Inquiry Into Inquiry

Theme One Program • Exposition 2

Posted on June 9, 2018 by Jon Awbrey

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.

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.

Resources

cc: Conceptual Graphs • Cybernetics • Laws of Form • Ontolog Forum
cc: FB | Theme One Program • Structural Modeling • Systems Science

Posted in Algorithms, Animata, Artificial Intelligence, Boolean Functions, C.S. Peirce, Cactus Graphs, Cognition, Computation, Constraint Satisfaction Problems, Data Structures, Differential Logic, Equational Inference, Formal Languages, Graph Theory, Inquiry Driven Systems, Laws of Form, Learning Theory, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Peirce, Propositional Calculus, Semiotics, Spencer Brown, Visualization | Tagged Algorithms, Animata, Artificial Intelligence, Boolean Functions, C.S. Peirce, Cactus Graphs, Cognition, Computation, Constraint Satisfaction Problems, Data Structures, Differential Logic, Equational Inference, Formal Languages, Graph Theory, Inquiry Driven Systems, Laws of Form, Learning Theory, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Peirce, Propositional Calculus, Semiotics, Spencer Brown, Visualization | 2 Comments

Theme One Program • Exposition 1

Posted on June 8, 2018 by Jon Awbrey

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

The program evolved over the course of an exploration into the integration of contrasting types of activities involved in learning and reasoning, especially the types of algorithms and data structures capable of supporting all sorts of inquiry processes, from everyday problem solving to scientific investigation. 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.

We begin by describing the class of graph-theoretic data structures used by the program, as determined by their local and global features. It will be the usual practice to shift around and 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 closely related to the graphs called cacti and conifers in graph theory, so we’ll 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. That structure is defined as a pair of Pascal data types by means of the following specifications.

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 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.

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

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 the more incidental graph structures until the points where they actually arise, focusing here on the particular varieties of cactoid graphs which constitute the main formal media of the program’s operation.

Resources

cc: Conceptual Graphs • Cybernetics • Laws of Form • Ontolog Forum
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Theme One Program • Motivation 6

Posted on June 2, 2018 by Jon Awbrey

Comments I made in reply to a correspondent’s questions about delimiters and tokenizing in the Learner module may be worth sharing here.

As a part of my M.A. work in psychology I applied my Theme One program to samples of data from my advisor’s funded research study on family dynamics. 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.

Exploratory Qualitative Analysis of Sequential Observation Data

In this application a phrase or string is a fixed-length sequence of qualitative features and a clause or strand is a sequence of such phrases ending 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.

Resource

Survey of Theme One Program

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Posted in Algorithms, Animata, Artificial Intelligence, Boolean Functions, C.S. Peirce, Cactus Graphs, Computation, Computational Complexity, Cybernetics, Data Structures, Differential Logic, Form, Formal Languages, Graph Theory, Inquiry, Inquiry Driven Systems, Intelligent Systems, Laws of Form, Learning, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Peirce, Pragmatics, Programming, Propositional Calculus, Propositional Equation Reasoning Systems, Reasoning, Semantics, Semiotics, Sign Relations, Spencer Brown, Syntax, Visualization | Tagged Algorithms, Animata, Artificial Intelligence, Boolean Functions, C.S. Peirce, Cactus Graphs, Computation, Computational Complexity, Cybernetics, Data Structures, Differential Logic, Form, Formal Languages, Graph Theory, Inquiry, Inquiry Driven Systems, Intelligent Systems, Laws of Form, Learning, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Peirce, Pragmatics, Programming, Propositional Calculus, Propositional Equation Reasoning Systems, Reasoning, Semantics, Semiotics, Sign Relations, Spencer Brown, Syntax, Visualization | 2 Comments

Memorial Day Homage to Abraham Lincoln

Posted on June 1, 2018 by Jon Awbrey

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

Posted in Abraham Lincoln, Gettysburg Address, Guest Post, Memorial Day, Susan Awbrey | Tagged Abraham Lincoln, Gettysburg Address, Guest Post, Memorial Day, Susan Awbrey | Leave a comment

Theme One Program • Motivation 5

Posted on May 30, 2018 by Jon Awbrey

Since I’m working from decades-old memories of 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 there’s 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.

