Inquiry Into Inquiry

Theme One Program • Exposition 5

Posted on June 23, 2022 by Jon Awbrey

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

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 | 6 Comments

Theme One Program • Exposition 4

Posted on June 20, 2022 by Jon Awbrey

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.

$\text{Figure 3. Parse Graph and Traverse String}$

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

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

Theme One Program • Exposition 3

Posted on June 17, 2022 by Jon Awbrey

Coding Logical Graphs

My earliest experiments coding logical graphs as dynamic “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 know as 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.

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 part of a larger polygon is called a 2‑gon or a bi‑gon. A terminal bi‑gon is called a spike.

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.

Resources

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

Theme One Program • Exposition 2

Posted on June 16, 2022 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.

Painted And Rooted Cacti

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

Theme One Program • Exposition 1

Posted on June 15, 2022 by Jon Awbrey

Theme One is a program for constructing 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 Idea↑Form Flag

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 as follows.

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
cc: FB | Theme One Program • Structural Modeling • Systems Science

Survey of Theme One Program • 4

Posted on June 12, 2022 by Jon Awbrey

This is a Survey of blog and wiki posts relating to the Theme One Program I worked on all through the 1980s. The aim was to develop fundamental algorithms and data structures for integrating empirical learning with logical reasoning. I had earlier developed separate programs for basic components of those tasks, namely, 2-level formal language learning and propositional constraint satisfaction, the latter using an extension of C.S. Peirce’s logical graphs as a syntax for propositional logic. Thus arose the question of how well it might be possible to get “empiricist” and “rationalist” modes of operation to cooperate. The long-term vision is the design and implementation of an Automated Research Tool able to double as a platform for Inquiry Driven Education.

Wiki Hub

Theme One Program • Overview

Documentation

Blog Series

Theme One Program

Exposition • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8)
Motivation • (1) • (2) • (3) • (4) • (5) • (6)
Discussion • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9)
Applications, Examples, Exercises

Jets and Sharks • (1) • (2) • (3)

Blog Dialogs

Theme One • A Program Of Inquiry • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10) • (11) • (12) • (13) • (14) • (15) • (16) • (17) • (18) • (19) • (20)

Applications

References

Awbrey, S.M., and Awbrey, J.L. (May 1991), “An Architecture for Inquiry • Building Computer Platforms for Discovery”, Proceedings of the Eighth International Conference on Technology and Education, Toronto, Canada, pp. 874–875. Online.
Awbrey, J.L., and Awbrey, S.M. (January 1991), “Exploring Research Data Interactively • Developing a Computer Architecture for Inquiry”, Poster presented at the Annual Sigma Xi Research Forum, University of Texas Medical Branch, Galveston, TX.
Awbrey, J.L., and Awbrey, S.M. (August 1990), “Exploring Research Data Interactively • Theme One : A Program of Inquiry”, Proceedings of the Sixth Annual Conference on Applications of Artificial Intelligence and CD-ROM in Education and Training, Society for Applied Learning Technology, Washington, DC, pp. 9–15. Online.

cc: Conceptual Graphs • Cybernetics (1) (2) • Laws of Form (1) (2) (3)
cc: Ontolog Forum (1) (2) (3) • Structural Modeling (1) (2) • Systems Science (1) (2) (3)
cc: Peirce List (Dec 2012) (Feb 2018) (Mar 2018) (Sep 2020) (Oct 2020) (Oct 2021)
cc: FB | Theme One Program

Functional Logic • Inquiry and Analogy • 21

Posted on June 9, 2022 by Jon Awbrey

Inquiry and Analogy • Generalized Umpire Operators

To get a better handle on the space of higher order propositions and continue developing our functional approach to quantification theory, we’ll need a number of specialized tools. To begin, we define a higher order operator $\Upsilon,$ called the umpire operator, which takes 1, 2, or 3 propositions as arguments and returns a single truth value as the result. Operators with optional numbers of arguments are called multigrade operators, typically defined as unions over function types. Expressing $\Upsilon$ in that form gives the following formula.

In contexts of application, that is, where a multigrade operator is actually being applied to arguments, the number of arguments in the argument list tells which of the optional types is “operative”. In the case of $\Upsilon,$ the first and last arguments appear as indices, the one in the middle serving as the main argument while the other two arguments serve to modify the sense of the operation in question. Thus, we have the following forms.

The operation $\Upsilon_p^r q$ evaluates the proposition $q$ on each model of the proposition $p$ and combines the results according to the method indicated by the connective parameter $r.$ In principle, the index $r$ may specify any logical connective on as many as $2^k$ arguments but in practice we usually have a much simpler form of combination in mind, typically either products or sums. By convention, each of the accessory indices $p, r$ is assigned a default value understood to be in force when the corresponding argument place is left blank, specifically, the constant proposition $1 : \mathbb{B}^k \to \mathbb{B}$ for the lower index $p$ and the continued conjunction or continued product operation $\textstyle\prod$ for the upper index $r.$ Taking the upper default value gives license to the following readings.

