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Jun 15, 2022, 9:30:32 AM6/15/22

to Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG

Cf: Theme One Program • Exposition 1

https://inquiryintoinquiry.com/2018/06/08/theme-one-program-exposition-1/

All,

Theme One is a program for building and transforming a particular species

of graph-theoretic data structures, forms designed to support a variety of

fundamental learning and reasoning tasks.

The program evolved over the course of an exploration into the integration of

contrasting types of activities involved in learning and reasoning, especially

the types of algorithms and data structures capable of supporting all sorts of

inquiry processes, from everyday problem solving to scientific investigation.

In its current state, Theme One integrates over a common data structure

fundamental algorithms for one type of inductive learning and one type

of deductive reasoning.

We begin by describing the class of graph-theoretic data structures

used by the program, as determined by their local and global aspects.

It will be the usual practice to shift around and to view these graphs

at many different levels of detail, from their abstract definition to

their concrete implementation, and many points in between.

The main work of the Theme One program is achieved by building and

transforming a single species of graph-theoretic data structures.

In their abstract form these structures are closely related to the

graphs that are called “cacti” and “conifers” in graph theory,

so we'll generally refer to them under those names.

The graph-theoretic data structures used by the program are built up from

a basic data structure called an “idea-form flag”. That structure is defined

as a pair of Pascal data types by means of the following specifications.

Box 1. Type Idea = ^Form

https://inquiryintoinquiry.files.wordpress.com/2022/06/theme-exposition-type-idea-5eform.png

An “idea” is a pointer to a “form”.

A “form” is a record consisting of:

A “sign” of type char;

Four pointers, as, up, on, by, of type idea;

A “code” of type numb, that is, an integer in [0, max integer].

Represented in terms of “digraphs”, or directed graphs, the combination

of an “idea” pointer and a “form” record is most easily pictured as an arc,

or directed edge, leading to a node that is labeled with the other data,

in this case, a letter and a number.

At the roughest but quickest level of detail, an idea of a form can be drawn like this.

Box 2. Idea^Form Node

https://inquiryintoinquiry.files.wordpress.com/2022/06/theme-exposition-idea5eform-node.png

When it is necessary to fill in more detail, the following schematic pattern can be used.

Box 3. Idea^Form Flag

https://inquiryintoinquiry.files.wordpress.com/2022/06/theme-exposition-idea5eform-flag.png

The idea-form type definition determines the local structure of

the whole host of graphs used by the program, including a motley

array of ephemeral buffers, temporary scratch lists, and other

graph-theoretic data structures used for their transient utilities

at specific points in the program.

I will put off discussing these more incidental graph structures

until the points where they actually arise, focusing here on the

particular varieties and the specific variants of cactoid graphs

that constitute the main formal media of the program's operation.

Regards,

Jon

https://inquiryintoinquiry.com/2018/06/08/theme-one-program-exposition-1/

All,

Theme One is a program for building and transforming a particular species

of graph-theoretic data structures, forms designed to support a variety of

fundamental learning and reasoning tasks.

The program evolved over the course of an exploration into the integration of

contrasting types of activities involved in learning and reasoning, especially

the types of algorithms and data structures capable of supporting all sorts of

inquiry processes, from everyday problem solving to scientific investigation.

In its current state, Theme One integrates over a common data structure

fundamental algorithms for one type of inductive learning and one type

of deductive reasoning.

We begin by describing the class of graph-theoretic data structures

used by the program, as determined by their local and global aspects.

It will be the usual practice to shift around and to view these graphs

at many different levels of detail, from their abstract definition to

their concrete implementation, and many points in between.

The main work of the Theme One program is achieved by building and

transforming a single species of graph-theoretic data structures.

In their abstract form these structures are closely related to the

graphs that are called “cacti” and “conifers” in graph theory,

so we'll generally refer to them under those names.

The graph-theoretic data structures used by the program are built up from

a basic data structure called an “idea-form flag”. That structure is defined

as a pair of Pascal data types by means of the following specifications.

Box 1. Type Idea = ^Form

https://inquiryintoinquiry.files.wordpress.com/2022/06/theme-exposition-type-idea-5eform.png

An “idea” is a pointer to a “form”.

A “form” is a record consisting of:

A “sign” of type char;

Four pointers, as, up, on, by, of type idea;

A “code” of type numb, that is, an integer in [0, max integer].

Represented in terms of “digraphs”, or directed graphs, the combination

of an “idea” pointer and a “form” record is most easily pictured as an arc,

or directed edge, leading to a node that is labeled with the other data,

in this case, a letter and a number.

At the roughest but quickest level of detail, an idea of a form can be drawn like this.

Box 2. Idea^Form Node

https://inquiryintoinquiry.files.wordpress.com/2022/06/theme-exposition-idea5eform-node.png

When it is necessary to fill in more detail, the following schematic pattern can be used.

Box 3. Idea^Form Flag

https://inquiryintoinquiry.files.wordpress.com/2022/06/theme-exposition-idea5eform-flag.png

The idea-form type definition determines the local structure of

the whole host of graphs used by the program, including a motley

array of ephemeral buffers, temporary scratch lists, and other

graph-theoretic data structures used for their transient utilities

at specific points in the program.

I will put off discussing these more incidental graph structures

until the points where they actually arise, focusing here on the

particular varieties and the specific variants of cactoid graphs

that constitute the main formal media of the program's operation.

Regards,

Jon

Jun 16, 2022, 11:45:13 AM6/16/22

to Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG

Cf: Theme One Program • Exposition 2

https://inquiryintoinquiry.com/2022/06/16/theme-one-program-exposition-2-2/

Re: Theme One Program • Exposition 1

https://inquiryintoinquiry.com/2022/06/15/theme-one-program-exposition-1-2/

All,

The previous post described the elementary data structure

used to represent nodes of graphs in the Theme One program.

This post describes the specific family of graphs employed

by the program.

Figure 1 shows a typical example of a “painted and rooted cactus”.

