Inquiry Into Inquiry

Cactus Language • Mechanics 5

Posted on October 4, 2025 by Jon Awbrey

The following Table summaries the mechanics of the parsing rules given in the previous post.

$\text{Algorithmic Translation Rules}$

A substructure of a painted and rooted cactus $C$ is defined recursively as follows. Starting from the root node of the cactus $C,$ each of its attachments is a substructure of $C.$ If a substructure is a blank or a paint then it constitutes a minimal substructure, meaning no further substructures of $C$ arise from it. If a substructure is a lobe then each of its appendants is also a substructure of $C$ and needs to be examined for further substructures.

The concept of substructure can be used to define the varieties of deletion and erasure operations which respect the structure of the abstract graph. In that application a blank symbol “ ” is treated as a primer, in other words, a clear paint or neutral tint, in effect letting $m_1 = p_0.$ In that frame of discussion it is useful to make the following distinction.

To delete a substructure is to replace it with an empty node, in effect, to reduce the whole structure to a trivial point.
To erase a substructure is to replace it with a blank symbol, in effect, to paint it out of the picture or overwrite it.

A bare PARC, loosely referred to as a bare cactus, is a painted and rooted cactus on the empty palette $\mathfrak{P} = \varnothing.$ A bare cactus can be described in various ways, depending on how the form arises in practice.

Leaning on the definition of a bare PARCE, a bare PARC can be described as the type of parse graph which arises from parsing a bare cactus expression, in other words, from parsing a sentence of the bare cactus language $\mathfrak{C}^0 = \mathrm{PARCE}^0.$

To express it more in its own terms, a bare PARC can be defined by tracing the recursive definition of a generic PARC, but then by detaching an independent form of description from the source of that analogy. The method is sufficiently sketched as follows.

A bare PARC is a PARC whose attachments are limited to blanks and bare lobes.
A bare lobe is a lobe whose appendants are limited to bare PARCs.

In practice a bare cactus is usually encountered in the process of analyzing or handling an arbitrary PARC, the circumstances of which frequently call for deleting or erasing its paints. Among other things, that generally makes it easier to observe the unadorned properties of its underlying graphical structure.

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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 • Mechanics 4

Posted on September 27, 2025 by Jon Awbrey

To develop a parser for cactus languages in a functional programming style takes a way to express the description of a PARC in terms of its nodes, by recursion from the root up. That requires each node to be specified by a functional expression, having a call of the generic function name $``\mathrm{Node}"$ be followed by a list of arguments naming the attachments of the node in question and having a call of the generic function name $``\mathrm{Lobe}"$ be followed by a list of arguments naming the appendants of the lobe in question. Thus one writes expressions of the following forms.

Working from a structural description of the cactus language, or any suitable formal grammar for $\mathfrak{C} (\mathfrak{P}),$ it is possible to give a recursive definition of a function called $``\mathrm{Parse}"$ which maps each expression in $\textsc{parce} (\mathfrak{P})$ to the corresponding graph in $\textsc{parc} (\mathfrak{P}).$ One way to do that proceeds as follows.

The Concatenation $\mathrm{Conc}_{j=1}^k$ of the $k$ sentences $(s_j)_{j=1}^k$ has a Parse defined as follows.

$\mathrm{Parse} (\mathrm{Conc}^0) ~=~ \mathrm{Node}^0.$
For $k > 0,$
$\mathrm{Parse} (\mathrm{Conc}_{j=1}^k s_j) ~=~ \mathrm{Node}_{j=1}^k \mathrm{Parse} (s_j).$

The Surcatenation $\mathrm{Surc}_{j=1}^k$ of the $k$ sentences $(s_j)_{j=1}^k$ has a Parse defined as follows.

$\mathrm{Parse} (\mathrm{Surc}^0) ~=~ \mathrm{Lobe}^0.$
For $k > 0,$
$\mathrm{Parse} (\mathrm{Surc}_{j=1}^k s_j) ~=~ \mathrm{Lobe}_{j=1}^k \mathrm{Parse} (s_j).$

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Cactus Language • Mechanics 3

Posted on September 25, 2025 by Jon Awbrey

Although the definition of a cactus graph lobe in terms of its intrinsic structural components is logically sufficient it is also useful to characterize the structure of a lobe in extrinsic relational terms, that is, to view the structure that typifies a lobe in relation to the structures of other PARCs and to mark the inclusion of the special type within the general run of PARCs.

That approach to the question of types results in a form of description that appears to be a bit more analytic, at least in mnemonic or prima facie terms, if not ultimately more revealing. Working in that vein, a lobe can be characterized as a special type of PARC called an unpainted root plant (UR‑plant).

An UR‑plant is a PARC of a simpler sort, at least, with respect to the recursive ordering of graph‑theoretic cacti being followed here. As a type, it is defined by the presence of two properties, that of being planted and that of having an unpainted root, defined as follows.

A PARC is planted if its list of attachments has just one PARC.
A PARC is UR if its list of attachments has no blanks or paints.

In short, an UR‑planted PARC has a single PARC as its only attachment, and since that attachment is prevented from being a blank or a paint, the single attachment at its root has to be another sort of structure, that which we call a lobe.