This means $\Upsilon_p (q) = 1$ if and only if $q$ holds for all models of $p.$ In propositional terms, this is tantamount to the assertion that $p \Rightarrow q,$ or that $\texttt{(} p \texttt{(} q \texttt{))} = 1.$

Throwing in the lower default value permits the following abbreviations.

This means $\Upsilon q = 1$ if and only if $q$ holds for the whole universe of discourse in question, that is, if and only $q$ is the constantly true proposition $1 : \mathbb{B}^k \to \mathbb{B}.$ The ambiguities of this usage are not a problem so long as we distinguish the context of definition from the context of application and restrict all shorthand notations to the latter.

Resources

Logic Syllabus
Boolean Function
Boolean-Valued Function
Logical Conjunction
Multigrade Operator
Parametric Operator
Minimal Negation Operator
Introduction to Inquiry Driven Systems
Functional Logic • Part 1 • Part 2 • Part 3
Cactus Language • Part 1 • Part 2 • Part 3 • References • Document History

cc: FB | Peirce Matters • Laws of Form • Mathstodon • Ontolog • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Posted in Abduction, Analogy, Argument, Aristotle, C.S. Peirce, Constraint, Deduction, Determination, Diagrammatic Reasoning, Diagrams, Differential Logic, Functional Logic, Hypothesis, Indication, Induction, Inference, Information, Inquiry, Logic, Logic of Science, Mathematics, Pragmatic Semiotic Information, Probable Reasoning, Propositional Calculus, Propositions, Reasoning, Retroduction, Semiotics, Sign Relations, Syllogism, Triadic Relations, Visualization | Tagged Abduction, Analogy, Argument, Aristotle, C.S. Peirce, Constraint, Deduction, Determination, Diagrammatic Reasoning, Diagrams, Differential Logic, Functional Logic, Hypothesis, Indication, Induction, Inference, Information, Inquiry, Logic, Logic of Science, Mathematics, Pragmatic Semiotic Information, Probable Reasoning, Propositional Calculus, Propositions, Reasoning, Retroduction, Semiotics, Sign Relations, Syllogism, Triadic Relations, Visualization | 2 Comments

Functional Logic • Inquiry and Analogy • 20

Posted on June 3, 2022 by Jon Awbrey

Inquiry and Analogy • Application of Higher Order Propositions to Quantification Theory

Table 21 provides a thumbnail sketch of the relationships discussed in this section.

$\text{Table 21. Relation of Quantifiers to Higher Order Propositions}$

Resources

Logic Syllabus
Boolean Function
Boolean-Valued Function
Logical Conjunction
Minimal Negation Operator
Introduction to Inquiry Driven Systems
Functional Logic • Part 1 • Part 2 • Part 3
Cactus Language • Part 1 • Part 2 • Part 3 • References • Document History

cc: FB | Peirce Matters • Laws of Form • Mathstodon • Ontolog • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Functional Logic • Inquiry and Analogy • 19

Posted on May 31, 2022 by Jon Awbrey

Inquiry and Analogy • Application of Higher Order Propositions to Quantification Theory

Reflection is turning a topic over in various aspects and in various lights so that nothing significant about it shall be overlooked — almost as one might turn a stone over to see what its hidden side is like or what is covered by it.

John Dewey • How We Think

Tables 19 and 20 present the same information as Table 18, sorting the rows in different orders to reveal other symmetries in the arrays.

$\text{Table 19. Simple Qualifiers of Propositions (Version 2)}$

$\text{Table 20. Simple Qualifiers of Propositions (Version 3)}$

Resources

Logic Syllabus
Boolean Function
Boolean-Valued Function
Logical Conjunction
Minimal Negation Operator
Introduction to Inquiry Driven Systems
Functional Logic • Part 1 • Part 2 • Part 3
Cactus Language • Part 1 • Part 2 • Part 3 • References • Document History

cc: FB | Peirce Matters • Laws of Form • Mathstodon • Ontolog • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Functional Logic • Inquiry and Analogy • 18

Posted on May 28, 2022 by Jon Awbrey

Inquiry and Analogy • Application of Higher Order Propositions to Quantification Theory

Last time we took up a fourfold scheme of quantified propositional forms traditionally known as a “Square of Opposition”, relating it to a quartet of higher order propositions which, depending on context, are also known as measures, qualifiers, or higher order indicator functions.

Table 18 develops the above ideas in further detail, expressing a larger set of quantified propositional forms by means of propositions about propositions.

$\text{Table 18. Simple Qualifiers of Propositions (Version 1)}$

Resources

Logic Syllabus
Boolean Function
Boolean-Valued Function
Logical Conjunction
Minimal Negation Operator
Introduction to Inquiry Driven Systems
Functional Logic • Part 1 • Part 2 • Part 3
Cactus Language • Part 1 • Part 2 • Part 3 • References • Document History

cc: FB | Peirce Matters • Laws of Form • Mathstodon • Ontolog • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

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Lexical, Literal, Logical

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Parsing Logical Graphs

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Coding Logical Graphs

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Painted And Rooted Cacti

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The Idea↑Form Flag

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Inquiry and Analogy • Generalized Umpire Operators

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