Figure 1. Painted And Rooted Cactus

https://inquiryintoinquiry.files.wordpress.com/2022/06/theme-exposition-painted-and-rooted-cactus-display.png

The graph itself is a mathematical object and does not inhabit the

page or other medium before our eyes, and it must not be confused

with any picture or other representation of it, anymore than we’d

want someone to confuse us with a picture of ourselves, but it’s

a fair enough picture, once we understand the conventions of

representation involved.

Let V(G) be the set of nodes in a graph G and let L be a set of identifiers.

We often find ourselves in situations where we have to consider many different

ways of associating the nodes of G with the identifiers in L. Various manners

of associating nodes with identifiers have been given conventional names by

different schools of graph theorists. I will give one way of describing

a few of the most common patterns of association.

• A graph is “painted” if there is a relation between its node set

and a set of identifiers, in which case the relation is called

a “painting” and the identifiers are called “paints”.

• A graph is “colored” if there is a function from its node set

to a set of identifiers, in which case the function is called

a “coloring” and the identifiers are called “colors”.

• A graph is “labeled” if there is a one-to-one mapping between

its node set and a set of identifiers, in which case the mapping

is called a “labeling” and the identifiers are called “labels”.

• A graph is said to be “rooted” if it has a unique distinguished node,

in which case the distinguished node is called the “root” of the graph.

The graph in Figure 1 has a root node marked by the “at” sign or amphora

symbol “@”.

The graph in Figure 1 has eight nodes plus the five paints in the

set {a, b, c, d, e}. The painting of nodes is indicated by drawing

the paints of each node next to the node they paint. Observe that

some nodes may be painted with an empty set of paints.

The structure of a painted and rooted cactus may be encoded in the form of

a character string called a “painted and rooted cactus expression”. For the

remainder of this discussion the terms “cactus” and “cactus expression” will

be used to mean the painted and rooted varieties. A cactus expression is

formed on an alphabet consisting of the relevant set of identifiers, the

“paints”, together with three punctuation marks: the left parenthesis,

the comma, and the right parenthesis.

Regards,

Jon

https://inquiryintoinquiry.com/2022/06/16/theme-one-program-exposition-2-2/

Re: Theme One Program • Exposition 1

https://inquiryintoinquiry.com/2022/06/15/theme-one-program-exposition-1-2/

All,

The previous post described the elementary data structure

used to represent nodes of graphs in the Theme One program.

This post describes the specific family of graphs employed

by the program.

Figure 1 shows a typical example of a “painted and rooted cactus”.

Figure 1. Painted And Rooted Cactus

https://inquiryintoinquiry.files.wordpress.com/2022/06/theme-exposition-painted-and-rooted-cactus-display.png

The graph itself is a mathematical object and does not inhabit the

page or other medium before our eyes, and it must not be confused

with any picture or other representation of it, anymore than we’d

want someone to confuse us with a picture of ourselves, but it’s

a fair enough picture, once we understand the conventions of

representation involved.

Let V(G) be the set of nodes in a graph G and let L be a set of identifiers.

We often find ourselves in situations where we have to consider many different

ways of associating the nodes of G with the identifiers in L. Various manners

of associating nodes with identifiers have been given conventional names by

different schools of graph theorists. I will give one way of describing

a few of the most common patterns of association.

• A graph is “painted” if there is a relation between its node set

and a set of identifiers, in which case the relation is called

a “painting” and the identifiers are called “paints”.

• A graph is “colored” if there is a function from its node set

to a set of identifiers, in which case the function is called

a “coloring” and the identifiers are called “colors”.

• A graph is “labeled” if there is a one-to-one mapping between

its node set and a set of identifiers, in which case the mapping

is called a “labeling” and the identifiers are called “labels”.

• A graph is said to be “rooted” if it has a unique distinguished node,

in which case the distinguished node is called the “root” of the graph.

The graph in Figure 1 has a root node marked by the “at” sign or amphora

symbol “@”.

The graph in Figure 1 has eight nodes plus the five paints in the

set {a, b, c, d, e}. The painting of nodes is indicated by drawing

the paints of each node next to the node they paint. Observe that

some nodes may be painted with an empty set of paints.

The structure of a painted and rooted cactus may be encoded in the form of

a character string called a “painted and rooted cactus expression”. For the

remainder of this discussion the terms “cactus” and “cactus expression” will

be used to mean the painted and rooted varieties. A cactus expression is

formed on an alphabet consisting of the relevant set of identifiers, the

“paints”, together with three punctuation marks: the left parenthesis,

the comma, and the right parenthesis.

Regards,

Jon

Jun 18, 2022, 2:00:19 PM6/18/22

to Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG

Cf: Theme One Program • Exposition 3

https://inquiryintoinquiry.com/2022/06/17/theme-one-program-exposition-3-2/

All,

My earliest experiments coding logical graphs as dynamic “pointer”

data structures taught me conceptual and computational efficiencies

of a critical sort could be achieved by generalizing their abstract

graphs from trees to the variety graph theorists know as “cacti”.

The genesis of that generalization is a tale worth telling another

time, but for now it's best to jump right in and proceed by way of

generic examples.

Figure 1 shows a typical example of a painted and rooted cactus.

Figure 1. Painted And Rooted Cactus

https://inquiryintoinquiry.files.wordpress.com/2022/06/theme-exposition-painted-and-rooted-cactus-display.png

Figure 2 shows a way to visualize the correspondence between

cactus graphs and cactus strings, demonstrated on the cactus

from Figure 1. By way of convenient terminology, the polygons

of a cactus graph are called its “lobes”. An edge not part of

a larger polygon is called a “2-gon” or a “bi-gon”. A terminal

bi-gon is called a “spike”.

Figure 2. Cactus Graph and Cactus Expression

https://inquiryintoinquiry.files.wordpress.com/2022/06/theme-exposition-cactus-graph-and-cactus-expression.png

The correspondence between a cactus graph and a cactus string is

obtained by an operation called “traversing” the graph in question.

