Inquiry Into Inquiry

Cactus Language • Preliminaries 16

Posted on May 27, 2025 by Jon Awbrey

The following definitions round out the concepts we need to begin applying formal grammar theory to the efficient description of formal languages, in particular, the family of cactus languages.

It is convenient to refer to the full set of symbols in $\{ ``S" \} \cup \mathfrak{Q} \cup \mathfrak{A}$ as the augmented alphabet of the candidate formal grammar for $\mathfrak{L}$ and thus to refer to the strings in $( \{ ``S" \} \cup \mathfrak{Q} \cup \mathfrak{A} )^*$ as the augmented strings of the grammar for $\mathfrak{L},$ in effect, articulating the forms superimposed on a language by one of its conceivable grammars.

In certain settings it becomes desirable to separate the augmented strings containing the symbol $``S"$ from all other cases of augmented strings. In those situations the strings in the disjoint union $\{ ``S" \} \cup (\mathfrak{Q} \cup \mathfrak{A} )^*$ are known as the sentential forms of the associated grammar.

In forming a grammar for a language, statements of the form $W :> W',$ where $W$ and $W'$ are augmented strings or sentential forms of specified types which depend on the style of the grammar being sought, are variously known as characterizations, covering rules, productions, rewrite rules, subsumptions, transformations, or typing rules. Statements of that form are collected together in a set $\mathfrak{K}$ which serves to complete the definition of the formal grammar in question.

The relation $S :> T$ has the converse form $T <: S$ which may be read as $T$ is covered by $S$ and understood in the sense that $T$ is of the type $S.$ Depending on the context, $T <: S$ can be taken in one of the following two ways.

Treating $T$ as a string variable, it means the individual string $T$ is an instance of the type $S.$
Treating $T$ as a type name, it means every string of the type $T$ is an instance of the type $S.$

In light of the above conventions, an expression of the form $t <: T$ can be read in all the ways one typically reads an expression of the form $t : T.$

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cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Posted in Automata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Differential Logic, Equational Inference, Formal Grammars, Formal Languages, Graph Theory, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Propositional Calculus, Visualization | Tagged Automata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Differential Logic, Equational Inference, Formal Grammars, Formal Languages, Graph Theory, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Propositional Calculus, Visualization | 2 Comments

Cactus Language • Preliminaries 15

Posted on May 25, 2025 by Jon Awbrey

A notation of the form $S :> T$ was introduced last time to indicate a category of grammatical relationships whose sense is suggested by any of the following readings.

$\begin{array}{l} S ~\mathrm{covers}~ T \\[2pt] S ~\mathrm{governs}~ T \\[2pt] S ~\mathrm{rules}~ T \\[2pt] S ~\mathrm{subsumes}~ T \\[2pt] S ~\mathrm{types~over}~ T \end{array}$

For the moment it’s enough to call $S :> T$ a covering relation, reading it as $S ~\mathrm{covers}~ T.$

In what follows the letter $``S"$ indicates the type of a sentence in the contemplated language $\mathfrak{L}.$ The letter $``S"$ is the initial symbol or sentence symbol of the candidate formal grammar for $\mathfrak{L}.$

Generally speaking, any number of letters like $``T",$ signifying other types of strings, will be found necessary for a reasonable account or rational reconstruction of the sentences in $\mathfrak{L}.$ The additional letters are known as intermediate symbols and collected together in the set $\mathfrak{Q}.$

Combining the singleton set $\{ ``S" \}$ whose sole member is the initial symbol with the set $\mathfrak{Q}$ of intermediate symbols results in the set $\{ ``S" \} \cup \mathfrak{Q}$ of non‑terminal symbols. Even though $\mathfrak{Q}$ is strictly only the set of intermediate symbols, it is handy to use $q$ as a typical variable ranging over the full set of non‑terminal symbols, $q \in \{ ``S" \} \cup \mathfrak{Q}.$

To complete the package, the alphabet $\mathfrak{A}$ of the language $\mathfrak{L}$ may also be referred to as the set of terminal symbols.

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cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Cactus Language • Preliminaries 14

Posted on May 23, 2025 by Jon Awbrey

A notation of the form $S :> T$ is introduced to indicate a category of grammatical relationships whose sense is suggested by any of the following readings.

