Cactus Language • Preliminaries
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Jon Awbrey
Mar 31, 2025, 11:15:22 AMMar 31
to Conceptual Graphs, Cybernetic Communications, Structural Modeling, SysSciWG
Cactus Language • Preliminaries 1
• https://inquiryintoinquiry.com/2025/03/30/cactus-language-preliminaries-1/
❝Picture two different configurations of such an irregular shape,
superimposed on each other in space, like a double exposure photograph.
Of the two images, the only part which coincides is the body. The two
different sets of quills stick out into very different regions of space.
The objective reality we see from within the first position, seemingly
so full and spherical, actually agrees with the shifted reality only in
the body of common knowledge. In every direction in which we look at all
deeply, the realm of discovered scientific truth could be quite different.
Yet in each of those two different situations, we would have thought the
world complete, firmly known, and rather round in its penetration of the
space of possible knowledge.❞
— Herbert J. Bernstein • “Idols of Modern Science”
The task before us is to describe the syntax of a family of formal languages
intended for use as a sentential calculus, and thus interpreted for the purpose
of reasoning about propositions and their logical relations.
To carry out our discussion we need a way of referring to signs as if they
were objects like any others, in other words, as the sorts of things which
can be named, indicated, described, discussed, and renamed if necessary,
which can be placed, arranged, and rearranged within a suitable medium of
expression — or else manipulated in the mind — which can be articulated and
decomposed into their elementary signs, and which can be strung together in
sequences to form complex signs.
Signs having signs as their objects are known as “higher order signs”,
a topic which demands an adequate level of formalization, but in due time.
The present discussion needs a quicker way to get into the subject, even if
it settles for informal means which cannot be rendered absolutely precise.
Resources —
Cactus Language • Preliminaries
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Cactus_Language_.E2.80.A2_Preliminaries
Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2024/03/18/survey-of-animated-logical-graphs-7/
Survey of Theme One Program
• https://inquiryintoinquiry.com/2024/02/26/survey-of-theme-one-program-6/
Regards,
Jon
• https://inquiryintoinquiry.com/2025/03/30/cactus-language-preliminaries-1/
❝Picture two different configurations of such an irregular shape,
superimposed on each other in space, like a double exposure photograph.
Of the two images, the only part which coincides is the body. The two
different sets of quills stick out into very different regions of space.
The objective reality we see from within the first position, seemingly
so full and spherical, actually agrees with the shifted reality only in
the body of common knowledge. In every direction in which we look at all
deeply, the realm of discovered scientific truth could be quite different.
Yet in each of those two different situations, we would have thought the
world complete, firmly known, and rather round in its penetration of the
space of possible knowledge.❞
— Herbert J. Bernstein • “Idols of Modern Science”
The task before us is to describe the syntax of a family of formal languages
intended for use as a sentential calculus, and thus interpreted for the purpose
of reasoning about propositions and their logical relations.
To carry out our discussion we need a way of referring to signs as if they
were objects like any others, in other words, as the sorts of things which
can be named, indicated, described, discussed, and renamed if necessary,
which can be placed, arranged, and rearranged within a suitable medium of
expression — or else manipulated in the mind — which can be articulated and
decomposed into their elementary signs, and which can be strung together in
sequences to form complex signs.
Signs having signs as their objects are known as “higher order signs”,
a topic which demands an adequate level of formalization, but in due time.
The present discussion needs a quicker way to get into the subject, even if
it settles for informal means which cannot be rendered absolutely precise.
Resources —
Cactus Language • Preliminaries
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Cactus_Language_.E2.80.A2_Preliminaries
Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2024/03/18/survey-of-animated-logical-graphs-7/
Survey of Theme One Program
• https://inquiryintoinquiry.com/2024/02/26/survey-of-theme-one-program-6/
Regards,
Jon
Jon Awbrey
Apr 7, 2025, 9:24:32 AMApr 7
to Conceptual Graphs, Cybernetic Communications, Structural Modeling, SysSciWG
Cactus Language • Preliminaries 2
• https://inquiryintoinquiry.com/2025/04/05/cactus-language-preliminaries-2/
As a temporary notation, let the relationship between
a particular sign s and a particular object o, namely,
the fact that s denotes o or the fact that o is denoted
by s, be symbolized in one of the following two ways.
1. s → o
2. o ← s
Now consider the following paradigm.
Cactus Language Display 1
• https://inquiryintoinquiry.files.wordpress.com/2025/04/cactus-language-display-1-blog.png
1. If “A” → Ann
that is, “A” denotes Ann
then A = Ann and Ann = A
Thus “Ann” → A
that is, “Ann” denotes A
2. If Bob ← “B”
that is, Bob is denoted by “B”
then Bob = B and B = Bob
Thus B ← “Bob”
that is, B is denoted by “Bob”
In the same vein, if we let the sign “blank” denote the sign “ ”
then the string of characters inside the first pair of quotation
marks will serve as another name for the string of characters
inside the second pair of quotation marks.
In other words, “blank” is a higher order sign whose object
is the sign “ ” and the string of five characters inside the
first pair of quotation marks is a sign at a higher level of
signification than the string of one character inside the
second pair of quotation marks. The relation in question
can be abbreviated in either one of the following two ways.
“blank” → “ ”
“ ” ← “blank”
Using the raised dot “∙” as a sign to mark the articulation of
a quoted string into a sequence of possibly shorter quoted strings,
and thus to mark the concatenation of a sequence of quoted strings
into a possibly larger quoted string, one can write the following
equation.
“ ” ← “blank” = “b” ∙ “l” ∙ “a” ∙ “n” ∙ “k”
The above tactic lets us refer to the blank as a type of character and
refer to any blank we choose as a token of that type, denoting either in
a markèd way, but without the use of quotation marks. As a blank is just
what the name “blank” names, it is possible to represent the denoting of
the sign “ ” by the name “blank” in the form of an identity between the
named objects, as follows.
“ ” = blank
Given the above identities it is possible to extend the use of
the “∙” sign to mark the articulation of either named or quoted
strings into both named and quoted strings. For example, we have
the following equations.
“ ” = “ ” ∙ “ ” = blank ∙ blank
“ blank” = “ ” ∙ “blank” = blank ∙ “blank”
“blank ” = “blank” ∙ “ ” = “blank” ∙ blank
• https://inquiryintoinquiry.com/2025/04/05/cactus-language-preliminaries-2/
As a temporary notation, let the relationship between
a particular sign s and a particular object o, namely,
the fact that s denotes o or the fact that o is denoted
by s, be symbolized in one of the following two ways.
1. s → o
2. o ← s
Now consider the following paradigm.