• One traverses a cactus graph by beginning at the left hand side

of the root node, reading off the list of paints one encounters

at that point. Since the order of elements at any node is not

significant, one may start the cactus string with that list of

paints or save them for the end. We have done the latter in

this case.

• One continues by climbing up the left hand side of the leftmost lobe,

marking the ascent by means of a left parenthesis, traversing whatever

cactus one happens to reach at the first node above the root, that done,

proceeding from left to right along the top side of the lobe, marking each

interlobal span by means of a comma, traversing each cactus in turn one meets

along the way, on completing the last of them climbing down the right hand side

of the lobe, marking the descent by means of a right parenthesis, then traversing

each cactus in turn, in left to right order, that is incident with the root node.

The string of letters, parentheses, and commas one obtains by this procedure

is called the “traversal string” of the graph, in this case, a “cactus string”.

Regards,

Jon

https://inquiryintoinquiry.com/2022/06/17/theme-one-program-exposition-3-2/

All,

My earliest experiments coding logical graphs as dynamic “pointer”

data structures taught me conceptual and computational efficiencies

of a critical sort could be achieved by generalizing their abstract

graphs from trees to the variety graph theorists know as “cacti”.

The genesis of that generalization is a tale worth telling another

time, but for now it's best to jump right in and proceed by way of

generic examples.

Figure 1 shows a typical example of a painted and rooted cactus.

Figure 1. Painted And Rooted Cactus

https://inquiryintoinquiry.files.wordpress.com/2022/06/theme-exposition-painted-and-rooted-cactus-display.png

cactus graphs and cactus strings, demonstrated on the cactus

from Figure 1. By way of convenient terminology, the polygons

of a cactus graph are called its “lobes”. An edge not part of

a larger polygon is called a “2-gon” or a “bi-gon”. A terminal

bi-gon is called a “spike”.

Figure 2. Cactus Graph and Cactus Expression

https://inquiryintoinquiry.files.wordpress.com/2022/06/theme-exposition-cactus-graph-and-cactus-expression.png

The correspondence between a cactus graph and a cactus string is

obtained by an operation called “traversing” the graph in question.

• One traverses a cactus graph by beginning at the left hand side

of the root node, reading off the list of paints one encounters

at that point. Since the order of elements at any node is not

significant, one may start the cactus string with that list of

paints or save them for the end. We have done the latter in

this case.

• One continues by climbing up the left hand side of the leftmost lobe,

marking the ascent by means of a left parenthesis, traversing whatever

cactus one happens to reach at the first node above the root, that done,

proceeding from left to right along the top side of the lobe, marking each

interlobal span by means of a comma, traversing each cactus in turn one meets

along the way, on completing the last of them climbing down the right hand side

of the lobe, marking the descent by means of a right parenthesis, then traversing

each cactus in turn, in left to right order, that is incident with the root node.

The string of letters, parentheses, and commas one obtains by this procedure

is called the “traversal string” of the graph, in this case, a “cactus string”.

Regards,

Jon

Jun 22, 2022, 10:15:19 AM6/22/22

to Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG

Cf: Theme One Program • Exposition 4

https://inquiryintoinquiry.com/2022/06/20/theme-one-program-exposition-4/

All,

It is possible to write a program that parses cactus expressions

into reasonable facsimiles of cactus graphs as pointer structures

in computer memory, making edges correspond to addresses and nodes

correspond to records. I did just that in the early forerunners of

the present program, but it turned out to be a more robust strategy

in the long run, despite the need for additional nodes at the outset,

to implement a more articulate but more indirect parsing algorithm,

one in which the punctuation marks are not just tacitly converted

to addresses in passing, but instead recorded as nodes in roughly

the same way as the ordinary identifiers, or “paints”.

Figure 3 illustrates the type of parsing paradigm used by the program,

showing the pointer graph obtained by parsing the cactus expression in

Figure 2. A traversal of this graph naturally reconstructs the cactus

string that parses into it.

Figure 2. Cactus Graph and Cactus Expression

https://inquiryintoinquiry.files.wordpress.com/2022/06/theme-exposition-cactus-graph-and-cactus-expression.png

Figure 3. Parse Graph and Traverse String

https://inquiryintoinquiry.files.wordpress.com/2022/06/theme-exposition-parse-graph-and-traverse-string.png

The pointer graph in Figure 3, namely, the parse graph of a cactus

expression, is the sort of thing we'll probably not be able to resist

calling a “cactus graph”, for all the looseness of that manner of speaking,

but we should keep in mind its level of abstraction lies a step further in

the direction of a concrete implementation than the last thing we called by

that name. While we have them before our mind's eyes, there are several other

distinctive features of cactus parse graphs we ought to notice before moving on.

In terms of idea-form structures, a cactus parse graph begins with a root idea

pointing into a “by”‑cycle of forms, each of whose “sign” fields bears either

a “paint”, in other words, a direct or indirect identifier reference, or an

opening left parenthesis indicating a link to a subtended lobe of the cactus.

A lobe springs from a form whose “sign” field bears a left parenthesis.

That stem form has an “on” idea pointing into a “by”‑cycle of forms,

exactly one of which has a “sign” field bearing a right parenthesis.

That last form has an “on” idea pointing back to the form bearing

the initial left parenthesis.

In the case of a lobe, aside from the single form bearing the closing

right parenthesis, the “by”‑cycle of a lobe may list any number of forms,

each of whose “sign” fields bears either a comma, a paint, or an opening

left parenthesis signifying a link to a more deeply subtended lobe.

Just to draw out one of the implications of this characterization and to

stress the point of it, the root node can be painted and bear many lobes,

but it cannot be segmented, that is, the “by”‑cycle corresponding to the

root node can bear no commas.

Regards,

Jon

https://inquiryintoinquiry.com/2022/06/20/theme-one-program-exposition-4/

All,

It is possible to write a program that parses cactus expressions

into reasonable facsimiles of cactus graphs as pointer structures

in computer memory, making edges correspond to addresses and nodes

correspond to records. I did just that in the early forerunners of

the present program, but it turned out to be a more robust strategy

in the long run, despite the need for additional nodes at the outset,

to implement a more articulate but more indirect parsing algorithm,

one in which the punctuation marks are not just tacitly converted

to addresses in passing, but instead recorded as nodes in roughly

the same way as the ordinary identifiers, or “paints”.

