Theme One • A Program Of Inquiry 8

Posted on March 10, 2018 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.

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.

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.

Resources

cc: Peirce List • (1) • (2)

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 , , , , , , , , , , , , , , , , , , , , , , , , , , | 9 Comments

Theme One • A Program Of Inquiry 7

Posted on March 3, 2018 by Jon Awbrey

Re: Peirce List • (1) • (2)

Discussion arose in the Laws Of Form Group about computational explorations of George Spencer Brown’s calculus of indications.

Readers of Peirce are generally aware Spencer Brown revived certain aspects of Peirce’s logical graphs, focusing on what Peirce called the Alpha level and its interpretation for Boolean Algebra and Propositional Calculus but adding hints of potential extension and generalization.  Spencer Brown used what amounts to Peirce’s entitative interpretation of the graphical forms in his exposition but he was clear about the abstract character of the forms themselves, as evidenced by their dual interpretations, dubbed entitative and existential by Peirce.

In computational contexts the question naturally arises how to code the abstract formal structures used by the calculi of CSP and GSB into the relatively concrete forms that a computer can process.

I began my response to that question as follows …

To be continued …

Resources

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 , , , , , , , , , , , , , , , , , , , , , , , , , , | 8 Comments

Theme One • A Program Of Inquiry 6

Posted on February 28, 2018 by Jon Awbrey

Programs are algorithms operating on data structures (Niklaus Wirth).  How do we turn abstract graphs like those used by Charles S. Peirce and G. Spencer Brown into concrete data structures algorithms can manipulate?  There are many ways to do this, but one very efficient way is through the use of “pointer data structures”.

The full documentation of my Theme One Program is still in progress, but here’s a link to a page of exposition, describing the family of graphs used in the program, how to code the graphs as strings of parentheses, commas, and letters, and how the program parses the strings into pointer structures that live in computer memory.

Here’s a link to a suitable point of entry for our present purpose:

Painted And Rooted Cacti And Conifers

Figure 1 depicts a typical example of a painted and rooted cactus (PARCA).

       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 very 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 depicted 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 (PARC) can be encoded in the form of a character string called a painted and rooted cactus expression (PARCE).  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

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 , , , , , , , , , , , , , , , , , , , , , , , , , , | 10 Comments

Theme One • A Program Of Inquiry 5

Posted on February 27, 2018 by Jon Awbrey

I started learning programming about the same time I first ran across C.S. Peirce’s Logical Graphs and Spencer Brown’s Laws of Form in the late 1960s and naturally tried each new language and each new set of skills I learned on writing processors and theorem provers for the propositional calculus levels of their graph-theoretic formalisms.  Using previous work I had done in Lisp, I spent the 1980s developing a series of Pascal programs that integrated aspects of sequential learning with aspects of propositional reasoning over an extension of the CSP–GSB systems.  I applied the program to a selection of observational data sets from one of my advisor’s research projects and got an M.A. in Quantitative Psych out of it.  People looking for contemporary applications of the general Peircean paradigm may find some of the directions in this work of interest.

The following blog post updates a list of links to what documentation and exposition I’ve put on the web so far.

Posted in Artificial Intelligence, C.S. Peirce, Cognition, Computation, Constraint Satisfaction Problems, Cybernetics, Formal Languages, Inquiry, Inquiry Driven Systems, Intelligent Systems, Learning Theory, Logic, Peirce, Semiotics | Tagged , , , , , , , , , , , , , | 9 Comments

Survey of Theme One Program • 2

Posted on February 25, 2018 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 to support an integrated learning and reasoning interface, looking toward the design of an Automated Research Tool able to double as a medium for Inquiry Driven Education.  I wrote up a pilot version of the program well enough to get a Master’s degree out of it but I’m still getting around to writing up the complete documentation.

Wiki Hub

Documentation in Progress

Applications

Blog Dialogs

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.
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 , , , , , , , , , , , , , , , , , , , , , , , , , , | 20 Comments

Inquiry Driven Systems • Comment 2

Posted on February 21, 2018 by Jon Awbrey

I just got reminded of an earlier blog post that more or less fits here.  It links to the bibliography I had in hand and mind when I went back to graduate school in systems engineering to synthesize all the unfinished projects I had been accumulating over the years and dedicated myself to the systems aspects of Peirce’s theory of inquiry in a more applied way.

I finally finished retyping the bibliography to my systems engineering proposal that had gotten lost in a move between computers, so here is a link to the OEIS Wiki copy.

This may be of interest to people working towards applications of Peirce’s theory of inquiry, especially the design of intelligent systems with a capacity for supporting scientific inquiry.

cc: Peirce List (1) (2)

Posted in Animata, Artificial Intelligence, C.S. Peirce, Cybernetics, Inquiry, Inquiry Driven Systems, Inquiry Into Inquiry, Intelligent Systems, Learning Systems, Logic, Logical Graphs, Peirce, Semiotics, Sign Relations, Visualization | Tagged , , , , , , , , , , , , , , | 8 Comments

Inquiry Driven Systems • Comment 1

Posted on February 18, 2018 by Jon Awbrey

Re: Peirce List • Daniel Everett • Edwina Taborsky

The role of acquired knowledge bases in inquiry, learning, and reasoning is discussed in the following article and sections.

cc: Peirce List (1) (2)

Posted in Animata, Artificial Intelligence, C.S. Peirce, Cybernetics, Inquiry, Inquiry Driven Systems, Inquiry Into Inquiry, Knowledge Bases, Learning, Logic, Peirce, Reasoning, Semiotics, Sign Relations, Systems Theory, Visualization | Tagged , , , , , , , , , , , , , , , | 8 Comments

Sign Relations • Comment 12

Posted on February 17, 2018 by Jon Awbrey

Re: Ontolog Discussion • HP

In the Peirce universe “the role that human institutions play in establishing grounding and associated frames of reference and standards” (Hans Polzer) is articulated by reference to “communities of inquiry” and “communities of interpretation”.  Invoking communities as extended agents of inquiry and interpretation equips us with a better handle on “contexts of interpretation” and the structures involved in this array of constructs are found to be of triadic sign relations all compact.