Cactus Language Display 1
• https://inquiryintoinquiry.files.wordpress.com/2025/04/cactus-language-display-1-blog.png
1. If “A” → Ann
that is, “A” denotes Ann
then A = Ann and Ann = A
Thus “Ann” → A
that is, “Ann” denotes A
2. If Bob ← “B”
that is, Bob is denoted by “B”
then Bob = B and B = Bob
Thus B ← “Bob”
that is, B is denoted by “Bob”
In the same vein, if we let the sign “blank” denote the sign “ ”
then the string of characters inside the first pair of quotation
marks will serve as another name for the string of characters
inside the second pair of quotation marks.
In other words, “blank” is a higher order sign whose object
is the sign “ ” and the string of five characters inside the
first pair of quotation marks is a sign at a higher level of
signification than the string of one character inside the
second pair of quotation marks. The relation in question
can be abbreviated in either one of the following two ways.
“blank” → “ ”
“ ” ← “blank”
Using the raised dot “∙” as a sign to mark the articulation of
a quoted string into a sequence of possibly shorter quoted strings,
and thus to mark the concatenation of a sequence of quoted strings
into a possibly larger quoted string, one can write the following
equation.
“ ” ← “blank” = “b” ∙ “l” ∙ “a” ∙ “n” ∙ “k”
The above tactic lets us refer to the blank as a type of character and
refer to any blank we choose as a token of that type, denoting either in
a markèd way, but without the use of quotation marks. As a blank is just
what the name “blank” names, it is possible to represent the denoting of
the sign “ ” by the name “blank” in the form of an identity between the
named objects, as follows.
“ ” = blank
Given the above identities it is possible to extend the use of
the “∙” sign to mark the articulation of either named or quoted
strings into both named and quoted strings. For example, we have
the following equations.
“ ” = “ ” ∙ “ ” = blank ∙ blank
“ blank” = “ ” ∙ “blank” = blank ∙ “blank”
“blank ” = “blank” ∙ “ ” = “blank” ∙ blank
Jon Awbrey
Apr 10, 2025, 9:24:28 AMApr 10
to Conceptual Graphs, Cybernetic Communications, Structural Modeling, SysSciWG
Cactus Language • Preliminaries 3
• https://inquiryintoinquiry.com/2025/04/09/cactus-language-preliminaries-3/
A few definitions from formal language theory are required at this point.
Note. Please see the blog post linked above for the
proper mathematical formats. The transcript below
uses ‡A‡ and ‡a‡ for Fraktur A and a, respectively.
• An “alphabet” is a finite set of signs, typically, ‡A‡ = {‡a‡₁, …, ‡a‡ₙ}.
• A “string” over an alphabet ‡A‡ is a finite sequence of signs from ‡A‡.
• The “length” of a string is just its length as a sequence of signs.
• The “empty string” is the unique sequence of length 0. It is sometimes
denoted by an empty pair of quotation marks (“”) but more often by the
Greek symbols epsilon or lambda.
• A sequence of length k > 0 is typically presented
in the following concatenated forms.
s₁ s₂ … sₖ₋₁ sₖ
or
s₁ ∙ s₂ ∙ … ∙ sₖ₋₁ ∙ sₖ
with sₘ ∈ ‡A‡ for all m = 1 … k.
The following notations provide useful alternatives.
• ε = “” = the empty string.
• η = {ε} = the language consisting of a single empty string.
Several operations on strings find sufficient application
to motivate the following definitions.
• To “erase” an appearance of a sign is to replace it with
an appearance of the blank symbol “ ”.
• To “delete” an appearance of a sign is to replace it with
an appearance of the empty string “”.
• If s is a string which ends with a sign t then s ∙ t⁻¹ is
the string which results by deleting the terminal t from s.
• A “token” is a particular appearance of a sign.
Finally —
• The “kleene star” ‡A‡* of alphabet ‡A‡ is the set of all strings over ‡A‡.
In particular, ‡A‡* includes among its elements the empty string ε.
• The “kleene plus” ‡A‡⁺ of an alphabet ‡A‡ is the set of all positive length
strings over ‡A‡, in other words, everything in ‡A‡* but the empty string.
• A “formal language” ‡L‡ over an alphabet ‡A‡ is a subset of ‡A‡*. In brief,
‡L‡ ⊆ ‡A‡*. If s is a string over ‡A‡ and s is an element of ‡L‡ then it
is customary to call s a “sentence” of ‡L‡. Thus, a formal language ‡L‡
is defined by specifying its elements, which amounts to saying what it
means to be a sentence of ‡L‡.
• https://inquiryintoinquiry.com/2025/04/09/cactus-language-preliminaries-3/
A few definitions from formal language theory are required at this point.
Note. Please see the blog post linked above for the
proper mathematical formats. The transcript below
uses ‡A‡ and ‡a‡ for Fraktur A and a, respectively.
• An “alphabet” is a finite set of signs, typically, ‡A‡ = {‡a‡₁, …, ‡a‡ₙ}.
• A “string” over an alphabet ‡A‡ is a finite sequence of signs from ‡A‡.
• The “length” of a string is just its length as a sequence of signs.
• The “empty string” is the unique sequence of length 0. It is sometimes
denoted by an empty pair of quotation marks (“”) but more often by the
Greek symbols epsilon or lambda.
• A sequence of length k > 0 is typically presented
in the following concatenated forms.
s₁ s₂ … sₖ₋₁ sₖ
or
s₁ ∙ s₂ ∙ … ∙ sₖ₋₁ ∙ sₖ
with sₘ ∈ ‡A‡ for all m = 1 … k.
The following notations provide useful alternatives.
• ε = “” = the empty string.
• η = {ε} = the language consisting of a single empty string.
Several operations on strings find sufficient application
to motivate the following definitions.
• To “erase” an appearance of a sign is to replace it with
an appearance of the blank symbol “ ”.
• To “delete” an appearance of a sign is to replace it with
an appearance of the empty string “”.
• If s is a string which ends with a sign t then s ∙ t⁻¹ is
the string which results by deleting the terminal t from s.
• A “token” is a particular appearance of a sign.
Finally —
• The “kleene star” ‡A‡* of alphabet ‡A‡ is the set of all strings over ‡A‡.
In particular, ‡A‡* includes among its elements the empty string ε.
• The “kleene plus” ‡A‡⁺ of an alphabet ‡A‡ is the set of all positive length
strings over ‡A‡, in other words, everything in ‡A‡* but the empty string.
• A “formal language” ‡L‡ over an alphabet ‡A‡ is a subset of ‡A‡*. In brief,
‡L‡ ⊆ ‡A‡*. If s is a string over ‡A‡ and s is an element of ‡L‡ then it
is customary to call s a “sentence” of ‡L‡. Thus, a formal language ‡L‡
is defined by specifying its elements, which amounts to saying what it
means to be a sentence of ‡L‡.