Figure 3 illustrates the type of parsing paradigm used by the program,

showing the pointer graph obtained by parsing the cactus expression in

Figure 2. A traversal of this graph naturally reconstructs the cactus

string that parses into it.

Figure 2. Cactus Graph and Cactus Expression

https://inquiryintoinquiry.files.wordpress.com/2022/06/theme-exposition-cactus-graph-and-cactus-expression.png

https://inquiryintoinquiry.files.wordpress.com/2022/06/theme-exposition-parse-graph-and-traverse-string.png

The pointer graph in Figure 3, namely, the parse graph of a cactus

expression, is the sort of thing we'll probably not be able to resist

calling a “cactus graph”, for all the looseness of that manner of speaking,

but we should keep in mind its level of abstraction lies a step further in

the direction of a concrete implementation than the last thing we called by

that name. While we have them before our mind's eyes, there are several other

distinctive features of cactus parse graphs we ought to notice before moving on.

In terms of idea-form structures, a cactus parse graph begins with a root idea

pointing into a “by”‑cycle of forms, each of whose “sign” fields bears either

a “paint”, in other words, a direct or indirect identifier reference, or an

opening left parenthesis indicating a link to a subtended lobe of the cactus.

A lobe springs from a form whose “sign” field bears a left parenthesis.

That stem form has an “on” idea pointing into a “by”‑cycle of forms,

exactly one of which has a “sign” field bearing a right parenthesis.

That last form has an “on” idea pointing back to the form bearing

the initial left parenthesis.

In the case of a lobe, aside from the single form bearing the closing

right parenthesis, the “by”‑cycle of a lobe may list any number of forms,

each of whose “sign” fields bears either a comma, a paint, or an opening

left parenthesis signifying a link to a more deeply subtended lobe.

Just to draw out one of the implications of this characterization and to

stress the point of it, the root node can be painted and bear many lobes,

but it cannot be segmented, that is, the “by”‑cycle corresponding to the

root node can bear no commas.

Regards,

Jon

Jun 23, 2022, 12:30:25 PM6/23/22

to Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG

Cf: Theme One Program • Exposition 5

https://inquiryintoinquiry.com/2022/06/23/theme-one-program-exposition-5/

All,

Lexical, Literal, Logical

=========================

Theme One puts cactus graphs to work in three distinct but related ways,

called their “lexical”, “literal”, and “logical” uses. Those three modes

of operation employ three distinct but overlapping subsets of the broader

species of cacti. Accordingly we find ourselves working with graphs, files,

and expressions of lexical, literal, and logical types, depending on the task

at hand.

The logical class of cacti is the broadest, encompassing the whole species

described above, of which we have already seen a typical example in its

several avatars as abstract graph, pointer data structure, and string

of characters suitable for storage in a text file.

Being a “logical cactus” is not just a matter of syntactic form —

it means being subject to meaningful interpretations as a sign of

a logical proposition. To enter the logical arena cactus expressions

must express something, a proposition true or false of something.

Fully addressing the logical, interpretive, semantic aspect of cactus graphs

normally requires a mind-boggling mass of preliminary work on the details of

their syntactic structure. Practical, pragmatic, and especially computational

considerations will eventually make that unavoidable. For the sake of the

present discussion, however, let’s put a pin in it and fast forward to the

logical substance.

Regards,

Jon

https://inquiryintoinquiry.com/2022/06/23/theme-one-program-exposition-5/

All,

Lexical, Literal, Logical

=========================

Theme One puts cactus graphs to work in three distinct but related ways,

called their “lexical”, “literal”, and “logical” uses. Those three modes

of operation employ three distinct but overlapping subsets of the broader

species of cacti. Accordingly we find ourselves working with graphs, files,

and expressions of lexical, literal, and logical types, depending on the task

at hand.

The logical class of cacti is the broadest, encompassing the whole species

described above, of which we have already seen a typical example in its

several avatars as abstract graph, pointer data structure, and string

of characters suitable for storage in a text file.

Being a “logical cactus” is not just a matter of syntactic form —

it means being subject to meaningful interpretations as a sign of

a logical proposition. To enter the logical arena cactus expressions

must express something, a proposition true or false of something.

Fully addressing the logical, interpretive, semantic aspect of cactus graphs

normally requires a mind-boggling mass of preliminary work on the details of

their syntactic structure. Practical, pragmatic, and especially computational

considerations will eventually make that unavoidable. For the sake of the

present discussion, however, let’s put a pin in it and fast forward to the

logical substance.

Regards,

Jon

Jun 24, 2022, 11:10:34 AM6/24/22

to Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG

Cf: Theme One Program • Discussion 7

http://inquiryintoinquiry.com/2022/06/24/theme-one-program-discussion-7/

Re: Ontolog Forum

https://groups.google.com/g/ontolog-forum/c/vlsQqvEiIkY

::: Alex Shkotin

https://groups.google.com/g/ontolog-forum/c/vlsQqvEiIkY/m/07uF-WTzDwAJ

<QUOTE AS:>

As we both like digraphs and looking at your way of rendering, let me

share my lazy way of using Graphviz ( https://graphviz.org/ ) on one of

the last pictures produced ( https://photos.app.goo.gl/pJEGBnNqJRBE7JUT9 ).

This is a picture of a derivation tree (aka AST) for the text of four

statements of context-free grammar of some kind. It is important that

this is a digraph with ordered children, and nodes have some attributes.

