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

[Jon Awbrey]

Cactus Language • Mechanics 3
• https://inquiryintoinquiry.com/2025/09/25/cactus-language-mechanics-3/

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”.

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/lenkBx
cc: https://www.researchgate.net/post/Cactus_Language_Mechanics

[Jon Awbrey]

Cactus Language • Mechanics 4
• https://inquiryintoinquiry.com/2025/09/27/cactus-language-mechanics-4/

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 “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 “Lobe” be followed by a
list of arguments naming the appendants of the lobe in question.
Thus one writes expressions of the following forms.

Cactus Language Mechanics Display 1
• https://inquiryintoinquiry.com/wp-content/uploads/2025/09/cactus-language-mechanics-display-1-3.0.png

Working from a structural description of the cactus language,
or any suitable formal grammar for ‡C‡(‡P‡), it is possible to
give a recursive definition of a function called “Parse” which
maps each expression in PARCE(‡P‡) to the corresponding graph
in PARC(‡P‡). One way to do that proceeds as follows.

Cactus Language Mechanics Display 2
• https://inquiryintoinquiry.com/wp-content/uploads/2025/09/cactus-language-mechanics-display-2-3.0.png

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/5A4oXr
cc: https://www.researchgate.net/post/Cactus_Language_Mechanics

[Jon Awbrey]

Cactus Language • Mechanics 5
• https://inquiryintoinquiry.com/2025/10/04/cactus-language-mechanics-5/

Re: Cactus Language • Mechanics 4
• https://inquiryintoinquiry.com/2025/09/27/cactus-language-mechanics-4/

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

Algorithmic Translation Rules
• https://inquiryintoinquiry.com/wp-content/uploads/2025/10/cactus-language-algorithmic-translation-rules.png

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 m₁ = p₀. 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 ‡P‡ = ⌀. 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 ‡C‡⁰ = PARCE⁰.

• 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.

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/5Rp2Bn
cc: https://www.researchgate.net/post/Cactus_Language_Mechanics