$\begin{array}{l} S ~\mathrm{covers}~ T \\[2pt] S ~\mathrm{governs}~ T \\[2pt] S ~\mathrm{rules}~ T \\[2pt] S ~\mathrm{subsumes}~ T \\[2pt] S ~\mathrm{types~over}~ T \end{array}$

The form $S :> T$ plays a number of roles in the description of formal languages by means of formal grammars, flexible enough to be read in the following variety of senses.

Individual: The named or quoted string $T$ is being typed as a sentence $S$ of the language $\mathfrak{L}.$
Extensional: Each member of the set $T$ also belongs to the set $S$ of sentences in the language $\mathfrak{L}.$
Intensional: The quality of being $T$ entails the quality $S$ of being a sentence in the language $\mathfrak{L}.$

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Cactus Language • Preliminaries 13

Posted on May 21, 2025 by Jon Awbrey

Consider what effects that might conceivably
have practical bearings you conceive the
objects of your conception to have. Then,
your conception of those effects is the
whole of your conception of the object.

Charles S. Peirce • Issues of Pragmaticism

We have before us what appears to be a maximally concise description of our subject matter.

The painted cactus language with paints in the set $\mathfrak{P} = \{ p_j : j \in J \}$ is the formal language $\mathfrak{L} = \mathfrak{C} (\mathfrak{P}) \subseteq \mathfrak{A}^* = (\mathfrak{M} \cup \mathfrak{P})^*$ defined as follows.

$\begin{array}{ll} \text{PC 1.} & \text{The blank symbol}~ m_1 ~\text{is a sentence.} \\ \text{PC 2.} & \text{The paint}~ p_j ~\text{is a sentence for each}~ j ~\text{in}~ J. \\ \text{PC 3.} & \mathrm{Conc}^0 ~\text{and}~ \mathrm{Surc}^0 ~\text{are sentences.} \\ \text{PC 4.} & \text{For each positive integer}~ n, \\ & \text{if}~ s_1, \ldots, s_n ~\text{are sentences} \\ & \text{then}~ \mathrm{Conc}_{k=1}^n s_k ~\text{is a sentence} \\ & \text{and}~ \mathrm{Surc}_{k=1}^n s_k ~\text{is a sentence.} \end{array}$

Here we encounter a problem. The very conciseness of that description presents an obstacle to understanding, glossing over infinities and divinities of detail which must be comprehended in effectively finite form, especially if we have in mind developing a fully computational parser.

A start in that direction, taking steps toward an effective description of cactus languages, a finitary conception of their membership conditions, and a bounded characterization of a typical sentence of that form, can be made by recasting the above description of cactus expressions into the pattern of what is called, more or less roughly, a formal grammar.

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Cactus Language • Preliminaries 12

Posted on May 17, 2025 by Jon Awbrey

We are engaged in teasing out the consequences of the following description of our subject.

Only one thing remains to cast that description of cactus language into a commonly acceptable form. As presently formulated, the principle PC 4 appears to be attempting to define an infinite number of new concepts all in a single step, at least, it appears to invoke the indefinitely long sequences of operators $\mathrm{Conc}^n$ and $\mathrm{Surc}^n$ for all $n > 0.$

As a general rule one prefers to work with effectively finite descriptions of conceptual objects. That means restricting each description to a finite number of schematic principles, each of which involves a finite number of schematic effects. In that way we hope to arrive at a finite number of schemata explicitly relating conditions to results.

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Cactus Language • Preliminaries 11

Posted on May 16, 2025 by Jon Awbrey

Given the idea of a Parce on $\mathfrak{P}$ as a member of the cactus language $\mathfrak{C}(\mathfrak{P})$ a number of frequently occurring sublanguages and commonly invoked operations on their expressions can now be given succinct definition.

A bare Parce, a bit loosely referred to as a bare cactus expression, is a Parce on the empty palette $\mathfrak{P} = \varnothing.$ A bare Parce is a sentence in the bare cactus language, variously notated as follows.

$\mathfrak{C}^0 = \mathfrak{C} (\varnothing) = \mathrm{Parce}^0 = \mathrm{Parce}(\varnothing)$

That set of strings, regarded as a formal language in its own right, is a sublanguage of every cactus language $\mathfrak{C}(\mathfrak{P}).$ A bare cactus expression is commonly encountered in practice when one has occasion to start with an arbitrary Parce and then finds reason to delete or erase all its paints.