In your case attributes are “sign”, “code”. In my case attributes are:

* node id,

* nonterminal,

** for syntactic nonterminal: rule id used for derivation,

** for lexical nonterminal: value taken from text.

</QUOTE>

Dear Alex,

Many thanks, the Graphviz suite looks very nice and I will

spend some time looking through the docs. I kept a few samples

of my old ASCII graphics, mostly from a sense of nostalgia, but

I've reached a point in reworking my Theme One Exposition where

I need to upgrade the graphics. My original aim was to have the

program display its own visuals, but it doesn't look like I'll

be the one doing that. Visualizing proofs requires animation —

I used to have an app for that bundled with CorelDraw but it

quit working in a previous platform change and I haven't gotten

around to hunting up a new one. At any rate, there's a sampler

of animated proofs in logical graphs on the following page.

* Proof Animations

( https://oeis.org/wiki/User:Jon_Awbrey/ANIMATION#Proof_Animations )

Regards,

Jon

http://inquiryintoinquiry.com/2022/06/24/theme-one-program-discussion-7/

Re: Ontolog Forum

https://groups.google.com/g/ontolog-forum/c/vlsQqvEiIkY

::: Alex Shkotin

https://groups.google.com/g/ontolog-forum/c/vlsQqvEiIkY/m/07uF-WTzDwAJ

<QUOTE AS:>

As we both like digraphs and looking at your way of rendering, let me

share my lazy way of using Graphviz ( https://graphviz.org/ ) on one of

the last pictures produced ( https://photos.app.goo.gl/pJEGBnNqJRBE7JUT9 ).

This is a picture of a derivation tree (aka AST) for the text of four

statements of context-free grammar of some kind. It is important that

this is a digraph with ordered children, and nodes have some attributes.

In your case attributes are “sign”, “code”. In my case attributes are:

* node id,

* nonterminal,

** for syntactic nonterminal: rule id used for derivation,

** for lexical nonterminal: value taken from text.

</QUOTE>

Dear Alex,

Many thanks, the Graphviz suite looks very nice and I will

spend some time looking through the docs. I kept a few samples

of my old ASCII graphics, mostly from a sense of nostalgia, but

I've reached a point in reworking my Theme One Exposition where

I need to upgrade the graphics. My original aim was to have the

program display its own visuals, but it doesn't look like I'll

be the one doing that. Visualizing proofs requires animation —

I used to have an app for that bundled with CorelDraw but it

quit working in a previous platform change and I haven't gotten

around to hunting up a new one. At any rate, there's a sampler

of animated proofs in logical graphs on the following page.

* Proof Animations

( https://oeis.org/wiki/User:Jon_Awbrey/ANIMATION#Proof_Animations )

Regards,

Jon

Jun 27, 2022, 3:45:23 PM6/27/22

to Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG

Cf: Theme One Program • Discussion 8

http://inquiryintoinquiry.com/2022/06/27/theme-one-program-discussion-8/

https://groups.google.com/g/ontolog-forum/c/vlsQqvEiIkY/m/m2ESSeyeBQAJ

Re: Theme One Program • Exposition 4

https://inquiryintoinquiry.com/2022/06/20/theme-one-program-exposition-4/

Re: Logical Graphs • Animated Proofs

https://oeis.org/wiki/User:Jon_Awbrey/ANIMATION#Proof_Animations

<QUOTE AS:>

The animation is mesmerizing: I would watch and watch. But without

the pause, next frame, and playback speed settings, it's just a work

of art that calls to see it step by step.

And on page [1]

( https://oeis.org/wiki/Propositional_Equation_Reasoning_Systems )

you start with the double negation theorem.

Have a look at a few of my comments as a reader, and only on the

graph topic taken separately. <...>

</QUOTE>

Dear Alex,

Thanks for viewing the animations and taking the time to work through that

first proof. Clearly a lot could be done to improve the production values —

what you see is what I got with whatever app I had umpteen years ago.

For the time being I'm focusing on the implementation layer of the

Theme One Program, which combines a learning component and a reasoning

component. The first implements a two‑level sequence learner and the

second implements a propositional calculator based on a variant of

Peirce's logical graphs. (I meant to say more about the learning

function this time around but I'm still working up to tackling that.)

To address your comments and questions we'll need to step back for a moment

to a more abstract, implementation-independent treatment of logical graphs.

There's a number of resources along those lines linked on the following

Survey page.

Survey of Animated Logical Graphs

https://inquiryintoinquiry.com/2021/05/01/survey-of-animated-logical-graphs-4/

The blog post “Logical Graphs • Formal Development”

( https://inquiryintoinquiry.com/2008/09/19/logical-graphs-2/ )

gives a quick but systematic account of the formal system

I use throughout. The OEIS wiki article “Logical Graphs”

( https://oeis.org/wiki/Logical_Graphs ) gives a more

detailed development.

Here's an excerpt from the above discussions, giving the four axioms

or “initials” which serve as graphical transformation rules, in effect,

the equational inference rules used to generate proofs and establish

theorems or “consequences”.

Axioms

======

The formal system of logical graphs is defined by a foursome of formal equations,

called “initials” when regarded purely formally, in abstraction from potential

interpretations, and called “axioms” when interpreted as logical equivalences.

There are two “arithmetic initials” and two “algebraic initials”, as shown below.

Arithmetic Initials

===================

Axiom I₁

https://inquiryintoinquiry.files.wordpress.com/2020/09/axiom-i1.png

Axiom I₂

https://inquiryintoinquiry.files.wordpress.com/2020/09/axiom-i2.png

Algebraic Initials

==================

Axiom J₁

https://inquiryintoinquiry.files.wordpress.com/2020/09/axiom-j1.png

Axiom J₂

https://inquiryintoinquiry.files.wordpress.com/2020/09/axiom-j2.png

The statement of the Double Negation Theorem is shown below.

C1. Double Negation Theorem

============================

The first theorem goes under the names of Consequence 1 (C₁),

the double negation theorem (DNT), or Reflection.

Double Negation Theorem

https://inquiryintoinquiry.files.wordpress.com/2021/02/double-negation-3.0.png

The following proof is adapted from the one given by George Spencer Brown

in his book Laws of Form and credited to two of his students, John Dawes

and D.A. Utting.