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Cactus Language • Mechanics 2

Posted on September 22, 2025 by Jon Awbrey

The structure of a painted cactus, insofar as it presents itself to the visual imagination, can be described as follows. The overall structure, as given by its underlying graph, falls within the species of graph commonly known as a rooted cactus, to which is added the idea that each of its nodes can be painted with a finite sequence of paints, chosen from a palette given by the parametric set $\{ ``\text{~}" \} \cup \mathfrak{P} = \{ m_1 \} \cup \{ p_1, \ldots, p_k \}.$

It is conceivable on purely graph‑theoretic grounds to have a class of cacti which are painted but not rooted, so it may occasionally be necessary, for the sake of precision, to more exactly pinpoint our target species of graphical structure as a painted and rooted cactus (PARC).

A painted cactus, as a rooted graph, has a distinguished node called its root. By starting from the root and working recursively, the rest of its structure can be described in the following fashion.

Each node of a PARC consists of a graphical point or vertex plus a finite sequence of attachments, described in relative terms as the attachments at or to that node. An empty sequence of attachments defines the empty node. Otherwise, each attachment is one of three kinds: a blank, a paint, or a type of PARC called a lobe.

Each lobe of a PARC consists of a directed graphical cycle plus a finite sequence of appendants, described in relative terms as the appendants of or on that lobe. Since every lobe comes already attached to a particular node, exactly one vertex of the corresponding cycle is the vertex at that node. The remaining vertices of the cycle have their definitions filled out according to the appendants of the lobe in question.

An empty sequence of appendants is structurally equivalent to a sequence containing a single empty node as its only appendant. Either way of looking at it defines a graph‑theoretic structure called a needle or a terminal edge. Otherwise, each appendant of a lobe is itself an arbitrary PARC.

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Cactus Language • Discussion 3

Posted on September 20, 2025 by Jon Awbrey

Re: Cactus Language • Stylistics 1
Re: Cybernetics • Shann Turnbull

ST:

How does your posting meet the test of being relevant to the Wiener definition of Cybernetic?

Cybernetics can explain how all living things are self‑regulating, self‑governing and to some extent self‑repairing. Models are not needed because they are illustrated in practice everywhere.

The web pages heading you referred to does not support cybernetics being hard science subject to empirical testing.

It states:

As a result, we can hardly conceive of how many possibilities there are for what we call objective reality. Our sharp quills of knowledge are so narrow and so concentrated in particular directions that with science there are myriads of totally different real worlds, each one accessible from the next simply by slight alterations — shifts of gaze — of every particular discipline and subspecialty.

May I suggest that you share your interest in semantics with only those dedicated to your topic? Refer to International Association of Literary Semantics.

Hopefully, the audience of this list, also interested in non‑testable science, will follow you to more efficient focus discussion on the cybcom list.

Thanks for your comments, Shann. It’s good to know one has a Reader.

It’s not usually necessary to give too weighty a justification for an epigraph like the one I used. They may be intended as nothing more than a bit of light relief from the daily te deums before turning back to the task at hand, a sidelong reflection on the broader scene, or even a counterpoint to the main theme in view. I can see how some of that may need to be developed as we go but I would not wish to get overly diverted by it. In the present time frame I have moved on to the topic of Cactus Language Mechanics, beginning with the following post.

Cactus Language • Mechanics 1

It appears I have run out of time for today. I’ll take up the rest of your comments next time.

Regards,
Jon

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Cactus Language • Mechanics 1

Posted on September 18, 2025 by Jon Awbrey

We are only now beginning to see how this works. Clearly one of the mechanisms for picking a reality is the sociohistorical sense of what is important — which research program, with all its particularity of knowledge, seems most fundamental, most productive, most penetrating. The very judgments which make us push narrowly forward simultaneously make us forget how little we know. And when we look back at history, where the lesson is plain to find, we often fail to imagine ourselves in a parallel situation. We ascribe the differences in world view to error, rather than to unexamined but consistent and internally justified choice.

Herbert J. Bernstein • “Idols of Modern Science”

The discussion to follow takes up the mechanics of parsing cactus language expressions into the corresponding computational data structures. Parsing translates each cactus expression into a computational form articulating its syntactic structure and preparing it for automated modes of processing and evaluation.

For present purposes it is necessary to describe the target data structures only at a fairly high level of abstraction, ignoring the details of address pointers and record structures and leaving the more operational aspects of implementation to the imagination of prospective programmers. In that way we may put off to another stage of elaboration and refinement the description of a program which creates those pointers and transforms those graph‑theoretic data structures.

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Charles Sanders Peirce, George Spencer Brown, and Me • 20

Posted on September 16, 2025 by Jon Awbrey

Re: Laws of Form • James Bowery

JB:: I’m interested in those who have approached the notion of self‑duality from the meta‑perspective of switching perspectives between Directed Cyclic Graphs of NiNAND and NiNOR gates (Ni for N‑Inputs à la boolean network theory). The burgeoning interest in what might be called “The Self‑Simulation Hypothesis” founded on the notion of self‑duality rather demands this meta‑perspective.