Jon Awbrey
Apr 12, 2025, 1:48:40 PMApr 12
to Conceptual Graphs, Cybernetic Communications, Structural Modeling, SysSciWG
Cactus Language • Preliminaries 4
• https://inquiryintoinquiry.com/2025/04/11/cactus-language-preliminaries-4/
The informal mechanisms illustrated in the preceding discussion equip us
with a description of “cactus language” adequate to providing conceptual
and computational representations for the minimal formal logical system
variously known as “propositional logic” or “sentential calculus”.
The “painted cactus language” ‡C‡ is actually a parameterized family of
languages, consisting of one language ‡C‡(‡P‡) for each set ‡P‡ of “paints”.
The alphabet ‡A‡ = ‡M‡ ∪ ‡P‡ is the disjoint union of the following
two sets of symbols.
‡M‡ is the alphabet of “markers”, the set of “punctuation marks”, or the
collection of “syntactic constants” common to all the languages ‡C‡(‡P‡).
Various ways of representing the elements of ‡M‡ are shown in the following
display.
Cactus Language Display 2
• https://inquiryintoinquiry.files.wordpress.com/2025/04/cactus-language-display-2-blog.png
• ‡M‡ = { ‡m‡₁ , ‡m‡₂ , ‡m‡₃ , ‡m‡₄ }
= { “ ” , “(” , “,” , “)” }
= { blank , links , comma , right }
‡P‡ is the “palette”, the alphabet of “paints”, or the collection of
“syntactic variables” peculiar to the language ‡C‡(‡P‡). The set of
signs in ‡P‡ may be enumerated as follows.
• ‡P‡ = {‡p‡ₖ : k ∈ K}.
• https://inquiryintoinquiry.com/2025/04/11/cactus-language-preliminaries-4/
The informal mechanisms illustrated in the preceding discussion equip us
with a description of “cactus language” adequate to providing conceptual
and computational representations for the minimal formal logical system
variously known as “propositional logic” or “sentential calculus”.
The “painted cactus language” ‡C‡ is actually a parameterized family of
languages, consisting of one language ‡C‡(‡P‡) for each set ‡P‡ of “paints”.
The alphabet ‡A‡ = ‡M‡ ∪ ‡P‡ is the disjoint union of the following
two sets of symbols.
‡M‡ is the alphabet of “markers”, the set of “punctuation marks”, or the
collection of “syntactic constants” common to all the languages ‡C‡(‡P‡).
Various ways of representing the elements of ‡M‡ are shown in the following
display.
Cactus Language Display 2
• https://inquiryintoinquiry.files.wordpress.com/2025/04/cactus-language-display-2-blog.png
• ‡M‡ = { ‡m‡₁ , ‡m‡₂ , ‡m‡₃ , ‡m‡₄ }
= { “ ” , “(” , “,” , “)” }
= { blank , links , comma , right }
‡P‡ is the “palette”, the alphabet of “paints”, or the collection of
“syntactic variables” peculiar to the language ‡C‡(‡P‡). The set of
signs in ‡P‡ may be enumerated as follows.
• ‡P‡ = {‡p‡ₖ : k ∈ K}.
Jon Awbrey
Apr 14, 2025, 1:08:10 PMApr 14
to Conceptual Graphs, Cybernetic Communications, Structural Modeling, SysSciWG
Cactus Language • Preliminaries 5
• https://inquiryintoinquiry.com/2025/04/13/cactus-language-preliminaries-5/
The easiest way to define the language ‡C‡(‡P‡) is to indicate the
general run of operations required to construct the greater share of
its sentences from the designated few which require a special election.
To do that we introduce a family of operations called “syntactic connectives”
on the strings of ‡A‡*. If the strings on which they operate are already
sentences of ‡C‡(‡P‡) then the operations amount to “sentential connectives”.
If the syntactic sentences, viewed as abstract strings of uninterpreted signs,
are provided with a semantics where they denote propositions, in other words,
indicator functions on a universe of discourse, then the operations amount
to “propositional connectives”.
Rather than presenting the most concise description of cactus languages
right from the beginning, it aids comprehension to develop a picture of
their forms in gradual stages, starting with the most natural ways of
viewing their elements, if somewhat at a distance, and working through
the most easily grasped impressions of their structures, if not always
the sharpest acquaintances with their details.
We begin by defining two sets of basic operations on strings of ‡A‡*.
Concatenation —
• The “concatenation” of one string s₁ is the string s₁
• The “concatenation” of two strings s₁, s₂ is the string s₁ ∙ s₂
• The “concatenation” of n strings (sₖ)ₖ₌₁…ₙ is the string s₁ ∙ … ∙ sₙ
Surcatenation —
• The “surcatenation” of one string s₁ is the string
“(” ∙ s₁ ∙ “)”
• The “surcatenation” of two strings s₁, s₂ is the string
“(” ∙ s₁ ∙ “,” ∙ s₂ ∙ “)”
• The “surcatenation” of n strings (sₖ)ₖ₌₁…ₙ is the string
“(” ∙ s₁ ∙ “,” ∙ … ∙ “,” ∙ sₙ ∙ “)”
Resources —
Cactus Language • Preliminaries
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Cactus_Language_.E2.80.A2_Preliminaries
Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2024/03/18/survey-of-animated-logical-graphs-7/
Survey of Theme One Program
• https://inquiryintoinquiry.com/2024/02/26/survey-of-theme-one-program-6/
Regards,
Jon
cc: https://www.academia.edu/community/lavx71
cc: https://www.researchgate.net/post/Cactus_Language_Preliminaries
• https://inquiryintoinquiry.com/2025/04/13/cactus-language-preliminaries-5/
The easiest way to define the language ‡C‡(‡P‡) is to indicate the
general run of operations required to construct the greater share of
its sentences from the designated few which require a special election.
To do that we introduce a family of operations called “syntactic connectives”
on the strings of ‡A‡*. If the strings on which they operate are already
sentences of ‡C‡(‡P‡) then the operations amount to “sentential connectives”.
If the syntactic sentences, viewed as abstract strings of uninterpreted signs,
are provided with a semantics where they denote propositions, in other words,
indicator functions on a universe of discourse, then the operations amount
to “propositional connectives”.
Rather than presenting the most concise description of cactus languages
right from the beginning, it aids comprehension to develop a picture of
their forms in gradual stages, starting with the most natural ways of
viewing their elements, if somewhat at a distance, and working through
the most easily grasped impressions of their structures, if not always
the sharpest acquaintances with their details.
We begin by defining two sets of basic operations on strings of ‡A‡*.