Double Negation Theorem • Proof

https://inquiryintoinquiry.files.wordpress.com/2021/02/double-negation-proof-3.0.png

That should fill in enough background to get started on your questions …

Regards,

Jon

http://inquiryintoinquiry.com/2022/06/27/theme-one-program-discussion-8/

https://groups.google.com/g/ontolog-forum/c/vlsQqvEiIkY/m/m2ESSeyeBQAJ

Re: Theme One Program • Exposition 4

https://inquiryintoinquiry.com/2022/06/20/theme-one-program-exposition-4/

Re: Logical Graphs • Animated Proofs

https://oeis.org/wiki/User:Jon_Awbrey/ANIMATION#Proof_Animations

<QUOTE AS:>

The animation is mesmerizing: I would watch and watch. But without

the pause, next frame, and playback speed settings, it's just a work

of art that calls to see it step by step.

And on page [1]

( https://oeis.org/wiki/Propositional_Equation_Reasoning_Systems )

you start with the double negation theorem.

Have a look at a few of my comments as a reader, and only on the

graph topic taken separately. <...>

</QUOTE>

Dear Alex,

Thanks for viewing the animations and taking the time to work through that

first proof. Clearly a lot could be done to improve the production values —

what you see is what I got with whatever app I had umpteen years ago.

For the time being I'm focusing on the implementation layer of the

Theme One Program, which combines a learning component and a reasoning

component. The first implements a two‑level sequence learner and the

second implements a propositional calculator based on a variant of

Peirce's logical graphs. (I meant to say more about the learning

function this time around but I'm still working up to tackling that.)

To address your comments and questions we'll need to step back for a moment

to a more abstract, implementation-independent treatment of logical graphs.

There's a number of resources along those lines linked on the following

Survey page.

Survey of Animated Logical Graphs

https://inquiryintoinquiry.com/2021/05/01/survey-of-animated-logical-graphs-4/

The blog post “Logical Graphs • Formal Development”

( https://inquiryintoinquiry.com/2008/09/19/logical-graphs-2/ )

gives a quick but systematic account of the formal system

I use throughout. The OEIS wiki article “Logical Graphs”

( https://oeis.org/wiki/Logical_Graphs ) gives a more

detailed development.

Here's an excerpt from the above discussions, giving the four axioms

or “initials” which serve as graphical transformation rules, in effect,

the equational inference rules used to generate proofs and establish

theorems or “consequences”.

Axioms

======

The formal system of logical graphs is defined by a foursome of formal equations,

called “initials” when regarded purely formally, in abstraction from potential

interpretations, and called “axioms” when interpreted as logical equivalences.

There are two “arithmetic initials” and two “algebraic initials”, as shown below.

Arithmetic Initials

===================

Axiom I₁

https://inquiryintoinquiry.files.wordpress.com/2020/09/axiom-i1.png

Axiom I₂

https://inquiryintoinquiry.files.wordpress.com/2020/09/axiom-i2.png

Algebraic Initials

==================

Axiom J₁

https://inquiryintoinquiry.files.wordpress.com/2020/09/axiom-j1.png

Axiom J₂

https://inquiryintoinquiry.files.wordpress.com/2020/09/axiom-j2.png

The statement of the Double Negation Theorem is shown below.

C1. Double Negation Theorem

============================

The first theorem goes under the names of Consequence 1 (C₁),

the double negation theorem (DNT), or Reflection.

Double Negation Theorem

https://inquiryintoinquiry.files.wordpress.com/2021/02/double-negation-3.0.png

The following proof is adapted from the one given by George Spencer Brown

in his book Laws of Form and credited to two of his students, John Dawes

and D.A. Utting.

Double Negation Theorem • Proof

https://inquiryintoinquiry.files.wordpress.com/2021/02/double-negation-proof-3.0.png

That should fill in enough background to get started on your questions …

Regards,

Jon

Jul 4, 2022, 1:40:27 PM7/4/22

to Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG

Cf: Theme One Program • Discussion 9

https://inquiryintoinquiry.com/2022/07/04/theme-one-program-discussion-9/

Re: Theme One Program • Exposition 5

https://inquiryintoinquiry.com/2022/06/23/theme-one-program-exposition-5/

https://groups.google.com/g/ontolog-forum/c/vlsQqvEiIkY/m/_IB_bNCzAAAJ

Dear Alex,

I know the material on sign relations I've been posting lately may seem

a digression from the Theme One Program already in progress but our last

discussions called for pulling the focus on logical graphs back to the

scene of Peirce's semiotics where those graphs work out their fates.

The Fourth is with us in the U.S. — it may be the middle of the week before

my ears stop ringing enough to think straight — but I did just want to touch

base and point out that bit of relevance, especially as I'll need to continue

with the background on sign relations a little longer.

Regards,

Jon

https://inquiryintoinquiry.com/2022/07/04/theme-one-program-discussion-9/

Re: Theme One Program • Exposition 5

https://inquiryintoinquiry.com/2022/06/23/theme-one-program-exposition-5/

https://groups.google.com/g/ontolog-forum/c/vlsQqvEiIkY/m/_IB_bNCzAAAJ

Dear Alex,

I know the material on sign relations I've been posting lately may seem

a digression from the Theme One Program already in progress but our last

discussions called for pulling the focus on logical graphs back to the

scene of Peirce's semiotics where those graphs work out their fates.

The Fourth is with us in the U.S. — it may be the middle of the week before

my ears stop ringing enough to think straight — but I did just want to touch

base and point out that bit of relevance, especially as I'll need to continue

with the background on sign relations a little longer.

Regards,

Jon

Oct 2, 2022, 3:00:32 PM10/2/22

to Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG

Cf: Theme One Program • Exposition 6

https://inquiryintoinquiry.com/2022/09/29/theme-one-program-exposition-6/

All,

Quickly recapping the discussion so far, we started with

a data structure called an “idea‑form flag” [1] and adopted

it as a building block for constructing a species of graph-

theoretic data structures called “painted and rooted cacti” [2].

