Cactus Language • Syntax
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Jon Awbrey
Jun 1, 2025, 11:30:27 AMJun 1
to Conceptual Graphs, Cybernetic Communications, Structural Modeling, SysSciWG
Cactus Language • Syntax 1
• https://inquiryintoinquiry.com/2025/05/31/cactus-language-syntax-1/
Grammar 1 —
Grammar 1 is something of a misnomer. It is nowhere near exemplifying
any kind of a standard form and it's put forth only as a starting point
for the initiation of more respectable grammars. Such as it is, it uses
the terminal alphabet ‡A‡ = ‡M‡ ∪ ‡P‡ coming with the territory of the
cactus language ‡C‡(‡P‡), it specifies ‡Q‡ = ⌀, in other words, it employs
no intermediate symbols, and it embodies the “covering set” ‡K‡ as listed in
the following display.
Cactus Language Grammar 1
• https://inquiryintoinquiry.files.wordpress.com/2025/05/cactus-language-grammar-1.png
The last two rules of Grammar 1 dictate the following typings.
• The concept of a sentence in ‡L‡ covers any concatenation
of sentences in ‡L‡, that is, any finite number of freely
chosen sentences available to be concatenated one after
another.
• The concept of a sentence in ‡L‡ covers any surcatenation
of sentences in ‡L‡, that is, any string opening with a “(”,
continuing with a sentence, possibly empty, following with
a finite number of phrases of the form “,” ∙ S, and closing
with a “)”.
The above appears to be just about the most concise description of
the cactus language ‡C‡(‡P‡) one can imagine but there are a couple
of problems commonly felt to afflict its style of presentation and
to make it less than completely acceptable. Briefly stated, the
problems turn on the following properties of the formulation.
• The invocation of the kleene star operation
is not reduced to a manifestly finitary form.
• The type S indicative of a sentence is allowed
to cover not only itself but also the empty string.
We'll discuss those issues at first in general, and especially in
regard to how the two features interact with one another, and then
we'll return to address in further detail the questions they engender
on their individual bases.
Resources —
Cactus Language • Syntax
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Cactus_Language_.E2.80.A2_Syntax
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/LYED0R
cc: https://www.researchgate.net/post/Cactus_Language_Syntax
• https://inquiryintoinquiry.com/2025/05/31/cactus-language-syntax-1/
Grammar 1 —
Grammar 1 is something of a misnomer. It is nowhere near exemplifying
any kind of a standard form and it's put forth only as a starting point
for the initiation of more respectable grammars. Such as it is, it uses
the terminal alphabet ‡A‡ = ‡M‡ ∪ ‡P‡ coming with the territory of the
cactus language ‡C‡(‡P‡), it specifies ‡Q‡ = ⌀, in other words, it employs
no intermediate symbols, and it embodies the “covering set” ‡K‡ as listed in
the following display.
Cactus Language Grammar 1
• https://inquiryintoinquiry.files.wordpress.com/2025/05/cactus-language-grammar-1.png
The last two rules of Grammar 1 dictate the following typings.
• The concept of a sentence in ‡L‡ covers any concatenation
of sentences in ‡L‡, that is, any finite number of freely
chosen sentences available to be concatenated one after
another.
• The concept of a sentence in ‡L‡ covers any surcatenation
of sentences in ‡L‡, that is, any string opening with a “(”,
continuing with a sentence, possibly empty, following with
a finite number of phrases of the form “,” ∙ S, and closing
with a “)”.
The above appears to be just about the most concise description of
the cactus language ‡C‡(‡P‡) one can imagine but there are a couple
of problems commonly felt to afflict its style of presentation and
to make it less than completely acceptable. Briefly stated, the
problems turn on the following properties of the formulation.
• The invocation of the kleene star operation
is not reduced to a manifestly finitary form.
• The type S indicative of a sentence is allowed
to cover not only itself but also the empty string.
We'll discuss those issues at first in general, and especially in
regard to how the two features interact with one another, and then
we'll return to address in further detail the questions they engender
on their individual bases.
Resources —
Cactus Language • Syntax
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Cactus_Language_.E2.80.A2_Syntax
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/LYED0R
cc: https://www.researchgate.net/post/Cactus_Language_Syntax
Jon Awbrey
Jun 2, 2025, 6:20:26 PMJun 2
to Conceptual Graphs, Cybernetic Communications, Structural Modeling, SysSciWG
Cactus Language • Syntax 2
• https://inquiryintoinquiry.com/2025/06/02/cactus-language-syntax-2/
Grammar 1 (cont.)
In the process of developing a grammar for a language we encounter
a number of organizational, pragmatic, and stylistic options whose
moment to moment choices decide the ongoing direction of the work
in progress and the impacts of whose evaluation work in tandem
to determine the shape of the grammar turned out in the end.
Most salient among the critical issues are three described
in the following way.
• The “degree of intermediate organization” in a grammar.
• The “distinction between empty and significant strings”,
and thus the “distinction between empty and significant
types of strings”.
• The “principle of intermediate significance”, a constraint
on the grammar which arises from considering the interaction
of the first two issues.
In responding to the collective issues, it is advisable at first
to proceed in a stepwise fashion, all the better to accommodate
the chances of pursuing a series of parallel developments in the
grammar, to allow for the possibility of reversing many steps in
its development, indeed, to take into account the almost certain
inevitability of having to revisit, revise, and reverse many prior
decisions about how to proceed toward an optimal description or
a satisfactory grammar for the language. Doing all that means
exploring the effects of various alterations and innovations
as independently from each other as possible.
Resources —
Cactus Language • Syntax
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Cactus_Language_.E2.80.A2_Syntax
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/V3PvxO
cc: https://www.researchgate.net/post/Cactus_Language_Syntax
• https://inquiryintoinquiry.com/2025/06/02/cactus-language-syntax-2/
Grammar 1 (cont.)
In the process of developing a grammar for a language we encounter
a number of organizational, pragmatic, and stylistic options whose
moment to moment choices decide the ongoing direction of the work
in progress and the impacts of whose evaluation work in tandem
to determine the shape of the grammar turned out in the end.
Most salient among the critical issues are three described
in the following way.
• The “degree of intermediate organization” in a grammar.
• The “distinction between empty and significant strings”,
and thus the “distinction between empty and significant
types of strings”.
• The “principle of intermediate significance”, a constraint
on the grammar which arises from considering the interaction
of the first two issues.
