Inquiry Into Inquiry

Re: Systems Science • Joseph Simpson

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 give just a pointer to one of many relevant discussions and hope to elaborate further at the next opportunity.

• Inquiry Driven Systems • Introduction
• Option of the Project • A Way Up To Inquiry
• Discussion of Formalization • General Topics
• A Formal Approach

cc: Ontolog Forum • Systems Science

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

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

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

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 be the set of nodes in a graph and let be a set of identifiers.  We often find ourselves in situations where we have to consider many different ways of associating the nodes of with the identifiers in   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 “”.

The graph in Figure 1 has eight nodes plus the five paints in the set   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

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

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

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

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

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