We showed how to code the abstract forms of cacti into character

strings called “cactus expressions” [3] and how to parse the

character strings into “pointer structures” [4] in computer memory.

1. https://inquiryintoinquiry.com/2022/06/15/theme-one-program-exposition-1-2/

2. https://inquiryintoinquiry.com/2022/06/16/theme-one-program-exposition-2-2/

3. https://inquiryintoinquiry.com/2022/06/17/theme-one-program-exposition-3-2/

4. https://inquiryintoinquiry.com/2022/06/20/theme-one-program-exposition-4/

At this point we had to choose between two expository strategies.

A full account of Theme One’s operation would describe its use of cactus

graphs in three distinct ways, called “lexical”, “literal”, and “logical”

applications [5]. The more logical order would approach the lexical and

literal tasks first. That is because the program’s formal language learner

must first acquire the vocabulary its propositional calculator interprets as

logical variables. The sequential learner operates at two levels, taking in

sequences of characters it treats as “strings” or “words” plus sequences of

words it treats as “strands” or “sentences”.

5. https://inquiryintoinquiry.com/2022/06/23/theme-one-program-exposition-5/

Finding ourselves more strongly attracted to the logical substance, however,

we leave the matter of grammar to another time and turn to Theme One’s use of

cactus graphs in its reasoning module to represent logical propositions on the

order of Peirce’s alpha graphs and Spencer Brown’s calculus of indications.

Logical Cacti

=============

Up till now we’ve been working to hammer out a two-edged sword of syntax,

honing the syntax of cactus graphs and cactus expressions and turning it

to use in taming the syntax of two-level formal languages.

But the purpose of a logical syntax is to support a logical semantics,

which means, for starters, to bear interpretation as sentential signs

capable of denoting objective propositions about a universe of objects.

One of the difficulties we face is that the words “interpretation”, “meaning”,

“semantics”, and their ilk take on so many different meanings from one moment

to the next of their use. A dedicated neologician might be able to think up

distinctive names for all the aspects of meaning and all the approaches to

them that concern us, but I will do the best I can with the common lot of

ambiguous terms, leaving it to context and intelligent interpreters to

sort it out as much as possible.

The formal language of cacti is formed at such a high level of abstraction that

its graphs bear at least two distinct interpretations as logical propositions.

The two interpretations concerning us here descend from the ones C.S. Peirce

called the “entitative” and the “existential” interpretations of his systems

of graphical logics.

Existential Interpretation

==========================

Table 1 illustrates the existential interpretation of cactus graphs

and cactus expressions by providing English translations for a few

of the most basic and commonly occurring forms.

Table 1. Existential Interpretation

https://inquiryintoinquiry.files.wordpress.com/2022/10/existential-interpretation.png

Entitative Interpretation

=========================

Table 2 illustrates the entitative interpretation of cactus graphs

and cactus expressions by providing English translations for a few

of the most basic and commonly occurring forms.

Table 2. Entitative Interpretation

https://inquiryintoinquiry.files.wordpress.com/2022/10/entitative-interpretation.png

Regards,

Jon

https://inquiryintoinquiry.com/2022/09/29/theme-one-program-exposition-6/

All,

Quickly recapping the discussion so far, we started with

a data structure called an “idea‑form flag” [1] and adopted

it as a building block for constructing a species of graph-

theoretic data structures called “painted and rooted cacti” [2].

We showed how to code the abstract forms of cacti into character

strings called “cactus expressions” [3] and how to parse the

character strings into “pointer structures” [4] in computer memory.

1. https://inquiryintoinquiry.com/2022/06/15/theme-one-program-exposition-1-2/

2. https://inquiryintoinquiry.com/2022/06/16/theme-one-program-exposition-2-2/

3. https://inquiryintoinquiry.com/2022/06/17/theme-one-program-exposition-3-2/

4. https://inquiryintoinquiry.com/2022/06/20/theme-one-program-exposition-4/

At this point we had to choose between two expository strategies.

A full account of Theme One’s operation would describe its use of cactus

graphs in three distinct ways, called “lexical”, “literal”, and “logical”

applications [5]. The more logical order would approach the lexical and

literal tasks first. That is because the program’s formal language learner

must first acquire the vocabulary its propositional calculator interprets as

logical variables. The sequential learner operates at two levels, taking in

sequences of characters it treats as “strings” or “words” plus sequences of

words it treats as “strands” or “sentences”.

5. https://inquiryintoinquiry.com/2022/06/23/theme-one-program-exposition-5/

Finding ourselves more strongly attracted to the logical substance, however,

we leave the matter of grammar to another time and turn to Theme One’s use of

cactus graphs in its reasoning module to represent logical propositions on the

order of Peirce’s alpha graphs and Spencer Brown’s calculus of indications.

Logical Cacti

=============

Up till now we’ve been working to hammer out a two-edged sword of syntax,

honing the syntax of cactus graphs and cactus expressions and turning it

to use in taming the syntax of two-level formal languages.

But the purpose of a logical syntax is to support a logical semantics,

which means, for starters, to bear interpretation as sentential signs

capable of denoting objective propositions about a universe of objects.

One of the difficulties we face is that the words “interpretation”, “meaning”,

“semantics”, and their ilk take on so many different meanings from one moment

to the next of their use. A dedicated neologician might be able to think up

distinctive names for all the aspects of meaning and all the approaches to

them that concern us, but I will do the best I can with the common lot of

ambiguous terms, leaving it to context and intelligent interpreters to

sort it out as much as possible.

The formal language of cacti is formed at such a high level of abstraction that

its graphs bear at least two distinct interpretations as logical propositions.

The two interpretations concerning us here descend from the ones C.S. Peirce

called the “entitative” and the “existential” interpretations of his systems

of graphical logics.

Existential Interpretation

==========================

Table 1 illustrates the existential interpretation of cactus graphs

and cactus expressions by providing English translations for a few

of the most basic and commonly occurring forms.

Table 1. Existential Interpretation

https://inquiryintoinquiry.files.wordpress.com/2022/10/existential-interpretation.png

Entitative Interpretation

=========================

Table 2 illustrates the entitative interpretation of cactus graphs

and cactus expressions by providing English translations for a few

of the most basic and commonly occurring forms.