In responding to the collective issues, it is advisable at first
to proceed in a stepwise fashion, all the better to accommodate
the chances of pursuing a series of parallel developments in the
grammar, to allow for the possibility of reversing many steps in
its development, indeed, to take into account the almost certain
inevitability of having to revisit, revise, and reverse many prior
decisions about how to proceed toward an optimal description or
a satisfactory grammar for the language. Doing all that means
exploring the effects of various alterations and innovations
as independently from each other as possible.
Resources —
Cactus Language • Syntax
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Cactus_Language_.E2.80.A2_Syntax
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_Syntax
Jon Awbrey
Jun 4, 2025, 10:30:41 AMJun 4
to Conceptual Graphs, Cybernetic Communications, Structural Modeling, SysSciWG
Cactus Language • Syntax 3
• https://inquiryintoinquiry.com/2025/06/04/cactus-language-syntax-3/
Grammar 1 (cont.)
The “degree of intermediate organization” in a grammar is
measured by the number of its intermediate symbols and the
complexity of their mutual interplay within the frame of
the grammar's productions.
Grammar 1 has no intermediate symbols at all, ‡Q‡ = ⌀, and so remains
at a trivial degree of intermediate organization. Some additions to
the list of intermediate symbols are practically obligatory in order
to arrive at any reasonable grammar at all. Other inclusions have
a more optional character, though useful from the standpoints of
clarity and ease of comprehension.
One of the troubles perceived to affect Grammar 1 is the way it wastes
so much of the available potential for efficient description in recounting
over and over again the simple fact that the empty string is present in the
language. The problem arises partly from the covering relation S :> S*,
which has the following implications.
• S :> S* = {ε} ∪ S ∪ S∙S ∪ S∙S∙S ∪ …
There is nothing wrong with the more expansive pan of the covered equation,
since it follows straightforwardly from the definition of the kleene star
operation. But the statement S :> S* is not a very productive piece of
information, in the sense of telling us much about the language falling
under the type of a sentence S.
In particular, S :> S* implies S :> {ε}. Since {ε} ∙ ‡L‡ = ‡L‡ ∙ {ε} = ‡L‡
for any formal language ‡L‡, the empty string ε is counted over and over in
every term of the union, and every non‑empty sentence under S appears again
and again in every term of the union following the initial appearance of S.
As a result, the overall style of characterization has to be classified as
“true but not very informative”.
If at all possible, one prefers to partition the language of interest into
a disjoint union of subsets, thereby accounting for each sentence under its
proper term, and one whose place under the sum serves as a useful parameter
of its character or its complexity. Such an ideal form of description is not
always possible to achieve but it is usually worth the trouble to actualize it
whenever one can.
Suppose one tries to deal with the problem by eliminating each use of
the kleene star operation, by reducing it to a purely finitary set of
steps, or by finding another way to cover the sublanguage it is used to
generate. That amounts, in effect, to “recognizing a type”, a complex
process involving the following steps.
• “Noticing” a category of strings which is generated by
iteration or recursion.
• “Acknowledging” the fact that it needs to be covered by
a non‑terminal symbol.
• “Making a note of it” by instituting an explicitly‑named
grammatical category.
In sum, one introduces a non‑terminal symbol for each type of sentence and
each part of speech or sentential component generated by means of iteration
or recursion under the ruling constraints of the grammar. To do that one
needs to analyze the iteration of each grammatical operation in a way which
is analogous to a mathematically inductive definition but further in a way
which is not forced explicitly to recognize a distinct and separate type of
expression merely to account for and recount every increment in the parameter
of iteration.
Resources —
Cactus Language • Syntax
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Cactus_Language_.E2.80.A2_Syntax
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/L2Oo3d
cc: https://www.researchgate.net/post/Cactus_Language_Syntax
• https://inquiryintoinquiry.com/2025/06/04/cactus-language-syntax-3/
Grammar 1 (cont.)
The “degree of intermediate organization” in a grammar is
measured by the number of its intermediate symbols and the
complexity of their mutual interplay within the frame of
the grammar's productions.
Grammar 1 has no intermediate symbols at all, ‡Q‡ = ⌀, and so remains
at a trivial degree of intermediate organization. Some additions to
the list of intermediate symbols are practically obligatory in order
to arrive at any reasonable grammar at all. Other inclusions have
a more optional character, though useful from the standpoints of
clarity and ease of comprehension.
One of the troubles perceived to affect Grammar 1 is the way it wastes
so much of the available potential for efficient description in recounting
over and over again the simple fact that the empty string is present in the
language. The problem arises partly from the covering relation S :> S*,
which has the following implications.
• S :> S* = {ε} ∪ S ∪ S∙S ∪ S∙S∙S ∪ …
There is nothing wrong with the more expansive pan of the covered equation,
since it follows straightforwardly from the definition of the kleene star
operation. But the statement S :> S* is not a very productive piece of
information, in the sense of telling us much about the language falling
under the type of a sentence S.
In particular, S :> S* implies S :> {ε}. Since {ε} ∙ ‡L‡ = ‡L‡ ∙ {ε} = ‡L‡
for any formal language ‡L‡, the empty string ε is counted over and over in
every term of the union, and every non‑empty sentence under S appears again
and again in every term of the union following the initial appearance of S.
As a result, the overall style of characterization has to be classified as
“true but not very informative”.
If at all possible, one prefers to partition the language of interest into
a disjoint union of subsets, thereby accounting for each sentence under its
proper term, and one whose place under the sum serves as a useful parameter
of its character or its complexity. Such an ideal form of description is not
always possible to achieve but it is usually worth the trouble to actualize it
whenever one can.
Suppose one tries to deal with the problem by eliminating each use of
the kleene star operation, by reducing it to a purely finitary set of
steps, or by finding another way to cover the sublanguage it is used to
generate. That amounts, in effect, to “recognizing a type”, a complex
process involving the following steps.
• “Noticing” a category of strings which is generated by
iteration or recursion.
• “Acknowledging” the fact that it needs to be covered by
a non‑terminal symbol.
• “Making a note of it” by instituting an explicitly‑named
grammatical category.
In sum, one introduces a non‑terminal symbol for each type of sentence and
each part of speech or sentential component generated by means of iteration
or recursion under the ruling constraints of the grammar. To do that one
needs to analyze the iteration of each grammatical operation in a way which
is analogous to a mathematically inductive definition but further in a way
which is not forced explicitly to recognize a distinct and separate type of
expression merely to account for and recount every increment in the parameter
of iteration.