Concatenation —
• The “concatenation” of one string s₁ is the string s₁
• The “concatenation” of two strings s₁, s₂ is the string s₁ ∙ s₂
• The “concatenation” of n strings (sₖ)ₖ₌₁…ₙ is the string s₁ ∙ … ∙ sₙ
Surcatenation —
• The “surcatenation” of one string s₁ is the string
“(” ∙ s₁ ∙ “)”
• The “surcatenation” of two strings s₁, s₂ is the string
“(” ∙ s₁ ∙ “,” ∙ s₂ ∙ “)”
• The “surcatenation” of n strings (sₖ)ₖ₌₁…ₙ is the string
“(” ∙ s₁ ∙ “,” ∙ … ∙ “,” ∙ sₙ ∙ “)”
Resources —
Cactus Language • Preliminaries
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Cactus_Language_.E2.80.A2_Preliminaries
Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2024/03/18/survey-of-animated-logical-graphs-7/
Survey of Theme One Program
• https://inquiryintoinquiry.com/2024/02/26/survey-of-theme-one-program-6/
Regards,
Jon
cc: https://www.researchgate.net/post/Cactus_Language_Preliminaries
Jon Awbrey
Apr 17, 2025, 12:12:30 PMApr 17
to Conceptual Graphs, Cybernetic Communications, Structural Modeling, SysSciWG
Cactus Language • Preliminaries 6
• https://inquiryintoinquiry.com/2025/04/16/cactus-language-preliminaries-6/
The definitions of the syntactic connectives can be made
a little more succinct by defining the following pair of
generic operators on strings.
Concatenation —
The “concatenation” Concₖ₌₁…ₙ of the sequence of
n strings (sₖ)ₖ₌₁…ₙ is defined recursively as follows.
For n = 1, Concₖ₌₁…ₙ sₖ = s₁
For n > 1, Concₖ₌₁…ₙ sₖ = Concₖ₌₁…ₙ₋₁ sₖ ∙ sₙ
Surcatenation —
The “surcatenation” Surcₖ₌₁…ₙ of the sequence of
n strings (sₖ)ₖ₌₁…ₙ is defined recursively as follows.
For n = 1, Surcₖ₌₁…ₙ sₖ = “(” ∙ s₁ ∙ “)”
For n > 1, Surcₖ₌₁…ₙ sₖ = Surcₖ₌₁…ₙ₋₁ sₖ ∙ (“)”)⁻¹ ∙ “,” ∙ sₙ ∙ “)”
Resources —
Cactus Language • Preliminaries
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Cactus_Language_.E2.80.A2_Preliminaries
Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2024/03/18/survey-of-animated-logical-graphs-7/
Survey of Theme One Program
• https://inquiryintoinquiry.com/2024/02/26/survey-of-theme-one-program-6/
Regards,
Jon
cc: https://www.academia.edu/community/VjWW16
cc: https://www.researchgate.net/post/Cactus_Language_Preliminaries
• https://inquiryintoinquiry.com/2025/04/16/cactus-language-preliminaries-6/
The definitions of the syntactic connectives can be made
a little more succinct by defining the following pair of
generic operators on strings.
Concatenation —
The “concatenation” Concₖ₌₁…ₙ of the sequence of
n strings (sₖ)ₖ₌₁…ₙ is defined recursively as follows.
For n = 1, Concₖ₌₁…ₙ sₖ = s₁
For n > 1, Concₖ₌₁…ₙ sₖ = Concₖ₌₁…ₙ₋₁ sₖ ∙ sₙ
Surcatenation —
The “surcatenation” Surcₖ₌₁…ₙ of the sequence of
n strings (sₖ)ₖ₌₁…ₙ is defined recursively as follows.
For n = 1, Surcₖ₌₁…ₙ sₖ = “(” ∙ s₁ ∙ “)”
For n > 1, Surcₖ₌₁…ₙ sₖ = Surcₖ₌₁…ₙ₋₁ sₖ ∙ (“)”)⁻¹ ∙ “,” ∙ sₙ ∙ “)”
Resources —
Cactus Language • Preliminaries
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Cactus_Language_.E2.80.A2_Preliminaries
Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2024/03/18/survey-of-animated-logical-graphs-7/
Survey of Theme One Program
• https://inquiryintoinquiry.com/2024/02/26/survey-of-theme-one-program-6/
Regards,
Jon
cc: https://www.researchgate.net/post/Cactus_Language_Preliminaries
Jon Awbrey
Apr 20, 2025, 12:00:33 PMApr 20
to Conceptual Graphs, Cybernetic Communications, Structural Modeling, SysSciWG
Cactus Language • Preliminaries 7
• https://inquiryintoinquiry.com/2025/04/19/cactus-language-preliminaries-7/
The array of syntactic operators may be put in more organized form
by making a few additional conventions and auxiliary definitions.
Concatenation —
The conception of concatenation permits extension to its natural “prequel”,
the corresponding operator on zero operands.
Conc⁰ = “” = the empty string
From that beginning the operation of concatenation may be broken into stages
by means of the following conceptions.
The “precatenation” Prec(s₁, s₂) of two strings s₁, s₂ is defined as follows.
Prec (s₁, s₂) = s₁ ∙ s₂
The “concatenation” of n strings s₁, …, sₙ may now be given a new definition
as the iterated precatenation of n+1 strings beginning with s₀ = Conc⁰ = “”
and continuing through the remaining n strings.
For n = 0, Concₖ₌₀…ₙ sₖ = Conc⁰ = “”
For n > 0, Concₖ₌₁…ₙ sₖ = Prec(Concₖ₌₁…ₙ₋₁ sₖ, sₙ)
Surcatenation —
The conception of surcatenation permits extension to its natural “prequel”,
the corresponding operator on zero operands.
Surc⁰ = “()”
From that beginning the operation of surcatenation may be broken into stages
by means of the following conceptions.
A “subclause” in ‡A‡* is a string ending with “)”
The “subcatenation” Subc(s₁, s₂) of a subclause s₁ by a string s₂
is defined as follows.
Subc(s₁, s₂) = s₁ ∙ (“)”)⁻¹ ∙ “,” ∙ s₂ ∙ “)”
The “surcatenation” of n strings s₁, …, sₙ may now be given a new definition
as the iterated subcatenation of n+1 strings beginning with s₀ = Surc⁰ = “()”
and continuing through the remaining n strings.
For n = 0, Surcₖ₌₀…ₙ sₖ = Surc⁰ = “()”
For n > 0, Surcₖ₌₁…ₙ sₖ = Subc(Surcₖ₌₁…ₙ₋₁ sₖ, sₙ)
Notice that the expressions Concₖ₌₀…₀ sₖ and Surcₖ₌₀…₀ sₖ are defined in
such a way that the respective operators Conc⁰ and Surc⁰ simply ignore,
in the manner of constants, whatever sequences of strings sₖ may be listed
as their ostensible arguments.
• https://inquiryintoinquiry.com/2025/04/19/cactus-language-preliminaries-7/
The array of syntactic operators may be put in more organized form
by making a few additional conventions and auxiliary definitions.
Concatenation —
The conception of concatenation permits extension to its natural “prequel”,
the corresponding operator on zero operands.
Conc⁰ = “” = the empty string
From that beginning the operation of concatenation may be broken into stages
by means of the following conceptions.