Table 2. Entitative Interpretation

https://inquiryintoinquiry.files.wordpress.com/2022/10/entitative-interpretation.png

Regards,

Jon

Oct 6, 2022, 8:12:26 AM10/6/22

to Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG

Cf: Theme One Program • Exposition 7

https://inquiryintoinquiry.com/2022/09/30/theme-one-program-exposition-7/

All,

Logical Cacti (cont.)

=====================

The main things to take away from the previous post

( https://inquiryintoinquiry.com/2022/09/29/theme-one-program-exposition-6/ )

are the following two ideas, one syntactic and one semantic:

• The compositional structures of cactus graphs and cactus expressions

are constructed from two kinds of connective operations.

• There are two ways of mapping these compositional structures into

the compositional structures of propositional sentences.

The two kinds of connective operations are described as follows.

• The “node connective” joins a number of component cacti C₁, …, Cₖ to a node:

Node Connective

https://inquiryintoinquiry.files.wordpress.com/2018/04/theme-one-exposition-e280a2-node-connective1.jpg

• The “lobe connective” joins a number of component cacti C₁, …, Cₖ to a lobe:

Lobe Connective

https://inquiryintoinquiry.files.wordpress.com/2018/04/theme-one-exposition-e280a2-lobe-connective1.jpg

The two ways of mapping cactus structures to logical meanings

are summarized in Table 3, which compares the entitative and

existential interpretations of the basic cactus structures,

in effect, the graphical constants and connectives.

Table 3. Logical Interpretations of Cactus Structures

https://inquiryintoinquiry.files.wordpress.com/2022/10/logical-interpretations-of-cactus-structures-e280a2-en-ex.png

Regards,

Jon

https://inquiryintoinquiry.com/2022/09/30/theme-one-program-exposition-7/

All,

Logical Cacti (cont.)

=====================

The main things to take away from the previous post

( https://inquiryintoinquiry.com/2022/09/29/theme-one-program-exposition-6/ )

are the following two ideas, one syntactic and one semantic:

• The compositional structures of cactus graphs and cactus expressions

are constructed from two kinds of connective operations.

• There are two ways of mapping these compositional structures into

the compositional structures of propositional sentences.

The two kinds of connective operations are described as follows.

• The “node connective” joins a number of component cacti C₁, …, Cₖ to a node:

Node Connective

https://inquiryintoinquiry.files.wordpress.com/2018/04/theme-one-exposition-e280a2-node-connective1.jpg

• The “lobe connective” joins a number of component cacti C₁, …, Cₖ to a lobe:

Lobe Connective

https://inquiryintoinquiry.files.wordpress.com/2018/04/theme-one-exposition-e280a2-lobe-connective1.jpg

The two ways of mapping cactus structures to logical meanings

are summarized in Table 3, which compares the entitative and

existential interpretations of the basic cactus structures,

in effect, the graphical constants and connectives.

Table 3. Logical Interpretations of Cactus Structures

https://inquiryintoinquiry.files.wordpress.com/2022/10/logical-interpretations-of-cactus-structures-e280a2-en-ex.png

Regards,

Jon

Oct 8, 2022, 12:12:13 PM10/8/22

to Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG

Cf: Theme One Program • Exposition 8

http://inquiryintoinquiry.com/2022/10/08/theme-one-program-exposition-8/

All,

Logical Cacti (cont.)

The abstract character of the cactus language relative to its logical

interpretations makes it possible to give abstract rules of equivalence

for transforming cacti among themselves which partition the space of cacti

into formal equivalence classes. The transformation rules and equivalence

classes are “purely formal” in the sense of being indifferent to the logical

interpretation, entitative or existential, one happens to choose.

Two definitions are useful here:

• A “reduction” is an equivalence transformation which applies in the

direction of decreasing graphical complexity.

• A “basic reduction” is a reduction which applies to a basic connective,

either a node connective or a lobe connective.

The two kinds of basic reductions are described as follows.

• A “node reduction” is permitted

if and only if

every component cactus joined to a node itself reduces to a node.

Node Reduction

https://inquiryintoinquiry.files.wordpress.com/2018/04/cactus-graph-e280a2-node-reduction.jpg

• A “lobe reduction” is permitted

if and only if

exactly one component cactus listed in a lobe reduces to an edge.

Lobe Reduction

https://inquiryintoinquiry.files.wordpress.com/2018/04/cactus-graph-e280a2-lobe-reduction.jpg

That is roughly the gist of the rules. More formal definitions

can wait for the day when we have to explain all this to a computer.

Regards,

Jon

http://inquiryintoinquiry.com/2022/10/08/theme-one-program-exposition-8/

All,

Logical Cacti (cont.)

The abstract character of the cactus language relative to its logical

interpretations makes it possible to give abstract rules of equivalence

for transforming cacti among themselves which partition the space of cacti

into formal equivalence classes. The transformation rules and equivalence

classes are “purely formal” in the sense of being indifferent to the logical

interpretation, entitative or existential, one happens to choose.

Two definitions are useful here:

• A “reduction” is an equivalence transformation which applies in the

direction of decreasing graphical complexity.

• A “basic reduction” is a reduction which applies to a basic connective,

either a node connective or a lobe connective.

The two kinds of basic reductions are described as follows.

• A “node reduction” is permitted

if and only if

every component cactus joined to a node itself reduces to a node.

Node Reduction

https://inquiryintoinquiry.files.wordpress.com/2018/04/cactus-graph-e280a2-node-reduction.jpg

• A “lobe reduction” is permitted

if and only if

exactly one component cactus listed in a lobe reduces to an edge.

Lobe Reduction

https://inquiryintoinquiry.files.wordpress.com/2018/04/cactus-graph-e280a2-lobe-reduction.jpg

That is roughly the gist of the rules. More formal definitions

can wait for the day when we have to explain all this to a computer.

Regards,

Jon

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