking each interlobal span by means of a comma, traversing each cactus in turn one meets along the way, on completing the last of them climbing down the right hand side of the lobe, marking the descent by means of a right parenthesis, and then traversing each cactus in turn, in left to right order, that is incident with the root node.

The string of letters, parentheses, and commas one obtains by this procedure is called the traversal string of the graph, in this case, a cactus string.

## Theme One Exposition • 2

The previous post described the elementary data structure used to represent nodes of graphs in the Theme One program.  This post describes the specific family of graphs employed by the program.

Figure 1 shows a typical example of a painted and rooted cactus.

```       o
a   |       d
o---o       o
\ /  b c   |
o----o----o b e
\       /
\     /
\   /
\ /
@ a c e

Figure 1.  Painted And Rooted Cactus
```

The graph itself is a mathematical object and does not inhabit the page or other medium before our eyes, and it must not be confused with any picture or other representation of it, anymore than we’d want someone to confuse us with a picture of ourselves, but it’s a fair enough picture, once we understand the conventions of representation involved.

Let $V(G)$ be the set of nodes in a graph $G$ and let $L$ be a set of identifiers.  We often find ourselves in situations where we have to consider many different ways of associating the nodes of $G$ with the identifiers in $L.$  Various manners of associating nodes with identifiers have been given conventional names by different schools of graph theorists.  I will give one way of describing a few of the most common patterns of association.

• A graph is painted if there is a relation between its node set and a set of identifiers, in which case the relation is called a painting and the identifiers are called paints.
• A graph is colored if there is a function from its node set to a set of identifiers, in which case the function is called a coloring and the identifiers are called colors.
• A graph is labeled if there is a one-to-one mapping between its node set and a set of identifiers, in which case the mapping is called a labeling and the identifiers are called labels.
• A graph is said to be rooted if it has a unique distinguished node, in which case the distinguished node is called the root of the graph.  The graph in Figure 1 has a root node marked by the “at” sign or amphora symbol “$\texttt{@}$”.

The graph in Figure 1 has eight nodes plus the five paints in the set $\{ a, b, c, d, e \}.$  The painting of nodes is indicated by drawing the paints of each node next to the node they paint.  Observe that some nodes may be painted with an empty set of paints.

The structure of a painted and rooted cactus can be encoded in the form of a character string called a painted and rooted cactus expression.  For the remainder of this discussion the terms cactus and cactus expression will be used to mean the painted and rooted varieties.  A cactus expression is formed on an alphabet consisting of the relevant set of identifiers, the paints, together with three punctuation marks:  the left parenthesis, the comma, and the right parenthesis.

## Theme One Exposition • 1

Re: Systems Science Discussions • (1)(2)
Re: Ontolog Forum Discussions • (1)(2)
Re: Laws Of Form Discussions • (1)(2)
Re: Structural Modeling Discussion • (1)

Theme One is a program for building and transforming a particular species of graph-theoretic data structures in memory, structures that have been designed to support a variety of fundamental learning and reasoning tasks.

The program was developed as a part of an exploration into the implementation and integration of different types of learning and reasoning procedures, concerned especially with the types of algorithms and data structures that might work in support of inquiry.  In its current state, Theme One integrates over a common data structure fundamental algorithms for one type of inductive learning and one type of deductive reasoning.

The first order of business is to describe the general class of graph-theoretic data structures that are used by the program, as they are determined in their local and their global aspects.

It will be the usual practice to shift around and to view these graphs at many different levels of detail, from their abstract definition to their concrete implementation, and many points in between.

The main work of the Theme One program is achieved by building and transforming a single species of graph-theoretic data structures.  In their abstract form these structures are most closely related to the graphs that are called cacti and conifers in graph theory, so I will generally refer to them under those names.

The graph-theoretic data structures used by the program are built up from a basic data structure called an idea-form flag.  This structure is defined as a pair of Pascal data types by means of the following specifications:

```   type    idea = ^form;
form = record
sign: char;
as, up, on, by: idea;
code: numb
end;
```
• An idea is a pointer to a form.
• A form is a record consisting of:
• A sign of type char;
• Four pointers, as, up, on, by, of type idea;
• A code of type numb, that is, an integer in [0, max integer].