The “precatenation” Prec(s₁, s₂) of two strings s₁, s₂ is defined as follows.
Prec (s₁, s₂) = s₁ ∙ s₂
The “concatenation” of n strings s₁, …, sₙ may now be given a new definition
as the iterated precatenation of n+1 strings beginning with s₀ = Conc⁰ = “”
and continuing through the remaining n strings.
For n = 0, Concₖ₌₀…ₙ sₖ = Conc⁰ = “”
For n > 0, Concₖ₌₁…ₙ sₖ = Prec(Concₖ₌₁…ₙ₋₁ sₖ, sₙ)
Surcatenation —
The conception of surcatenation permits extension to its natural “prequel”,
the corresponding operator on zero operands.
Surc⁰ = “()”
From that beginning the operation of surcatenation may be broken into stages
by means of the following conceptions.
A “subclause” in ‡A‡* is a string ending with “)”
The “subcatenation” Subc(s₁, s₂) of a subclause s₁ by a string s₂
is defined as follows.
Subc(s₁, s₂) = s₁ ∙ (“)”)⁻¹ ∙ “,” ∙ s₂ ∙ “)”
The “surcatenation” of n strings s₁, …, sₙ may now be given a new definition
as the iterated subcatenation of n+1 strings beginning with s₀ = Surc⁰ = “()”
and continuing through the remaining n strings.
For n = 0, Surcₖ₌₀…ₙ sₖ = Surc⁰ = “()”
For n > 0, Surcₖ₌₁…ₙ sₖ = Subc(Surcₖ₌₁…ₙ₋₁ sₖ, sₙ)
Notice that the expressions Concₖ₌₀…₀ sₖ and Surcₖ₌₀…₀ sₖ are defined in
such a way that the respective operators Conc⁰ and Surc⁰ simply ignore,
in the manner of constants, whatever sequences of strings sₖ may be listed
as their ostensible arguments.
Jon Awbrey
Apr 23, 2025, 10:40:27 AMApr 23
to Conceptual Graphs, Cybernetic Communications, Structural Modeling, SysSciWG
Cactus Language • Preliminaries 8
• https://inquiryintoinquiry.com/2025/04/22/cactus-language-preliminaries-8/
Defining the basic operations of concatenation and surcatenation on
arbitrary strings gives them operational meaning for the all‑inclusive
language ‡L‡ = ‡A‡*. With that in hand it is time to adjoin the notion
of a more discriminating grammaticality, in other words, a more properly
restrictive concept of a sentence.
If ‡L‡ is an arbitrary formal language over an alphabet of the type we
have been discussing, that is, an alphabet of the form ‡A‡ = ‡M‡ ∪ ‡P‡,
then there are a number of basic structural relations which can be defined
on the strings of ‡L‡.
Concatenation —
s is the “concatenation” of s₁ and s₂ in ‡L‡
if and only if
s₁ is a sentence of ‡L‡, s₂ is a sentence of ‡L‡,
and
s = s₁ ∙ s₂
s is the “concatenation” of the n strings s₁, …, sₙ in ‡L‡
if and only if
sₖ is a sentence of ‡L‡ for all k = 1 … n
and
s = Concₖ₌₁…ₙ sₖ = s₁ ∙ … ∙ sₙ
Discatenation —
s is the “discatenation” of s₁ by t
if and only if
s₁ is a sentence of ‡L‡, t is an element of ‡A‡,
and
s₁ = s ∙ t
in which case we more commonly write
s = s₁ ∙ t⁻¹
Subclause —
s is a “subclause” of ‡L‡
if and only if
s is a sentence of ‡L‡
and
s ends with a “)”
Subcatenation —
s is the “subcatenation” of s₁ by s₂
if and only if
s₁ is a subclause of ‡L‡, s₂ is a sentence of ‡L‡,
and
s = s₁ ∙ (“)”)⁻¹ ∙ “,” ∙ s₂ ∙ “)”
Surcatenation —
s is the “surcatenation” of the n strings s₁, …, sₙ in ‡L‡
if and only if
sₖ is a sentence of ‡L‡ for all k = 1 … n
and
s = Surcₖ₌₁…ₙ sₖ = “(” ∙ s₁ ∙ “,” ∙ … ∙ “,” ∙ sₙ ∙ “)”
The converses of the above decomposition relations amount to
the corresponding composition operations. As complementary
forms of analysis and synthesis they make it possible to
articulate the structures of strings and sentences in
two directions.
Resources —
Cactus Language • Preliminaries
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Cactus_Language_.E2.80.A2_Preliminaries
Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2024/03/18/survey-of-animated-logical-graphs-7/
Survey of Theme One Program
• https://inquiryintoinquiry.com/2024/02/26/survey-of-theme-one-program-6/
Regards,
Jon
cc: https://www.academia.edu/community/VqWbWE
cc: https://www.researchgate.net/post/Cactus_Language_Preliminaries
• https://inquiryintoinquiry.com/2025/04/22/cactus-language-preliminaries-8/
Defining the basic operations of concatenation and surcatenation on
arbitrary strings gives them operational meaning for the all‑inclusive
language ‡L‡ = ‡A‡*. With that in hand it is time to adjoin the notion
of a more discriminating grammaticality, in other words, a more properly
restrictive concept of a sentence.
If ‡L‡ is an arbitrary formal language over an alphabet of the type we
have been discussing, that is, an alphabet of the form ‡A‡ = ‡M‡ ∪ ‡P‡,
then there are a number of basic structural relations which can be defined
on the strings of ‡L‡.
Concatenation —
s is the “concatenation” of s₁ and s₂ in ‡L‡
if and only if
s₁ is a sentence of ‡L‡, s₂ is a sentence of ‡L‡,
and
s = s₁ ∙ s₂
s is the “concatenation” of the n strings s₁, …, sₙ in ‡L‡
if and only if
sₖ is a sentence of ‡L‡ for all k = 1 … n
and
s = Concₖ₌₁…ₙ sₖ = s₁ ∙ … ∙ sₙ
Discatenation —
s is the “discatenation” of s₁ by t
if and only if
s₁ is a sentence of ‡L‡, t is an element of ‡A‡,
and
s₁ = s ∙ t
in which case we more commonly write
s = s₁ ∙ t⁻¹
Subclause —
s is a “subclause” of ‡L‡
if and only if
s is a sentence of ‡L‡
and
s ends with a “)”
Subcatenation —
s is the “subcatenation” of s₁ by s₂
if and only if
s₁ is a subclause of ‡L‡, s₂ is a sentence of ‡L‡,
and
s = s₁ ∙ (“)”)⁻¹ ∙ “,” ∙ s₂ ∙ “)”
Surcatenation —
s is the “surcatenation” of the n strings s₁, …, sₙ in ‡L‡
if and only if
sₖ is a sentence of ‡L‡ for all k = 1 … n
and
s = Surcₖ₌₁…ₙ sₖ = “(” ∙ s₁ ∙ “,” ∙ … ∙ “,” ∙ sₙ ∙ “)”
The converses of the above decomposition relations amount to
the corresponding composition operations. As complementary
forms of analysis and synthesis they make it possible to
articulate the structures of strings and sentences in
two directions.