Resources —
Cactus Language • Syntax
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Cactus_Language_.E2.80.A2_Syntax
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_Syntax
Jon Awbrey
Jun 6, 2025, 9:00:18 AMJun 6
to Conceptual Graphs, Cybernetic Communications, Structural Modeling, SysSciWG
Cactus Language • Syntax 4
• https://inquiryintoinquiry.com/2025/06/05/cactus-language-syntax-4/
Grammar 1 (concl.)
Cactus Language Grammar 1
• https://inquiryintoinquiry.files.wordpress.com/2025/06/cactus-language-grammar-1-2.0.png
Returning to the case of the cactus language, the process of
recognizing iterative or recursive types can be illustrated in
the following way. The operative phrases in the simplest form of
recursive definition are its “initial part” and its “generic part”.
For the cactus language ‡C‡(‡P‡), one has the following definitions of
concatenation as iterated precatenation and surcatenation as iterated
subcatenation, respectively.
Cactus Language Grammar 1 Display 1
• https://inquiryintoinquiry.files.wordpress.com/2025/06/cactus-language-grammar-1-display-1.png
Transforming the recursive definitions into grammar rules is
accomplished by introducing a new pair of intermediate symbols,
Conc and Surc, corresponding to the operations of the same names
but ignoring the inflexions of their individual parameters j and k.
Recognizing the type of a sentence by means of the initial symbol S
and interpreting Conc and Surc as names for the types of strings
generated by concatenation and by surcatenation, respectively, one
arrives at the following transformation of the ruling operator
definitions into the form of covering grammar rules.
Cactus Language Grammar 1 Display 2
• https://inquiryintoinquiry.files.wordpress.com/2025/06/cactus-language-grammar-1-display-2.png
The draft of a grammar reached at this point represents a measure
of improvement. Still, it exhibits a couple of features which are
desirable to amend.
• Given the covering S :> Conc, the covering rule Conc :> Conc ∙ S
says no more than the covering rule Conc :> S ∙ S. Consequently,
all the information contained in those two covering rules is already
covered by the statement that S :> S ∙ S.
• Grammar rules invoking notions of discatenation, deletion, erasure,
and other forms of retrograde production may be felt to lack due elegance.
If for the sake of the aesthetic in question one entertains for a moment
keeping open the option of adopting that style of critique, it becomes
necessary to backtrack a little bit, to experiment with withdrawing
all forms of elliptical operators, but without, of course, eliding
the record of having considered them.
• https://inquiryintoinquiry.com/2025/06/05/cactus-language-syntax-4/
Grammar 1 (concl.)
Cactus Language Grammar 1
• https://inquiryintoinquiry.files.wordpress.com/2025/06/cactus-language-grammar-1-2.0.png
Returning to the case of the cactus language, the process of
recognizing iterative or recursive types can be illustrated in
the following way. The operative phrases in the simplest form of
recursive definition are its “initial part” and its “generic part”.
For the cactus language ‡C‡(‡P‡), one has the following definitions of
concatenation as iterated precatenation and surcatenation as iterated
subcatenation, respectively.
Cactus Language Grammar 1 Display 1
• https://inquiryintoinquiry.files.wordpress.com/2025/06/cactus-language-grammar-1-display-1.png
Transforming the recursive definitions into grammar rules is
accomplished by introducing a new pair of intermediate symbols,
Conc and Surc, corresponding to the operations of the same names
but ignoring the inflexions of their individual parameters j and k.
Recognizing the type of a sentence by means of the initial symbol S
and interpreting Conc and Surc as names for the types of strings
generated by concatenation and by surcatenation, respectively, one
arrives at the following transformation of the ruling operator
definitions into the form of covering grammar rules.
Cactus Language Grammar 1 Display 2
• https://inquiryintoinquiry.files.wordpress.com/2025/06/cactus-language-grammar-1-display-2.png
The draft of a grammar reached at this point represents a measure
of improvement. Still, it exhibits a couple of features which are
desirable to amend.
• Given the covering S :> Conc, the covering rule Conc :> Conc ∙ S
says no more than the covering rule Conc :> S ∙ S. Consequently,
all the information contained in those two covering rules is already
covered by the statement that S :> S ∙ S.
• Grammar rules invoking notions of discatenation, deletion, erasure,
and other forms of retrograde production may be felt to lack due elegance.
If for the sake of the aesthetic in question one entertains for a moment
keeping open the option of adopting that style of critique, it becomes
necessary to backtrack a little bit, to experiment with withdrawing
all forms of elliptical operators, but without, of course, eliding
the record of having considered them.
Jon Awbrey
Jun 8, 2025, 1:48:40 PMJun 8
to Conceptual Graphs, Cybernetic Communications, Structural Modeling, SysSciWG
Cactus Language • Syntax 5
• https://inquiryintoinquiry.com/2025/06/08/cactus-language-syntax-5/
Grammar 2 —
One way to analyze the surcatenation of any number of sentences
is to introduce an auxiliary type of string, not itself a sentence
but a proper component of any sentence formed by surcatenation.
Doing that brings one to the following definition.
A “tract” is a concatenation of a finite sequence of sentences, with
a literal comma “,” interpolated between each pair of adjacent sentences.
Thus, a typical tract T takes the following form.
• T = S₁ ∙ “,” ∙ … ∙ “,” ∙ Sₖ
A tract of that type must be distinguished from the abstract sequence
of sentences, S₁, …, Sₖ, where the commas coming to mind, as if being
called up to separate the successive sentences of the sequence, remain
as partially abstract conceptions, or as signs retaining a disengaged
status on the borderline between text and the mind.
The kinds of commas appearing in the abstract sequence continue to exist
as conceptual abstractions and fail to be cognized in a wholly explicit
fashion, either as concrete tokens in the object language or as marks in
the text engaging one's parsing attention.
Returning to the painted cactus language ‡L‡ = ‡C‡(‡P‡), it is possible to
put the assembled pieces of grammar together in the light of the adopted
canons of style to refine our analysis of the fact that the concept of
a sentence covers any concatenation of sentences and any surcatenation
of sentences, and so arrive at the following form of grammar.
Cactus Language Grammar 2
• https://inquiryintoinquiry.files.wordpress.com/2025/06/cactus-language-grammar-2.png
In that rendition, a string of type T is not in general a sentence itself
but a proper “part of speech”, that is, a strictly lesser component of
a sentence in any suitable ordering of sentences and their components.