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Cactus Language • Preliminaries 10

Posted on May 15, 2025 by Jon Awbrey

Last time we arrived at the following definition of our subject matter.

A sentence of $\mathfrak{C} (\mathfrak{P})$ is known as a painted and rooted cactus expression on the palette $\mathfrak{P},$ more briefly as a Parce on $\mathfrak{P},$ or simply as a cactus expression when the context is clear.

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Cactus Language • Discussion 2

Posted on May 13, 2025 by Jon Awbrey

Re: Cactus Language • Preliminaries 9
Re: Cactus Language • Discussion 1
Re: Alex Shkotin

AS:

Interesting language. It’s unusual to treat “ ” as a sentence. Usually it is just a separator for other lexemes.

Are the following correct?

One blank in brackets i.e. $``\texttt{(~)}"$ is a sentence.
Two blanks in brackets i.e. $``\texttt{(~~)}"$ is not a sentence.

Are the three strings below sentences?

$\texttt{(,,)}$
$\texttt{(~,~,~)}$
$\texttt{((),(),())}$

And at last. You use your own notation to define formal language. Is it correct that this language is context‑free?

Thanks for the questions, Alex,

As I mentioned in the first discussion post, the current presentation of Cactus Language is rather abstract and formal because that’s what we need to implement a fully computational parser for the family of languages we have in mind. That is all well and good but it does leave us hanging when it comes to motivation and remembering why we are bothering with such a mass of formal detail.

When I find myself getting lost in syntactic abstractions it’s a good idea to stop and remind myself what led me to explore the computational powers of cactus graphs in the first place. In my case it began with C.S. Peirce’s and Spencer Brown’s systems of logical graphs. Two developments of that work which take up the story much nearer to scratch can be found on the following pages.

Short answers to the above questions —

One blank in brackets i.e. $``\texttt{(~)}"$ is a sentence.
Two blanks in brackets i.e. $``\texttt{(~~)}"$ is a sentence.
(Blanks can be concatenated any number of times.)
All three of the other strings listed above are sentences.

Finally, I think cactus languages are context‑free as I think the last best grammars I constructed for them are context‑free, but that is one of those hazy memories I’ll need to check out on the current pass through the material.

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Cactus Language • Discussion 1

Posted on May 7, 2025 by Jon Awbrey

Re: Cactus Language • Preliminaries 9
Re: Cybernetics • Joe Bury

JB:: What does subcatenation and surcatenation mean? Their definitions are not found in a dictionary. I get the formulas you wrote but I don’t understand the meaning.

Thanks for the question, Joe,

The current presentation of Cactus Language is rather abstract and formal because that’s what we need for a fully computational parsing algorithm, and there’s quite a bit more to do on that score as we go, but I have written more intuitive introductions to the same material various times before — You might try one of the following for starters.

Keeping it short and simple as possible —

Under the Existential Interpretation

The syntactic connective of Concatenation is interpreted as the Logical Conjunction, which says all of its operands are true.
The syntactic connective of Surcatenation is interpreted as the Minimal Negation Operation, which says exactly one of its operands is false.

Under the Entitative Interpretation

The syntactic connective of Concatenation is interpreted as the Logical Disjunction, which says some of its operands are true.
The syntactic connective of Surcatenation is interpreted as the Dual of Minimal Negation, which says not just one of its operands is true.

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Survey of Theme One Program • 7

Posted on May 6, 2025 by Jon Awbrey

This is a Survey of resources 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, in particular, two‑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 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

Motivation • (1) • (2) • (3) • (4) • (5) • (6)
Exposition • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9)
Discussion • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10)
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.

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Posted in Algorithms, Animata, Artificial Intelligence, Automated Research Tools, Boolean Functions, C.S. Peirce, Cactus Graphs, 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, Propositional Calculus, Propositional Equation Reasoning Systems, Spencer Brown, Visualization | Tagged Algorithms, Animata, Artificial Intelligence, Automated Research Tools, Boolean Functions, C.S. Peirce, Cactus Graphs, 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, Propositional Calculus, Propositional Equation Reasoning Systems, Spencer Brown, Visualization | 80 Comments

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