Resources —
Cactus Language • Preliminaries
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Cactus_Language_.E2.80.A2_Preliminaries
Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2024/03/18/survey-of-animated-logical-graphs-7/
Survey of Theme One Program
• https://inquiryintoinquiry.com/2024/02/26/survey-of-theme-one-program-6/
Regards,
Jon
cc: https://www.researchgate.net/post/Cactus_Language_Preliminaries
Jon Awbrey
Apr 26, 2025, 5:40:47 PMApr 26
to Conceptual Graphs, Cybernetic Communications, Structural Modeling, SysSciWG
Cactus Language • Preliminaries 9
• https://inquiryintoinquiry.com/2025/04/25/cactus-language-preliminaries-9/
We now have the materials in place to formulate a definition of our subject.
The “painted cactus language” with “paints” in the set ‡P‡ = {pₖ : k ∈ K} is
the formal language ‡L‡ = ‡C‡(‡P‡) ⊆ ‡A‡* = (‡M‡ ∪ ‡P‡)* defined as follows.
PC 1. The blank symbol m_1 is a sentence.
PC 2. The paint pₖ is a sentence for each k ∈ K.
PC 3. Conc^0 and Surc^0 are sentences.
PC 4. For each positive integer n,
if s₁, …, sₙ are sentences
then Concₖ₌₁…ₙ sₖ is a sentence
and Surcₖ₌₁…ₙ sₖ is a sentence.
In the idiom of formal language theory, a string s is called a “sentence
of ‡L‡” if and only if it belongs to ‡L‡, or simply a “sentence” if the
language ‡L‡ is understood. A sentence of ‡C‡(‡P‡) is referred to as
a “painted and rooted cactus expression” on the palette ‡P‡, or
a “cactus expression” for short.
Resources —
Cactus Language • Preliminaries
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Cactus_Language_.E2.80.A2_Preliminaries
Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2024/03/18/survey-of-animated-logical-graphs-7/
Survey of Theme One Program
• https://inquiryintoinquiry.com/2024/02/26/survey-of-theme-one-program-6/
Regards,
Jon
cc: https://www.academia.edu/community/l80YQz
cc: https://www.researchgate.net/post/Cactus_Language_Preliminaries
• https://inquiryintoinquiry.com/2025/04/25/cactus-language-preliminaries-9/
We now have the materials in place to formulate a definition of our subject.
The “painted cactus language” with “paints” in the set ‡P‡ = {pₖ : k ∈ K} is
the formal language ‡L‡ = ‡C‡(‡P‡) ⊆ ‡A‡* = (‡M‡ ∪ ‡P‡)* defined as follows.
PC 1. The blank symbol m_1 is a sentence.
PC 2. The paint pₖ is a sentence for each k ∈ K.
PC 3. Conc^0 and Surc^0 are sentences.
PC 4. For each positive integer n,
if s₁, …, sₙ are sentences
then Concₖ₌₁…ₙ sₖ is a sentence
and Surcₖ₌₁…ₙ sₖ is a sentence.
In the idiom of formal language theory, a string s is called a “sentence
of ‡L‡” if and only if it belongs to ‡L‡, or simply a “sentence” if the
language ‡L‡ is understood. A sentence of ‡C‡(‡P‡) is referred to as
a “painted and rooted cactus expression” on the palette ‡P‡, or
a “cactus expression” for short.
Resources —
Cactus Language • Preliminaries
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Cactus_Language_.E2.80.A2_Preliminaries
Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2024/03/18/survey-of-animated-logical-graphs-7/
Survey of Theme One Program
• https://inquiryintoinquiry.com/2024/02/26/survey-of-theme-one-program-6/
Regards,
Jon
cc: https://www.researchgate.net/post/Cactus_Language_Preliminaries
Joe Bury
Apr 27, 2025, 5:14:26 AMApr 27
to cyb...@googlegroups.com
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.
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Jon Awbrey
May 15, 2025, 6:30:25 PMMay 15
to Conceptual Graphs, Cybernetic Communications, Structural Modeling, SysSciWG
Cactus Language • Preliminaries 10
• https://inquiryintoinquiry.com/2025/05/15/cactus-language-preliminaries-10/
Last time we arrived at the following definition of our subject matter.
The “painted cactus language” with “paints” in the set ‡P‡ = {pₖ : k ∈ K}
is the formal language ‡L‡ = ‡C‡(‡P‡) ⊆ ‡A‡* = (‡M‡ ∪ ‡P‡)* defined as follows.
PC 1. The blank symbol m₁ is a sentence.
PC 2. The paint pₖ is a sentence for each k ∈ K.
PC 3. Conc⁰ and Surc⁰ are sentences.
PC 4. For each positive integer n,
if s₁, …, sₙ are sentences
then Concₖ₌₁…ₙ sₖ is a sentence
and Surcₖ₌₁…ₙ sₖ is a sentence.
A sentence of ‡C‡(‡P‡) is known as a “painted and rooted cactus expression” on
the palette ‡P‡, or a “Parce” on ‡P‡, or even more simply as a “cactus expression”
when the context is understood.
Resources —
Cactus Language • Preliminaries
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Cactus_Language_.E2.80.A2_Preliminaries
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/lavp8O
cc: https://www.researchgate.net/post/Cactus_Language_Preliminaries
• https://inquiryintoinquiry.com/2025/05/15/cactus-language-preliminaries-10/
Last time we arrived at the following definition of our subject matter.
The “painted cactus language” with “paints” in the set ‡P‡ = {pₖ : k ∈ K}
is the formal language ‡L‡ = ‡C‡(‡P‡) ⊆ ‡A‡* = (‡M‡ ∪ ‡P‡)* defined as follows.
PC 2. The paint pₖ is a sentence for each k ∈ K.
PC 4. For each positive integer n,
if s₁, …, sₙ are sentences
then Concₖ₌₁…ₙ sₖ is a sentence
and Surcₖ₌₁…ₙ sₖ is a sentence.
the palette ‡P‡, or a “Parce” on ‡P‡, or even more simply as a “cactus expression”
when the context is understood.
Resources —
Cactus Language • Preliminaries
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Cactus_Language_.E2.80.A2_Preliminaries
Survey of Animated Logical Graphs
Survey of Theme One Program
Regards,
Jon
cc: https://www.academia.edu/community/lavp8O
cc: https://www.researchgate.net/post/Cactus_Language_Preliminaries
Jon Awbrey
May 16, 2025, 2:08:34 PMMay 16
to Conceptual Graphs, Cybernetic Communications, Structural Modeling, SysSciWG
Cactus Language • Preliminaries 11
• https://inquiryintoinquiry.com/2025/05/16/cactus-language-preliminaries-11/
Given the idea of a “Parce” on ‡P‡ as a member of the cactus language ‡C‡(‡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 ‡P‡ = ⌀. A bare Parce is a sentence in
the “bare cactus language”, variously notated as follows.