In order to see how the grammatical category T gets off the ground,
that is, to detect its minimal strings and to discover how its ensuing
generations take their start from those, it is useful to observe that
the covering rule T :> S means that T inherits all the initial conditions
of S, namely, T :> ⌀, m₁, pₕ. In accord with those simple beginnings it
comes to pass that the rule T :> T ∙ “,” ∙ S, with the substitutions
T = ⌀ and S = ⌀, bears the germinal implication that T :> “,”.
Grammar 2 achieves a portion of its success through a higher degree of
intermediate organization. The level of organization is reflected in
the size of the intermediate alphabet ‡Q‡ = {“T”} but the number of
symbols alone does not give a full account, as intermediate symbols
are taken to serve a purpose, a purpose which is easy to recognize but
not so easy to pin down and specify exactly. Nevertheless, it is worth
the trouble to explore the intermediate level of organization and its
development a little further.
Resources —
Cactus Language • Syntax
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Cactus_Language_.E2.80.A2_Syntax
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/Vv7w7n
cc: https://www.researchgate.net/post/Cactus_Language_Syntax
• https://inquiryintoinquiry.com/2025/06/08/cactus-language-syntax-5/
Grammar 2 —
One way to analyze the surcatenation of any number of sentences
is to introduce an auxiliary type of string, not itself a sentence
but a proper component of any sentence formed by surcatenation.
Doing that brings one to the following definition.
A “tract” is a concatenation of a finite sequence of sentences, with
a literal comma “,” interpolated between each pair of adjacent sentences.
Thus, a typical tract T takes the following form.
• T = S₁ ∙ “,” ∙ … ∙ “,” ∙ Sₖ
A tract of that type must be distinguished from the abstract sequence
of sentences, S₁, …, Sₖ, where the commas coming to mind, as if being
called up to separate the successive sentences of the sequence, remain
as partially abstract conceptions, or as signs retaining a disengaged
status on the borderline between text and the mind.
The kinds of commas appearing in the abstract sequence continue to exist
as conceptual abstractions and fail to be cognized in a wholly explicit
fashion, either as concrete tokens in the object language or as marks in
the text engaging one's parsing attention.
Returning to the painted cactus language ‡L‡ = ‡C‡(‡P‡), it is possible to
put the assembled pieces of grammar together in the light of the adopted
canons of style to refine our analysis of the fact that the concept of
a sentence covers any concatenation of sentences and any surcatenation
of sentences, and so arrive at the following form of grammar.
Cactus Language Grammar 2
• https://inquiryintoinquiry.files.wordpress.com/2025/06/cactus-language-grammar-2.png
In that rendition, a string of type T is not in general a sentence itself
but a proper “part of speech”, that is, a strictly lesser component of
a sentence in any suitable ordering of sentences and their components.
In order to see how the grammatical category T gets off the ground,
that is, to detect its minimal strings and to discover how its ensuing
generations take their start from those, it is useful to observe that
the covering rule T :> S means that T inherits all the initial conditions
of S, namely, T :> ⌀, m₁, pₕ. In accord with those simple beginnings it
comes to pass that the rule T :> T ∙ “,” ∙ S, with the substitutions
T = ⌀ and S = ⌀, bears the germinal implication that T :> “,”.
Grammar 2 achieves a portion of its success through a higher degree of
intermediate organization. The level of organization is reflected in
the size of the intermediate alphabet ‡Q‡ = {“T”} but the number of
symbols alone does not give a full account, as intermediate symbols
are taken to serve a purpose, a purpose which is easy to recognize but
not so easy to pin down and specify exactly. Nevertheless, it is worth
the trouble to explore the intermediate level of organization and its
development a little further.
Resources —
Cactus Language • Syntax
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Cactus_Language_.E2.80.A2_Syntax
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_Syntax
Jon Awbrey
Jun 11, 2025, 8:08:20 AMJun 11
to Conceptual Graphs, Cybernetic Communications, Structural Modeling, SysSciWG
Cactus Language • Syntax 6
• https://inquiryintoinquiry.com/2025/06/10/cactus-language-syntax-6/
Grammar 3 —
It is possible to organize the materials of our developing grammar
in a more easily graspable fashion by recognizing two recurring types
of strings appearing in typical cactus expressions. In doing so one
arrives at the next two definitions.
A “rune” is a string of blanks and paints concatenated together.
Thus, a typical rune R is a string over {m₁} ∪ ‡P‡, possibly
the empty string.
• R ∈ ({m₁} ∪ ‡P‡)*
When there is no risk of confusion, the letter “R” may be used
either as a string variable ranging over the set of runes or as
a type name for the class of runes. The latter reading amounts
to the recruitment of a new intermediate symbol “R” ∈ ‡Q‡ to
form a new grammar for ‡C‡(‡P‡). Thus “R” accords grammatical
recognition to any rune forming a part of a sentence in ‡C‡(‡P‡).
In situations where the variant usages are likely to be confused
the types of strings may be indicated by way of expressions like
r <: R and W <: R.
A “foil” is a string of the form “(” ∙ T ∙ “)”, where T is a tract,
giving a foil F the following form.
• F = “(” ∙ S₁ ∙ “,” ∙ … ∙ “,” ∙ Sₖ ∙ “)”
Thus a foil is nothing other than the surcatenation of a sequence
of sentences S₁, …, Sₖ. In the case where the sequence of sentences
is empty and thus where the tract T is the empty string, we have the
minimal foil F₀ = “()”.
Explicitly marking each foil F embodied in a cactus expression
is tantamount to recognizing a new intermediate symbol, “F” ∈ ‡Q‡,
further articulating the structure of expressions and expanding the
grammar for the language ‡C‡(‡P‡). All the same remarks about the
versatile uses of intermediate symbols, as string variables and
as type names, apply again to the letter “F”.
Cactus Language Grammar 3
• https://inquiryintoinquiry.files.wordpress.com/2025/06/cactus-language-grammar-3.png
Resources —
Cactus Language • Syntax
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Cactus_Language_.E2.80.A2_Syntax
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/Lp1BWp
cc: https://www.researchgate.net/post/Cactus_Language_Syntax
• https://inquiryintoinquiry.com/2025/06/10/cactus-language-syntax-6/
Grammar 3 —
It is possible to organize the materials of our developing grammar
in a more easily graspable fashion by recognizing two recurring types
of strings appearing in typical cactus expressions. In doing so one
arrives at the next two definitions.
