Cactus Language • Mechanics

[Jon Awbrey]

Cactus Language • Mechanics 1
• https://inquiryintoinquiry.com/2025/09/18/cactus-language-mechanics-1/

❝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 the sentences
of a cactus language into the corresponding computational data structures.
Parsing provides each sentence of the language with a translation 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.

Resources —

Cactus Language • Mechanics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Cactus_Language_.E2.80.A2_Mechanics

Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2025/05/02/survey-of-animated-logical-graphs-8/

Survey of Theme One Program
• https://inquiryintoinquiry.com/2025/05/06/survey-of-theme-one-program-7/

Regards,

Jon

cc: https://www.academia.edu/community/lJx4ZA

[Jon Awbrey]

Cactus Language • Mechanics 2
• https://inquiryintoinquiry.com/2025/09/22/cactus-language-mechanics-2/

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
{“ ”} ∪ ‡P‡ = {m₁} ∪ {p₁, …, pₖ}.

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.

Resources —

Cactus Language • Mechanics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Cactus_Language_.E2.80.A2_Mechanics

Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2025/05/02/survey-of-animated-logical-graphs-8/

Survey of Theme One Program
• https://inquiryintoinquiry.com/2025/05/06/survey-of-theme-one-program-7/

Regards,

Jon

cc: https://www.academia.edu/community/LZ1jGd
cc: https://www.researchgate.net/post/Cactus_Language_Mechanics