Represented in terms of digraphs, or directed graphs, the combination of an idea pointer and a form record is most easily pictured as an arc, or directed edge, leading to a node that is labeled with the other data, in this case, a letter and a number.

At the roughest but quickest level of detail, an idea of a form can be drawn like this:

```      o a 1
^
|
@
```

When it is necessary to fill in more detail, the following schematic pattern can be used:

```    ^ ^     ^
as\|up on|
o-----o by
| a 1 |--->
o-----o
^
|
@
```

The idea-form type definition determines the local structure of the whole host of graphs used by the program, including a motley array of ephemeral buffers, temporary scratch lists, and other graph-theoretic data structures used for their transient utilities at specific points in the program.

I will put off discussing these more incidental graph structures until the points where they actually arise, focusing here on the particular varieties and the specific variants of cactoid graphs that constitute the main formal media of the program’s operation.

## Theme One Motivation • 6

Re: Sys Sci Group Discussion • (1)
Re: Ontolog Forum Discussion • (1)
Re: Laws Of Form Discussion • (1)

As a part of my Master’s work in psychology I applied my program to a few samples of data from my advisor’s funded research study on family interactions.  In one phase of the study observers viewed video-taped sessions of family members (parent and child) interacting in various modes (play or work) and coded qualitative features of each moment’s activity over a period of time.

The following page describes the application in more detail and reflects on its implications for the conduct of scientific inquiry in general.

In this application a “word” or “string” is a fixed-length sequence of qualitative features and a “sentence” or “strand” is a sequence of words that ends with what the observer judges to be a significant pause in activity.

In the qualitative research phases of the study one is simply attempting to discern any significant or recurring patterns in the data one possibly can.

In this case the observers are tokenizing the observations according to a codebook that has passed enough intercoder reliability studies to afford them all a measure of confidence it captures meaningful aspects of whatever reality is passing before their eyes and ears.

## Memorial Day Homage to Abraham Lincoln

Two centuries, two score and two years ago our fathers brought forth on this continent, a new nation, conceived in Liberty, and dedicated to the proposition that all people are created equal.

Now we are engaged in a great cultural war, testing whether that nation, or any
nation so conceived and so dedicated, can long endure.  We are met on a great battle-field of that war.  We have dedicated countless fields, here and abroad, as final resting places for those who gave their lives that that nation might live.  It is altogether fitting and proper that we should do this.

But in a larger sense, we can not dedicate—we can not consecrate—we can not hallow—these grounds.  The brave people, living and dead, who fought, have consecrated them, far above our poor power to add or detract.  It is for us the living, rather, to be dedicated to the unfinished work which they who fought have thus far so nobly advanced.  It is rather for us to be dedicated to the great task remaining before us—that from these honored dead we take increased devotion to that cause for which they gave the last full measure of devotion—that we highly resolve that these dead shall not have died in vain—that this nation, under God, shall have a new birth of freedom and understanding of what it means to be truly great—and that government of the people, by the people, for the people, shall not perish from the earth.

With thanks to Abraham Lincoln for his profound words,

Susan Awbrey
June 1, 2018

## Theme One Motivation • 5

Re: Laws Of Form DiscussionJB
Re: Ontolog Forum Discussion • (1)
Re: Sys Sci Group Discussion • (1)

As I’m working from 40-year-old memories of these first inklings I thought I might peruse the web for current information about Zipf’s Law.  I see there is now something called the Zipf–Mandelbrot (and sometimes –Pareto) Law and that was interesting because my wife Susan Awbrey made use of Mandelbrot’s ideas about self-similarity in her dissertation and communicated with him about it.  So more to read up on …

Just off-hand, though, I think my Learner is dealing with a different problem.  It has more to do with the savings in effort a learner gets by anticipating future experiences based on its record of past experiences than the savings it gets by minimizing bits of storage as far as mechanically possible.  There is still a type of compression involved but it’s more like Korzybski’s “time-binding” than space-savings proper.  (Speaking of old memories …)

The other difference I see is that Zipf’s Law applies to an established and preferably large corpus of linguistic material, while my Learner has to start from scratch, accumulating experience over time, making the best of whatever data it has at the outset and every moment thereafter.