A “rune” is a string of blanks and paints concatenated together.
Thus, a typical rune R is a string over {m₁} ∪ ‡P‡, possibly
the empty string.
• R ∈ ({m₁} ∪ ‡P‡)*
When there is no risk of confusion, the letter “R” may be used
either as a string variable ranging over the set of runes or as
a type name for the class of runes. The latter reading amounts
to the recruitment of a new intermediate symbol “R” ∈ ‡Q‡ to
form a new grammar for ‡C‡(‡P‡). Thus “R” accords grammatical
recognition to any rune forming a part of a sentence in ‡C‡(‡P‡).
In situations where the variant usages are likely to be confused
the types of strings may be indicated by way of expressions like
r <: R and W <: R.
A “foil” is a string of the form “(” ∙ T ∙ “)”, where T is a tract,
giving a foil F the following form.
• F = “(” ∙ S₁ ∙ “,” ∙ … ∙ “,” ∙ Sₖ ∙ “)”
Thus a foil is nothing other than the surcatenation of a sequence
of sentences S₁, …, Sₖ. In the case where the sequence of sentences
is empty and thus where the tract T is the empty string, we have the
minimal foil F₀ = “()”.
Explicitly marking each foil F embodied in a cactus expression
is tantamount to recognizing a new intermediate symbol, “F” ∈ ‡Q‡,
further articulating the structure of expressions and expanding the
grammar for the language ‡C‡(‡P‡). All the same remarks about the
versatile uses of intermediate symbols, as string variables and
as type names, apply again to the letter “F”.
Cactus Language Grammar 3
• https://inquiryintoinquiry.files.wordpress.com/2025/06/cactus-language-grammar-3.png
Resources —
Cactus Language • Syntax
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Cactus_Language_.E2.80.A2_Syntax
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_Syntax
Jon Awbrey
Jun 13, 2025, 9:48:49 AMJun 13
to Conceptual Graphs, Cybernetic Communications, Structural Modeling, SysSciWG
Cactus Language • Syntax 7
• https://inquiryintoinquiry.com/2025/06/12/cactus-language-syntax-7/
Grammar 3 (cont.)
Cactus Language Grammar 3
• https://inquiryintoinquiry.files.wordpress.com/2025/06/cactus-language-grammar-3.png
In Grammar 3, the first three Rules say a sentence (a string of type S),
is either a rune (a string of type R), a foil (a string of type F), or
formed by concatenating strings of those two types.
Rules 4 through 7 say a rune R is either an empty string ε,
a blank symbol m₁, a paint pₕ, or formed by concatenating
strings of those three types.
Rule 8 characterizes a foil F as a string of the form “(” ∙ T ∙ “)”,
where T is a tract.
The last two Rules say a tract T is either a sentence S or the
concatenation of a tract, a comma, and a sentence, in that order.
At this point in the succession of grammars for ‡C‡(‡P‡), the problematic
applications of indefinite iterations, like the kleene star operator, are
now reduced to finite forms of concatenation but the problems stemming from
permitting non‑terminal symbols to cover both themselves and empty strings
have yet to be resolved.
A moment's reflection on the issue raises the general question:
What is a practical strategy for accounting for the empty string
in the organization of any formal language which counts it among
its sentences?
One answer presenting itself is the following: If the empty string
belongs to a formal language, it suffices to count it once at the
beginning of the formal account which enumerates its sentences and
then move on to more interesting materials.
Resources —
Cactus Language • Syntax
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Cactus_Language_.E2.80.A2_Syntax
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/l80e7o
cc: https://www.researchgate.net/post/Cactus_Language_Syntax
• https://inquiryintoinquiry.com/2025/06/12/cactus-language-syntax-7/
Grammar 3 (cont.)
Cactus Language Grammar 3
• https://inquiryintoinquiry.files.wordpress.com/2025/06/cactus-language-grammar-3.png
is either a rune (a string of type R), a foil (a string of type F), or
formed by concatenating strings of those two types.
Rules 4 through 7 say a rune R is either an empty string ε,
a blank symbol m₁, a paint pₕ, or formed by concatenating
strings of those three types.
Rule 8 characterizes a foil F as a string of the form “(” ∙ T ∙ “)”,
where T is a tract.
The last two Rules say a tract T is either a sentence S or the
concatenation of a tract, a comma, and a sentence, in that order.
At this point in the succession of grammars for ‡C‡(‡P‡), the problematic
applications of indefinite iterations, like the kleene star operator, are
now reduced to finite forms of concatenation but the problems stemming from
permitting non‑terminal symbols to cover both themselves and empty strings
have yet to be resolved.
A moment's reflection on the issue raises the general question:
What is a practical strategy for accounting for the empty string
in the organization of any formal language which counts it among
its sentences?
One answer presenting itself is the following: If the empty string
belongs to a formal language, it suffices to count it once at the
beginning of the formal account which enumerates its sentences and
then move on to more interesting materials.
Resources —
Cactus Language • Syntax
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Cactus_Language_.E2.80.A2_Syntax
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_Syntax
Jon Awbrey
Jun 16, 2025, 1:20:25 PMJun 16
to Conceptual Graphs, Cybernetic Communications, Structural Modeling, SysSciWG
Cactus Language • Syntax 8
• https://inquiryintoinquiry.com/2025/06/15/cactus-language-syntax-8/
Grammar 3 (concl.)
Returning to the cactus language ‡C‡(‡P‡) and fixing the
parameter ‡P‡ for the moment, we have a language Parce
of painted and rooted cactus expressions. It serves the
purpose of efficient accounting to divide the language
into two sublanguages.
1. The “emptily painted and rooted cactus expressions”
make up the language EParce which consists of
a single empty string as its only sentence.
• EParce = η = {ε}
2. The “significantly painted and rooted cactus expressions”
make up the language SParce which consists of everything
else, namely, all the non‑empty strings in the language
Parce.
• SParce = Parce \ ε
Marking the distinction between empty and significant sentences
effectively categorizes each of the three classes of strings as
an entity unto itself and conceives the whole of its membership
as falling under a distinctive symbol, thereby obtaining the
following equation among the three sublanguages.
• SParce = Parce - EParce
That makes Parce the disjoint union of EParce and SParce.