Going through my notes I see I blogged fairly extensively on the En‑Ex duality last year. There’s so much going on in the world right now I’m having trouble maintaining focus so I’ll just post a collection of links to the relevant title searches against the day when I can review those series and bring them to bear on the topics above.

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Posted in Abstraction, Amphecks, Analogy, Animata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Cybernetics, Deduction, Differential Analytic Turing Automata, Differential Logic, Duality, Form, Graph Theory, Inquiry, Inquiry Driven Systems, Laws of Form, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Model Theory, Proof Theory, Propositional Calculus, Semiotics, Sign Relational Manifolds, Sign Relations, Spencer Brown, Theorem Proving, Time, Topology | Tagged Abstraction, Amphecks, Analogy, Animata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Cybernetics, Deduction, Differential Analytic Turing Automata, Differential Logic, Duality, Form, Graph Theory, Inquiry, Inquiry Driven Systems, Laws of Form, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Model Theory, Proof Theory, Propositional Calculus, Semiotics, Sign Relational Manifolds, Sign Relations, Spencer Brown, Theorem Proving, Time, Topology | 1 Comment

Charles Sanders Peirce, George Spencer Brown, and Me • 19

Posted on September 10, 2025 by Jon Awbrey

Re: Laws of Form • James Bowery

JB:: I’m interested in those who have approached the notion of self‑duality from the meta‑perspective of switching perspectives between Directed Cyclic Graphs of NiNAND and NiNOR gates (Ni for N‑Inputs à la boolean network theory). The burgeoning interest in what might be called “The Self‑Simulation Hypothesis” founded on the notion of self‑duality rather demands this meta‑perspective.

As far as logical graphs and boolean functions go, the main form of duality occupying me for the last half century has been the duality between existential and entitative interpretations of logical graphs for propositional calculus. That duality points to a deeper mathematical unity underlying the diversity of logical interpretation. To my way of thinking, that unity of form is the most significant fact Peirce discovered about the relationship between mathematics and logic. Sheffer, Huntington, Spencer Brown, and all the most perceptive writers who followed Peirce, have been able to appreciate its fundamental status.

Various folks who are likely to be reading this now will recognize how various subsets of us have been through the whole array of tangent topics and issues at varying levels of attention and interest for a very long time, so I will take a moment to check my notes and see what seems most salient for current concerns.

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Charles Sanders Peirce, George Spencer Brown, and Me • 18

Posted on September 7, 2025 by Jon Awbrey

Re: Laws of Form • James Bowery

JB:: I’m interested in those who have approached the notion of self‑duality from the meta‑perspective of switching perspectives between Directed Cyclic Graphs of NiNAND and NiNOR gates (Ni for N‑Inputs à la boolean network theory). The burgeoning interest in what might be called “The Self‑Simulation Hypothesis” founded on the notion of self‑duality rather demands this meta‑perspective.

Hi James, it’s been a while … Picking up this thread again always leads me through a maze of reminiscence — I’m used to that — but it’s taking more time than usual to sort out what bears on the topics you raise, more from the richness of the embedding matrix than any lack of content … but I will keep at it … here’s a first bit …

Parametrized families of logical operators like the ones you mention are some of the first things I remember discussing with one of my former logic professors, Herb Hendry, who told me they are called “multigrade operators”. Herb taught in the philosophy department at Michigan State and became an early adopter of instructional technology for teaching logic, developing a software package by the name of CALL for Computer Assisted Logic Lessons. It was only natural that I would come to have many discussions with him about my own adventures in computing for logic.

Then as now I came at everything from a Peircean direction and I had early on learned about the operators Peirce described as the ampheck $\curlywedge$ and its dual $\bar\curlywedge$ — McCulloch would later refer to both as amphecks for reasons we’ll get to eventually — and others of my teachers called NNOR and NAND, respectively.

Making a long story as short as possible, the natural extensions of NAND and NNOR to finite numbers of variables are represented by logical graphs of the following forms.

Working under what amounts to Peirce’s existential interpretation, an expression of the form $\texttt{(} e_1 \ldots e_n \texttt{)}$ is a negation of a conjunction while an expression of the form $\texttt{(} e_1 \texttt{)} \ldots \texttt{(} e_n \texttt{)}$ is a negation of a disjunction, which is also a conjunction of many negations.

For concreteness of orientation, the corresponding venn diagrams for the case where $n = 3$ are shown below.

References

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Charles Sanders Peirce, George Spencer Brown, and Me • 17

Posted on August 22, 2025 by Jon Awbrey

Re: Laws of Form • James Bowery

JB:: I’m interested in those who have approached the notion of self‑duality from the meta‑perspective of switching perspectives between Directed Cyclic Graphs of NiNAND and NiNOR gates (Ni for N‑Inputs à la boolean network theory). The burgeoning interest in what might be called “The Self‑Simulation Hypothesis” founded on the notion of self‑duality rather demands this meta‑perspective.

Thanks, James, I’ve been needing a break about here — the series on Cactus Language is very important but it can be rough going at times — as far as the current thread goes it’s been a long time passing so allow me a bit of anamnesis to remember why I walked into the room.

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