Over the years I have found the hardest thing to convey about sign relations has been what it’s like to think and work within an extended sign relational environment.  A “setting” like that consists of a large number of individual sign-relational triples called “elementary sign relations”, each having the form (o, s, i), where o is the object, s is the sign, and i is the interpretant sign of the triple.

This means that any given sign relation L is a subset of a cartesian product O \times S \times I, where O is the object domain, S is the sign domain, and I is the interpretant sign domain of the sign relation L in view.

Taking this point of view on sign relations makes a big difference in the conjoined theories of inquiry and interpretation that develop from this point on.

Posted in C.S. Peirce, Inquiry, Logic of Relatives, Peirce, Relation Theory, Semiotics, Sign Relations | Tagged , , , , , , | 16 Comments

Sign Relations • Comment 11

Posted on February 14, 2018 by Jon Awbrey

Re: Peirce List • Jon Alan Schmidt

When you ask a question about what something is, you are asking a question about its ontology.  But signhood is not a matter of ontology, it is a form of relation.

Re: Peirce List • Edwina Taborsky • Helmut Raulien • Neal Bruss

Here again is that budget of excerpts on Determination, mostly Peirce with a few others before and after his time, all of which I collected back when I was turning my hand to the cybernetic and intelligent systems engineering prospects of Peirce’s theories of information, inquiry, and signs.

Contemporary conceptions of determination and determinacy in mathematics, physics, computer science, and engineering are covered by the concept of constraint and generalize beyond absolute determinism to degrees and measures of determination, ranging from none at all to totality.

Posted in C.S. Peirce, Inquiry, Logic of Relatives, Peirce, Relation Theory, Semiotics, Sign Relations | Tagged , , , , , , | 15 Comments

Sign Relations • Comment 10

Posted on February 9, 2018 by Jon Awbrey

Re: Peirce List • John Sowa

Three-Headed Dogs and Triadic Sign Relations

Peirce’s “Sop to Cerberus” got tossed about quite a bit in our discussions across the Web this millennium.  Here’s a record of one occasion from the days when our discussions bridged over multiple perspectives, in this instance the Peirce List and its parallel Arisbe List, the French SemioCom, and the Standard Upper Ontology Working Group:

There is a critical passage where Peirce explains the relationship between his popular illustrations and his technical theory of signs.

It is clearly indispensable to start with an accurate and broad analysis of the nature of a Sign.  I define a Sign as anything which is so determined by something else, called its Object, and so determines an effect upon a person, which effect I call its Interpretant, that the latter is thereby mediately determined by the former.  My insertion of “upon a person” is a sop to Cerberus, because I despair of making my own broader conception understood.  (Peirce 1908, Selected Writings, p. 404).

I have long connected this passage with Peirce’s much earlier “metaphorical argument” where he changes the addressee of a word — that to which it stands for something — from a person, to that person’s memory, to “a particular remembrance or image in that memory”, to wit, “the one which is the mental equivalent of the word … in short, its interpretant.”

Here is a passage from Peirce that is decisive in clearing up the relationship between the interpreter and the interpretant …

I think we need to reflect upon the circumstance that every word implies some proposition or, what is the same thing, every word, concept, symbol has an equivalent term — or one which has become identified with it, — in short, has an interpretant.

Consider, what a word or symbol is;  it is a sort of representation.  Now a representation is something which stands for something.  I will not undertake to analyze, this evening, this conception of standing for something — but, it is sufficiently plain that it involves the standing to something for something.  A thing cannot stand for something without standing to something for that something.  Now, what is this that a word stands to?  Is it a person?

We usually say that the word homme stands to a Frenchman for man.  It would be a little more precise to say that it stands to the Frenchman’s mind — to his memory.  It is still more accurate to say that it addresses a particular remembrance or image in that memory.  And what image, what remembrance?  Plainly, the one which is the mental equivalent of the word homme — in short, its interpretant.  Whatever a word addresses then or stands to, is its interpretant or identified symbol.  …

The interpretant of a term, then, and that which it stands to are identical.  Hence, since it is of the very essence of a symbol that it should stand to something, every symbol — every word and every conception — must have an interpretant — or what is the same thing, must have information or implication.  (Peirce 1866, Chronological Edition 1, pp. 466–467).

As I read the long arc of Peirce’s work, the greater significance of the transformation he suggests at these points is not the shift from one type of interpreter to another, however compelling the consideration of life-forms in general as sign-processing agents may be, but the change of perspective that pulls our exclusive focus on representative agents of semiosis back to a properly relational point of view and the triadic sign relations that generate competent semiotic conduct.  But Peirce made this transformation early on in his work, and even more strikingly in its first trials.  Viewed in that light I think I share Peirce’s despair that its full impact has yet to be felt.

References

  • Peirce, C.S. (1866), “The Logic of Science, or, Induction and Hypothesis”, Lowell Lectures of 1866, pp. 357–504 in Writings of Charles S. Peirce : A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.
  • Peirce, C.S. (1908), “Letters to Lady Welby”, Chapter 24, pp. 380–432 in Charles S. Peirce : Selected Writings (Values in a Universe of Chance), Edited with Introduction and Notes by Philip P. Wiener, Dover Publications, New York, NY, 1966.

Resources

Posted in C.S. Peirce, Inquiry, Logic of Relatives, Peirce, Relation Theory, Semiotics, Sign Relations | Tagged , , , , , , | 16 Comments