• ‡C‡⁰ = ‡C‡(⌀) = Parce⁰ = Parce(⌀)
That set of strings, regarded as a formal language in its own right,
is a sublanguage of every cactus language ‡C‡(‡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.
Resources —
Cactus Language • Preliminaries
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Cactus_Language_.E2.80.A2_Preliminaries
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/LxQYm1
cc: https://www.researchgate.net/post/Cactus_Language_Preliminaries
• https://inquiryintoinquiry.com/2025/05/16/cactus-language-preliminaries-11/
Given the idea of a “Parce” on ‡P‡ as a member of the cactus language ‡C‡(‡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 ‡P‡ = ⌀. A bare Parce is a sentence in
the “bare cactus language”, variously notated as follows.
• ‡C‡⁰ = ‡C‡(⌀) = Parce⁰ = Parce(⌀)
That set of strings, regarded as a formal language in its own right,
is a sublanguage of every cactus language ‡C‡(‡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.
Resources —
Cactus Language • Preliminaries
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Cactus_Language_.E2.80.A2_Preliminaries
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.researchgate.net/post/Cactus_Language_Preliminaries
Jon Awbrey
May 17, 2025, 2:30:35 PMMay 17
to Conceptual Graphs, Cybernetic Communications, Structural Modeling, SysSciWG
Cactus Language • Preliminaries 12
• https://inquiryintoinquiry.com/2025/05/17/cactus-language-preliminaries-12/
We are engaged in teasing out the consequences of the following description of our subject.
The “painted cactus language” with “paints” in the set ‡P‡ = {pₖ : k ∈ K}
is the formal language ‡L‡ = ‡C‡(‡P‡) ⊆ ‡A‡* = (‡M‡ ∪ ‡P‡)* defined as follows.
PC 1. The blank symbol m₁ is a sentence.
PC 2. The paint pₖ is a sentence for each k ∈ K.
PC 3. Conc⁰ and Surc⁰ are sentences.
PC 4. For each positive integer n,
if s₁, …, sₙ are sentences
then Concₖ₌₁…ₙ sₖ is a sentence
and Surcₖ₌₁…ₙ sₖ is a sentence.
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 Concⁿ and Surcⁿ 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.
We'll begin work on that task next time.
Resources —
Cactus Language • Preliminaries
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Cactus_Language_.E2.80.A2_Preliminaries
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/lz0vKp
cc: https://www.researchgate.net/post/Cactus_Language_Preliminaries
• https://inquiryintoinquiry.com/2025/05/17/cactus-language-preliminaries-12/
We are engaged in teasing out the consequences of the following description of our subject.
The “painted cactus language” with “paints” in the set ‡P‡ = {pₖ : k ∈ K}
is the formal language ‡L‡ = ‡C‡(‡P‡) ⊆ ‡A‡* = (‡M‡ ∪ ‡P‡)* defined as follows.
PC 1. The blank symbol m₁ is a sentence.
PC 2. The paint pₖ is a sentence for each k ∈ K.
PC 3. Conc⁰ and Surc⁰ are sentences.
PC 4. For each positive integer n,
if s₁, …, sₙ are sentences
then Concₖ₌₁…ₙ sₖ is a sentence
and Surcₖ₌₁…ₙ sₖ is a sentence.
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 Concⁿ and Surcⁿ 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.
We'll begin work on that task next time.
Resources —
Cactus Language • Preliminaries
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Cactus_Language_.E2.80.A2_Preliminaries
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.researchgate.net/post/Cactus_Language_Preliminaries
Jon Awbrey
May 21, 2025, 2:30:48 PMMay 21
to Conceptual Graphs, Cybernetic Communications, Structural Modeling, SysSciWG
Cactus Language • Preliminaries 13
• https://inquiryintoinquiry.com/2025/05/21/cactus-language-preliminaries-13/
❝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
• https://www.jstor.org/stable/27899620
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 ‡P‡ = {pₕ : h ∈ H}
PC 3. Conc⁰ and Surc⁰ are sentences.
PC 4. For each positive integer n,
if s₁, …, sₙ are sentences
then Concₖ₌₁…ₙ sₖ is a sentence
and Surcₖ₌₁…ₙ sₖ is a sentence.
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”.
• https://inquiryintoinquiry.com/2025/05/21/cactus-language-preliminaries-13/
❝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
• https://www.jstor.org/stable/27899620
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 ‡P‡ = {pₕ : h ∈ H}
is the formal language ‡L‡ = ‡C‡(‡P‡) ⊆ ‡A‡* = (‡M‡ ∪ ‡P‡)* defined as follows.
PC 1. The blank symbol m₁ is a sentence.
PC 2. The paint pₕ is a sentence for each h ∈ H.
PC 1. The blank symbol m₁ is a sentence.
PC 3. Conc⁰ and Surc⁰ are sentences.
PC 4. For each positive integer n,
if s₁, …, sₙ are sentences
then Concₖ₌₁…ₙ sₖ is a sentence
and Surcₖ₌₁…ₙ sₖ is a sentence.
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”.
Jon Awbrey
May 23, 2025, 10:15:35 AMMay 23
to Conceptual Graphs, Cybernetic Communications, Structural Modeling, SysSciWG
Cactus Language • Preliminaries 14
• https://inquiryintoinquiry.com/2025/05/23/cactus-language-preliminaries-14/
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.
• S covers T
• S governs T
• S rules T
• S subsumes T
• S types over T
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 ‡L‡.
Extensional
• Each member of the set T also belongs to the set S of sentences
in the language ‡L‡.
Intensional
• The quality of being T entails the quality S of being a sentence
in the language ‡L‡.
Resources —
Cactus Language • Preliminaries
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Cactus_Language_.E2.80.A2_Preliminaries
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/LYENQD
cc: https://www.researchgate.net/post/Cactus_Language_Preliminaries
• https://inquiryintoinquiry.com/2025/05/23/cactus-language-preliminaries-14/
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.
• S covers T
• S governs T
• S rules T
• S subsumes T
• S types over T
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 ‡L‡.
Extensional
• Each member of the set T also belongs to the set S of sentences
in the language ‡L‡.
Intensional
• The quality of being T entails the quality S of being a sentence
in the language ‡L‡.