Resource

Survey of Theme One Program

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Theme One • A Program Of Inquiry 16

Posted on May 28, 2018 by Jon Awbrey

It looks like my time will be at the mercy of contractors and realtors most of the summer but I’ll continue with the informal exposition as time permits. I’ve also returned to work on the formal documentation, organizing the commentary by individual functions and groups of related functions, still mostly low-level primitives at this point. The current version is here:

Theme One Program • Commentary 2005

cc: Ontolog Forum • Peirce List (1) (2) (3) (4) • Systems Science

Theme One Program • Motivation 4

Posted on May 20, 2018 by Jon Awbrey

From Zipf’s Law and the category of “things that vary inversely to frequency” I got my first brush with the idea that keeping track of usage frequencies is part and parcel of building efficient codes.

In its first application the environment the Learner had to learn was the usage behavior of its user, as given by finite sequences of characters from a finite alphabet which might as well be called words and as given by finite sequences of those words which might as well be called phrases or sentences. In other words, Job One for the Learner was the job of constructing a user model.

In that frame of mind we are not seeking anything so grand as a Universal Induction Algorithm but simply looking for any approach that gives us a leg up, complexity wise, in Interactive Real Time.

Resource

Survey of Theme One Program

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Theme One Program • Motivation 3

Posted on May 19, 2018 by Jon Awbrey

Sometime around 1970 John B. Eulenberg came from Stanford to direct Michigan State’s Artificial Language Lab, where I would come to spend many interesting hours hanging out all through the 70s and 80s. Along with its research program the lab did a lot of work on augmentative communication technology for limited mobility users and the observations I made there prompted the first inklings of my Learner program.

Early in that period I visited John’s course in mathematical linguistics, which featured Laws of Form among its readings, along with the more standard fare of Wall, Chomsky, Jackendoff, and the Unified Science volume by Charles Morris which credited Peirce with pioneering the pragmatic theory of signs. I learned about Zipf’s Law relating the lengths of codes to their usage frequencies and I named the earliest avatar of my Learner program XyPh, partly after Zipf and playing on the xylem and phloem of its tree data structures.

Resource

Survey of Theme One Program

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Theme One Program • Motivation 2

Posted on May 17, 2018 by Jon Awbrey

A side-effect of working on the Theme One program over the course of a decade was the measure of insight it gave me into the reasons why empiricists and rationalists have so much trouble understanding each other, even when those two styles of thinking inhabit the very same soul.

The way it came about was this. The code from which the program is currently assembled initially came from two distinct programs, ones I developed in alternate years, at first only during the summers.

In the Learner program I sought to implement a Humean empiricist style of learning algorithm for the adaptive uptake of coded sequences of occurrences in the environment, say, as codified in a formal language. I knew all the theorems from formal language theory telling how limited any such strategy must ultimately be in terms of its generative capacity, but I wanted to explore the boundaries of that capacity in concrete computational terms.

In the Modeler program I aimed to implement a variant of Peirce’s graphical syntax for propositional logic, making use of graph-theoretic extensions I had developed over the previous decade.

As I mentioned, work on those two projects proceeded in a parallel series of fits and starts through interwoven summers for a number of years, until one day it dawned on me how the Learner, one of whose aliases was Index, could be put to work helping with sundry substitution tasks the Modeler needed to carry out.

So I began integrating the functions of the Learner and the Modeler, at first still working on the two component modules in an alternating manner, but devoting a portion of effort to amalgamating their principal data structures, bringing them into convergence with each other, and unifying them over a common basis.

Another round of seasons and many changes of mind and programming style, I arrived at a unified graph-theoretic data structure, strung like a wire through the far‑flung pearls of my programmed wit. But the pearls I polished in alternate years maintained their shine along axes of polarization whose grains remained skew in regard to each other. To put it more plainly, the strategies I imagined were the smartest tricks to pull from the standpoint of optimizing the program’s performance on the Learning task I found the next year were the dumbest moves to pull from the standpoint of its performance on the Reasoning task. I gradually came to appreciate that trade-off as a discovery …

Resource

Survey of Theme One Program

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Theme One Program • Motivation 1

Posted on May 15, 2018 by Jon Awbrey

The main idea behind the Theme One program is the efficient use of graph-theoretic data structures for the tasks of “learning” and “reasoning”.

I am thinking of learning in the sense of learning about an environment, in essence, gaining information about the nature of an environment and being able to apply the information acquired to a specific purpose.

Under the heading of reasoning I am simply lumping together all the ordinary sorts of practical activities which would probably occur to most people under that name.

There is a natural relation between the tasks. Learning the character of an environment leads to the recognition of laws which govern the environment and making full use of that recognition requires the ability to reason logically about those laws in abstract terms.

Resource

Survey of Theme One Program

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cc: FB | Theme One Program • Laws of Form (1) (2)

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