• Parce = EParce ∪ SParce
For brevity in the present case, and to serve as a generic device
in similar situations, let S be the type of an arbitrary sentence,
possibly empty, and let S′ be the type of a non‑empty sentence.
In addition, let η be the type of the empty sentence, in effect,
the language η = {ε} containing a single empty string, and let
a plus sign “+” signify a disjoint union of types. In the most
general type of situation, where the type S is permitted to include
the empty string, one notes the following relation among types.
• S = η + S′
With the distinction between empty and significant expressions
in mind, we return to the analysis of the cactus language ‡L‡ =
‡C‡(‡P‡) = Parce(‡P‡) afforded by Grammar 2, and, taking that as
a point of departure, explore other avenues of possible improvement
in the comprehension of its expressions.
To observe the effects of the alteration as clearly as possible in
isolation from other factors it is useful to strip away the higher
levels of intermediate organization presented by Grammar 3 and start
again with a single intermediate symbol, as used in Grammar 2. One way
to execute that strategy leads to a style of grammar we take up next.
Resources —
Cactus Language • Syntax
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Cactus_Language_.E2.80.A2_Syntax
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/VoBDJn
cc: https://www.researchgate.net/post/Cactus_Language_Syntax
• https://inquiryintoinquiry.com/2025/06/15/cactus-language-syntax-8/
Grammar 3 (concl.)
Returning to the cactus language ‡C‡(‡P‡) and fixing the
parameter ‡P‡ for the moment, we have a language Parce
of painted and rooted cactus expressions. It serves the
purpose of efficient accounting to divide the language
into two sublanguages.
1. The “emptily painted and rooted cactus expressions”
make up the language EParce which consists of
a single empty string as its only sentence.
• EParce = η = {ε}
2. The “significantly painted and rooted cactus expressions”
make up the language SParce which consists of everything
else, namely, all the non‑empty strings in the language
Parce.
• SParce = Parce \ ε
Marking the distinction between empty and significant sentences
effectively categorizes each of the three classes of strings as
an entity unto itself and conceives the whole of its membership
as falling under a distinctive symbol, thereby obtaining the
following equation among the three sublanguages.
• SParce = Parce - EParce
That makes Parce the disjoint union of EParce and SParce.
• Parce = EParce ∪ SParce
For brevity in the present case, and to serve as a generic device
in similar situations, let S be the type of an arbitrary sentence,
possibly empty, and let S′ be the type of a non‑empty sentence.
In addition, let η be the type of the empty sentence, in effect,
the language η = {ε} containing a single empty string, and let
a plus sign “+” signify a disjoint union of types. In the most
general type of situation, where the type S is permitted to include
the empty string, one notes the following relation among types.
• S = η + S′
With the distinction between empty and significant expressions
in mind, we return to the analysis of the cactus language ‡L‡ =
‡C‡(‡P‡) = Parce(‡P‡) afforded by Grammar 2, and, taking that as
a point of departure, explore other avenues of possible improvement
in the comprehension of its expressions.
To observe the effects of the alteration as clearly as possible in
isolation from other factors it is useful to strip away the higher
levels of intermediate organization presented by Grammar 3 and start
again with a single intermediate symbol, as used in Grammar 2. One way
to execute that strategy leads to a style of grammar we take up next.
Resources —
Cactus Language • Syntax
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Cactus_Language_.E2.80.A2_Syntax
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_Syntax
Jon Awbrey
Jun 18, 2025, 7:36:20 AMJun 18
to Conceptual Graphs, Cybernetic Communications, Structural Modeling, SysSciWG
Cactus Language • Syntax 9
• https://inquiryintoinquiry.com/2025/06/17/cactus-language-syntax-9/
Grammar 4 —
If one imposes the distinction between empty and significant types on
each non‑terminal symbol in Grammar 2 then the symbols “S” and “T” give
rise to the expanded set of non‑terminal symbols “S”, “S′ ”, “T”, “T′ ”,
leaving the last three to form a new intermediate alphabet.
Grammar 4 has the intermediate alphabet ‡Q‡ = {“S′ ”, “T”, “T′ ”}
with the set ‡K‡ of covering rules listed in the next display.
Cactus Language Grammar 4
• https://inquiryintoinquiry.files.wordpress.com/2025/06/cactus-language-grammar-4.png
Grammar 4 partitions the intermediate type T as T = η + T′
in parallel fashion with the division of its overlying type
as S = η + S′. That is an option we will close off for now
but leave open to consider at a later point. It suffices to
give a brief discussion of the considerations involved in
choosing between grammars at this point, and then move on
to the next alternative.
Resources —
Cactus Language • Syntax
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Cactus_Language_.E2.80.A2_Syntax
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/
cc: https://www.academia.edu/community/VqWO8d
cc: https://www.researchgate.net/post/Cactus_Language_Syntax
• https://inquiryintoinquiry.com/2025/06/17/cactus-language-syntax-9/
Grammar 4 —
If one imposes the distinction between empty and significant types on
each non‑terminal symbol in Grammar 2 then the symbols “S” and “T” give
rise to the expanded set of non‑terminal symbols “S”, “S′ ”, “T”, “T′ ”,
leaving the last three to form a new intermediate alphabet.
Grammar 4 has the intermediate alphabet ‡Q‡ = {“S′ ”, “T”, “T′ ”}
with the set ‡K‡ of covering rules listed in the next display.
Cactus Language Grammar 4
• https://inquiryintoinquiry.files.wordpress.com/2025/06/cactus-language-grammar-4.png
Grammar 4 partitions the intermediate type T as T = η + T′
in parallel fashion with the division of its overlying type
as S = η + S′. That is an option we will close off for now
but leave open to consider at a later point. It suffices to
give a brief discussion of the considerations involved in
choosing between grammars at this point, and then move on
to the next alternative.
Resources —
Cactus Language • Syntax
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Cactus_Language_.E2.80.A2_Syntax
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/
cc: https://www.researchgate.net/post/Cactus_Language_Syntax
Jon Awbrey
Jun 22, 2025, 5:40:54 PMJun 22
to Conceptual Graphs, Cybernetic Communications, Structural Modeling, SysSciWG
Cactus Language • Syntax 10
• https://inquiryintoinquiry.com/2025/06/22/cactus-language-syntax-10/
Grammar 4 (concl.)
Cactus Language Grammar 4
• https://inquiryintoinquiry.files.wordpress.com/2025/06/cactus-language-grammar-4.png
As we have seen, Grammar 4 partitions the intermediate type T as T = η + T′
and then moving on to the next alternative.