Resources —
Cactus Language • Preliminaries
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Cactus_Language_.E2.80.A2_Preliminaries
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.researchgate.net/post/Cactus_Language_Preliminaries
Jon Awbrey
May 25, 2025, 2:45:27 PMMay 25
to Conceptual Graphs, Cybernetic Communications, Structural Modeling, SysSciWG
Cactus Language • Preliminaries 15
• https://inquiryintoinquiry.com/2025/05/25/cactus-language-preliminaries-15/
A notation of the form S :> T was introduced last time to indicate
reading it as “S covers T”.
In what follows the letter “S” indicates the type of a sentence in the
contemplated language ‡L‡. The letter “S” is the “initial symbol” or
“sentence symbol” of the candidate formal grammar for ‡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 ‡L‡. The additional letters are known
as “intermediate symbols” and collected together in the set ‡Q‡.
Combining the singleton set {“S”} whose sole member is the initial symbol
with the set ‡Q‡ of intermediate symbols results in the set {“S”} ∪ ‡Q‡
of “non‑terminal symbols”. Even though ‡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 ∈ {“S”} ∪ ‡Q‡.
To complete the package, the alphabet ‡A‡ of the language ‡L‡ may also
be referred to as the set of “terminal symbols”.
Resources —
Cactus Language • Preliminaries
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Cactus_Language_.E2.80.A2_Preliminaries
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/VDbOxr
cc: https://www.researchgate.net/post/Cactus_Language_Preliminaries
• https://inquiryintoinquiry.com/2025/05/25/cactus-language-preliminaries-15/
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.
• S covers T
• S governs T
• S rules T
• S subsumes T
• S types over T
For the moment it's enough to call S :> T a “covering relation”,
by any of the following readings.
• S covers T
• S governs T
• S rules T
• S subsumes T
• S types over T
reading it as “S covers T”.
In what follows the letter “S” indicates the type of a sentence in the
contemplated language ‡L‡. The letter “S” is the “initial symbol” or
“sentence symbol” of the candidate formal grammar for ‡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 ‡L‡. The additional letters are known
as “intermediate symbols” and collected together in the set ‡Q‡.
Combining the singleton set {“S”} whose sole member is the initial symbol
with the set ‡Q‡ of intermediate symbols results in the set {“S”} ∪ ‡Q‡
of “non‑terminal symbols”. Even though ‡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 ∈ {“S”} ∪ ‡Q‡.
To complete the package, the alphabet ‡A‡ of the language ‡L‡ may also
be referred to as the set of “terminal symbols”.
Resources —
Cactus Language • Preliminaries
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Cactus_Language_.E2.80.A2_Preliminaries
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.researchgate.net/post/Cactus_Language_Preliminaries
Jon Awbrey
May 28, 2025, 8:08:16 AMMay 28
to Conceptual Graphs, Cybernetic Communications, Structural Modeling, SysSciWG
Cactus Language • Preliminaries 16
• https://inquiryintoinquiry.com/2025/05/27/cactus-language-preliminaries-16/
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”} ∪ ‡Q‡ ∪ ‡A‡
as the “augmented alphabet” of the candidate formal grammar for ‡L‡ and thus
to refer to the strings in ({“S”} ∪ ‡Q‡ ∪ ‡A‡)* as the “augmented strings”
of the grammar for ‡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”} ∪ (‡Q‡ ∪ ‡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 into a set ‡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.
Resources —
Cactus Language • Preliminaries
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Cactus_Language_.E2.80.A2_Preliminaries
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/Lb2QeD
cc: https://www.researchgate.net/post/Cactus_Language_Preliminaries
• https://inquiryintoinquiry.com/2025/05/27/cactus-language-preliminaries-16/
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”} ∪ ‡Q‡ ∪ ‡A‡
as the “augmented alphabet” of the candidate formal grammar for ‡L‡ and thus
to refer to the strings in ({“S”} ∪ ‡Q‡ ∪ ‡A‡)* as the “augmented strings”
of the grammar for ‡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”} ∪ (‡Q‡ ∪ ‡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 into a set ‡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.
Resources —
Cactus Language • Preliminaries
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Cactus_Language_.E2.80.A2_Preliminaries
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.researchgate.net/post/Cactus_Language_Preliminaries
Jon Awbrey
May 29, 2025, 11:00:23 AMMay 29
to Conceptual Graphs, Cybernetic Communications, Structural Modeling, SysSciWG
Cactus Language • Preliminaries 17
• https://inquiryintoinquiry.com/2025/05/29/cactus-language-preliminaries-17/
A certain degree of flexibility in the use of covering relations
is typically allowed in practice. Where there is little danger
of confusion we may allow symbols to stand equivocally either
for individual strings or for their types.
There is a measure of consistency to this practice, considering the fact
that perfectly individual entities are rarely if ever grasped by means of
signs and finite expressions, since every appearance of an apparent token
is only a type of more particular tokens, ultimately leaving no recourse
but to the order of discerning interpretation which has to decide exactly
how every sign is intended.
Thus we have is sufficient license for expressions of the form t <: T
and T <: S, where the symbols t, T, S may be taken to signify either
the tokens or the subtypes of their covering types.
Note. For some time to come in the discussion that follows, although we will
continue to focus on the cactus language as our principal object example, our
more general purpose will be to develop the subject matter of formal languages
and grammars in general. We will do this by taking up a particular method of
“stepwise refinement” which leads us to extract a rigorous formal grammar for
the cactus language, starting with little more than a rough description of the
target language and applying a systematic analysis to develop a sequence of
increasingly more effective and exact approximations to the desired grammar.
Resources —
Cactus Language • Preliminaries
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Cactus_Language_.E2.80.A2_Preliminaries
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/V1a92m
cc: https://www.researchgate.net/post/Cactus_Language_Preliminaries
• https://inquiryintoinquiry.com/2025/05/29/cactus-language-preliminaries-17/
A certain degree of flexibility in the use of covering relations
is typically allowed in practice. Where there is little danger
of confusion we may allow symbols to stand equivocally either
for individual strings or for their types.
There is a measure of consistency to this practice, considering the fact
that perfectly individual entities are rarely if ever grasped by means of
signs and finite expressions, since every appearance of an apparent token
is only a type of more particular tokens, ultimately leaving no recourse
but to the order of discerning interpretation which has to decide exactly
how every sign is intended.
Thus we have is sufficient license for expressions of the form t <: T
and T <: S, where the symbols t, T, S may be taken to signify either
the tokens or the subtypes of their covering types.
Note. For some time to come in the discussion that follows, although we will
continue to focus on the cactus language as our principal object example, our
more general purpose will be to develop the subject matter of formal languages
and grammars in general. We will do this by taking up a particular method of
“stepwise refinement” which leads us to extract a rigorous formal grammar for
the cactus language, starting with little more than a rough description of the
target language and applying a systematic analysis to develop a sequence of
increasingly more effective and exact approximations to the desired grammar.
Resources —
Cactus Language • Preliminaries
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Cactus_Language_.E2.80.A2_Preliminaries
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.researchgate.net/post/Cactus_Language_Preliminaries
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