There does not appear to be anything radically wrong with trying
the above approach to types. It is reasonable and consistent in
its underlying principle and it provides a rational and uniform
strategy toward all parts of speech. But it does require an extra
amount of conceptual overhead in that every non‑trivial type has to
be split into two parts and comprehended in two stages. Consequently,
in view of the largely practical difficulties of making the required
distinctions for every intermediate symbol, it is a common convention,
whenever possible, to restrict intermediate types to covering exclusively
non‑empty strings.
It is convenient to refer to the above restriction on intermediate symbols
as the “intermediate significance” constraint. It may be given compact form
as a condition on the relations between non‑terminal symbols q ∈ {“S”} ∪ ‡Q‡
and sentential forms W ∈ {“S”} ∪ (‡Q‡ ∪ ‡A‡)*.
Cactus Language Grammar 4 Display 1
• https://inquiryintoinquiry.files.wordpress.com/2025/06/cactus-language-grammar-4-display-1.png
If that begins to sound like a monotone condition then it is not absurd
to sharpen the resemblance and render the likeness more acute. That is
achieved by declaring a couple of ordering relations, denoting them under
variant interpretations by the same sign, “<”.
• The ordering “<” on the collection of non‑terminal symbols q ∈ {“S”} ∪ ‡Q‡
ordains the initial symbol “S” to be strictly prior to every intermediate symbol.
That amounts to an axiom of the form “S” < q for all q ∈ ‡Q‡.
• The ordering “<” on the collection of sentential forms W ∈ {“S”} ∪ (‡Q‡ ∪ ‡A‡)*
ordains the empty string to be strictly less than every other sentential form.
That amounts to an axiom of the form ε < W for every non‑empty sentential form W.
Given the above orderings, the constraint on intermediate significance
may be stated as follows.
Cactus Language Grammar 4 Display 2
• https://inquiryintoinquiry.files.wordpress.com/2025/06/cactus-language-grammar-4-display-2.png
A grammar respecting intermediate significance will normally require a more
detailed account of the initial setting of each type, both with regard to
the type of context inciting its appearance and also with respect to the
minimal strings arising under the type in question. In order to find
covering productions satisfying the intermediate significance condition,
one must be prepared to consider a wider variety of calling contexts or
inciting situations observed to surround each recognized type and also
to enumerate a larger number of minimal cases observed to fall under
the significant types.
Resources —
Cactus Language • Syntax
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Cactus_Language_.E2.80.A2_Syntax
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
• https://inquiryintoinquiry.com/2025/06/22/cactus-language-syntax-10/
Grammar 4 (concl.)
Cactus Language Grammar 4
• https://inquiryintoinquiry.files.wordpress.com/2025/06/cactus-language-grammar-4.png
in parallel fashion with the division of its overlying type as S = η + S′.
That is an option we will close off for now but leave open to consider at
a later point, noting only the issues involved in choosing between grammars,
That is an option we will close off for now but leave open to consider at
and then moving on to the next alternative.
There does not appear to be anything radically wrong with trying
the above approach to types. It is reasonable and consistent in
its underlying principle and it provides a rational and uniform
strategy toward all parts of speech. But it does require an extra
amount of conceptual overhead in that every non‑trivial type has to
be split into two parts and comprehended in two stages. Consequently,
in view of the largely practical difficulties of making the required
distinctions for every intermediate symbol, it is a common convention,
whenever possible, to restrict intermediate types to covering exclusively
non‑empty strings.
It is convenient to refer to the above restriction on intermediate symbols
as the “intermediate significance” constraint. It may be given compact form
as a condition on the relations between non‑terminal symbols q ∈ {“S”} ∪ ‡Q‡
and sentential forms W ∈ {“S”} ∪ (‡Q‡ ∪ ‡A‡)*.
Cactus Language Grammar 4 Display 1
• https://inquiryintoinquiry.files.wordpress.com/2025/06/cactus-language-grammar-4-display-1.png
If that begins to sound like a monotone condition then it is not absurd
to sharpen the resemblance and render the likeness more acute. That is
achieved by declaring a couple of ordering relations, denoting them under
variant interpretations by the same sign, “<”.
• The ordering “<” on the collection of non‑terminal symbols q ∈ {“S”} ∪ ‡Q‡
ordains the initial symbol “S” to be strictly prior to every intermediate symbol.
That amounts to an axiom of the form “S” < q for all q ∈ ‡Q‡.
• The ordering “<” on the collection of sentential forms W ∈ {“S”} ∪ (‡Q‡ ∪ ‡A‡)*
ordains the empty string to be strictly less than every other sentential form.
That amounts to an axiom of the form ε < W for every non‑empty sentential form W.
Given the above orderings, the constraint on intermediate significance
may be stated as follows.
Cactus Language Grammar 4 Display 2
• https://inquiryintoinquiry.files.wordpress.com/2025/06/cactus-language-grammar-4-display-2.png
A grammar respecting intermediate significance will normally require a more
detailed account of the initial setting of each type, both with regard to
the type of context inciting its appearance and also with respect to the
minimal strings arising under the type in question. In order to find
covering productions satisfying the intermediate significance condition,
one must be prepared to consider a wider variety of calling contexts or
inciting situations observed to surround each recognized type and also
to enumerate a larger number of minimal cases observed to fall under
the significant types.
Resources —
Cactus Language • Syntax
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Cactus_Language_.E2.80.A2_Syntax
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/
Jon
Jon Awbrey
Jun 24, 2025, 12:12:31 PMJun 24
to Conceptual Graphs, Cybernetic Communications, Structural Modeling, SysSciWG
Cactus Language • Syntax 11
• https://inquiryintoinquiry.com/2025/06/23/cactus-language-syntax-11/
Grammar 5 —
With the foregoing array of considerations in mind, one is gradually led
to a grammar for ‡L‡ = ‡C‡(‡P‡) in which all of the covering productions
have one of the following two forms.
• S :> ε
• q :> W, with q ∈ {“S”} ∪ ‡Q‡ and W ∈ (‡Q‡ ∪ ‡A‡)⁺
A grammar fitting that description is called a “context‑free grammar”.
The first type of rewrite rule is called a “special production” while
the second type of rewrite rule is called an “ordinary production”.
An “ordinary derivation” is one composed solely of ordinary productions.
An ordinary production q :> W always replaces q with a non‑empty string W,
so the lengths of the augmented strings or sentential forms which follow one
another in an ordinary derivation never decrease at any stage of the process,
up to and including the terminal string finally generated by the grammar.
The feature just described is known as the “non‑contracting property”
of productions, derivations, and grammars. A grammar is said to have
the property if all of its covering productions, with the possible
exception of S :> ε, are non‑contracting.
In particular, “context‑free grammars” are special cases of non‑contracting
grammars. The presence of the non‑contracting property in a grammar makes
the length of the augmented string available as a parameter to figure into
mathematical induction and motivate recursive proofs. A handle like that
on the generative process makes it possible to establish results about the
generated language which are not easy to achieve in more general cases, nor
even by other means in the context‑free case.
Grammar 5 is a context‑free grammar for the painted cactus language ‡L‡ = ‡C‡(‡P‡)
with the intermediate alphabet ‡Q‡ = {“S′ ”, “T”} and the set ‡K‡ of covering rules
• https://inquiryintoinquiry.files.wordpress.com/2025/06/cactus-language-grammar-5.png
Finally, it is worth trying to bring together the separate advantages of diverse
styles of grammar, to the extent they are compatible. To do that, a prospective
grammar must be capable of maintaining a high level of intermediate organization,
like that exhibited by Grammar 2, while respecting the principle of intermediate
significance, thus accumulating the benefits of the context‑free style in Grammar 5.
A plausible synthesis of those features is given in Grammar 6.
Resources —
Cactus Language • Syntax
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Cactus_Language_.E2.80.A2_Syntax
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/VjWEab
cc: https://www.researchgate.net/post/Cactus_Language_Syntax
• https://inquiryintoinquiry.com/2025/06/23/cactus-language-syntax-11/
Grammar 5 —
With the foregoing array of considerations in mind, one is gradually led
to a grammar for ‡L‡ = ‡C‡(‡P‡) in which all of the covering productions
have one of the following two forms.
• S :> ε
• q :> W, with q ∈ {“S”} ∪ ‡Q‡ and W ∈ (‡Q‡ ∪ ‡A‡)⁺
A grammar fitting that description is called a “context‑free grammar”.
The first type of rewrite rule is called a “special production” while
the second type of rewrite rule is called an “ordinary production”.
An “ordinary derivation” is one composed solely of ordinary productions.
An ordinary production q :> W always replaces q with a non‑empty string W,
so the lengths of the augmented strings or sentential forms which follow one
another in an ordinary derivation never decrease at any stage of the process,
up to and including the terminal string finally generated by the grammar.
The feature just described is known as the “non‑contracting property”
of productions, derivations, and grammars. A grammar is said to have
the property if all of its covering productions, with the possible
exception of S :> ε, are non‑contracting.
In particular, “context‑free grammars” are special cases of non‑contracting
grammars. The presence of the non‑contracting property in a grammar makes
the length of the augmented string available as a parameter to figure into
mathematical induction and motivate recursive proofs. A handle like that
on the generative process makes it possible to establish results about the
generated language which are not easy to achieve in more general cases, nor
even by other means in the context‑free case.
Grammar 5 is a context‑free grammar for the painted cactus language ‡L‡ = ‡C‡(‡P‡)
with the intermediate alphabet ‡Q‡ = {“S′ ”, “T”} and the set ‡K‡ of covering rules
listed in the next display.
Cactus Language Grammar 5
• https://inquiryintoinquiry.files.wordpress.com/2025/06/cactus-language-grammar-5.png
Finally, it is worth trying to bring together the separate advantages of diverse
styles of grammar, to the extent they are compatible. To do that, a prospective
grammar must be capable of maintaining a high level of intermediate organization,
like that exhibited by Grammar 2, while respecting the principle of intermediate
significance, thus accumulating the benefits of the context‑free style in Grammar 5.
A plausible synthesis of those features is given in Grammar 6.
Resources —
Cactus Language • Syntax
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Cactus_Language_.E2.80.A2_Syntax
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_Syntax
Jon Awbrey
Jun 25, 2025, 5:36:29 PMJun 25
to Conceptual Graphs, Cybernetic Communications, Structural Modeling, SysSciWG
Cactus Language • Syntax 12
• https://inquiryintoinquiry.com/2025/06/25/cactus-language-syntax-12/
Grammar 6 —
Grammar 6 has the intermediate alphabet ‡Q‡ = {“S′ ”, “F”, “R”, “T”}
with the set ‡K‡ of covering rules listed in the next display.
Cactus Language Grammar 6
• https://inquiryintoinquiry.files.wordpress.com/2025/06/cactus-language-grammar-6.png
Our exploration of the grammar space for the language ‡C‡(‡P‡) shows how
an initially effective and succinct definition of a formal language can be
terse to the point of forcing its interpreters to spend exorbitant amounts
of time developing its consequences, but that it can be converted to a form
more efficient from the operational point of view, however ungainly in regard
to its elegance.
The main idea behind the grammar‑grinding remains the same,
to give concrete implementation to the following general rule.
Cactus Language General Rule
• https://inquiryintoinquiry.files.wordpress.com/2025/06/cactus-language-general-rule.png
Resources —
Cactus Language • Syntax
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Cactus_Language_.E2.80.A2_Syntax
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/lQwAOp
cc: https://www.researchgate.net/post/Cactus_Language_Syntax
• https://inquiryintoinquiry.com/2025/06/25/cactus-language-syntax-12/
Grammar 6 —
Grammar 6 has the intermediate alphabet ‡Q‡ = {“S′ ”, “F”, “R”, “T”}
with the set ‡K‡ of covering rules listed in the next display.
Cactus Language Grammar 6
• https://inquiryintoinquiry.files.wordpress.com/2025/06/cactus-language-grammar-6.png
Our exploration of the grammar space for the language ‡C‡(‡P‡) shows how
an initially effective and succinct definition of a formal language can be
terse to the point of forcing its interpreters to spend exorbitant amounts
of time developing its consequences, but that it can be converted to a form
more efficient from the operational point of view, however ungainly in regard
to its elegance.
The main idea behind the grammar‑grinding remains the same,
to give concrete implementation to the following general rule.
Cactus Language General Rule
• https://inquiryintoinquiry.files.wordpress.com/2025/06/cactus-language-general-rule.png
Resources —
Cactus Language • Syntax
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Cactus_Language_.E2.80.A2_Syntax
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_Syntax
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