3 views

Skip to first unread message

Jun 10, 2024, 12:48:32 PMJun 10

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

Theme One Program • Exposition 1

• https://inquiryintoinquiry.com/2024/06/09/theme-one-program-exposition-1-b/

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

a range of inquiry processes, from everydayproblem 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 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.

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 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.

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.

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

making up the main formal media of the program's operation.

Resources —

Theme One Program • Overview

• https://oeis.org/wiki/Theme_One_Program_%E2%80%A2_Overview

Theme One Program • Exposition

• https://oeis.org/wiki/Theme_One_Program_%E2%80%A2_Exposition

Theme One Program • User Guide

• https://www.academia.edu/5211369/Theme_One_Program_User_Guide

Survey of Theme One Program

• https://inquiryintoinquiry.com/2024/02/26/survey-of-theme-one-program-6/

Regards,

Jon

cc: https://www.academia.edu/community/Vv83Xj

cc: https://mathstodon.xyz/@Inquiry/110039613333677509

• https://inquiryintoinquiry.com/2024/06/09/theme-one-program-exposition-1-b/

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

a range of inquiry processes, from everydayproblem 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 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.

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 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.

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.

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

making up the main formal media of the program's operation.

Resources —

Theme One Program • Overview

• https://oeis.org/wiki/Theme_One_Program_%E2%80%A2_Overview

Theme One Program • Exposition

• https://oeis.org/wiki/Theme_One_Program_%E2%80%A2_Exposition

Theme One Program • User Guide

• https://www.academia.edu/5211369/Theme_One_Program_User_Guide

Survey of Theme One Program

• https://inquiryintoinquiry.com/2024/02/26/survey-of-theme-one-program-6/

Regards,

Jon

cc: https://www.academia.edu/community/Vv83Xj

cc: https://mathstodon.xyz/@Inquiry/110039613333677509

Jun 11, 2024, 12:00:28 PMJun 11

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

Theme One Program • Exposition 2

• https://inquiryintoinquiry.com/2024/06/10/theme-one-program-exposition-2-b/

Re: Theme One Program • Exposition 1

• https://inquiryintoinquiry.com/2024/06/09/theme-one-program-exposition-1-b/

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.

Resources —

Theme One Program • Overview

• https://oeis.org/wiki/Theme_One_Program_%E2%80%A2_Overview

Theme One Program • Exposition

• https://oeis.org/wiki/Theme_One_Program_%E2%80%A2_Exposition

Theme One Program • User Guide

• https://www.academia.edu/5211369/Theme_One_Program_User_Guide

Survey of Theme One Program

• https://inquiryintoinquiry.com/2024/02/26/survey-of-theme-one-program-6/

Regards,

Jon

cc: https://www.academia.edu/community/Lbr4M3

cc: https://mathstodon.xyz/@Inquiry/110039613333677509

• https://inquiryintoinquiry.com/2024/06/10/theme-one-program-exposition-2-b/

Re: Theme One Program • Exposition 1

• https://inquiryintoinquiry.com/2024/06/09/theme-one-program-exposition-1-b/

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.

Resources —

Theme One Program • Overview

• https://oeis.org/wiki/Theme_One_Program_%E2%80%A2_Overview

Theme One Program • Exposition

• https://oeis.org/wiki/Theme_One_Program_%E2%80%A2_Exposition

Theme One Program • User Guide

• https://www.academia.edu/5211369/Theme_One_Program_User_Guide

Survey of Theme One Program

• https://inquiryintoinquiry.com/2024/02/26/survey-of-theme-one-program-6/

Regards,

Jon

cc: https://mathstodon.xyz/@Inquiry/110039613333677509

Jun 12, 2024, 1:00:33 PMJun 12

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

Theme One Program • Exposition 3

• https://inquiryintoinquiry.com/2024/06/12/theme-one-program-exposition-3-b/

All,

My earliest experiments coding logical graphs as dynamic “pointer” data

structures taught me that 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 the left hand side of the leftmost

lobe, marking the ascent with 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, and 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”.

Resources —

Theme One Program • Overview

• https://oeis.org/wiki/Theme_One_Program_%E2%80%A2_Overview

Theme One Program • Exposition

• https://oeis.org/wiki/Theme_One_Program_%E2%80%A2_Exposition

Theme One Program • User Guide

• https://www.academia.edu/5211369/Theme_One_Program_User_Guide

Survey of Theme One Program

• https://inquiryintoinquiry.com/2024/02/26/survey-of-theme-one-program-6/

Regards,

Jon

cc: https://www.academia.edu/community/V1P00E

cc: https://mathstodon.xyz/@Inquiry/110039613333677509

• https://inquiryintoinquiry.com/2024/06/12/theme-one-program-exposition-3-b/

All,

My earliest experiments coding logical graphs as dynamic “pointer” data

structures taught me that 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

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 the left hand side of the leftmost

lobe, marking the ascent with 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, and 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”.

Resources —

Theme One Program • Overview

• https://oeis.org/wiki/Theme_One_Program_%E2%80%A2_Overview

Theme One Program • Exposition

• https://oeis.org/wiki/Theme_One_Program_%E2%80%A2_Exposition

Theme One Program • User Guide

• https://www.academia.edu/5211369/Theme_One_Program_User_Guide

Survey of Theme One Program

• https://inquiryintoinquiry.com/2024/02/26/survey-of-theme-one-program-6/

Regards,

Jon

cc: https://mathstodon.xyz/@Inquiry/110039613333677509

Jun 13, 2024, 1:36:25 PMJun 13

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

Theme One Program • Exposition 4

• https://inquiryintoinquiry.com/2024/06/13/theme-one-program-exposition-4-b/

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 probably won't 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.

Resources —

Theme One Program • Overview

• https://oeis.org/wiki/Theme_One_Program_%E2%80%A2_Overview

Theme One Program • Exposition

• https://oeis.org/wiki/Theme_One_Program_%E2%80%A2_Exposition

Theme One Program • User Guide

• https://www.academia.edu/5211369/Theme_One_Program_User_Guide

Survey of Theme One Program

• https://inquiryintoinquiry.com/2024/02/26/survey-of-theme-one-program-6/

Regards,

Jon

cc: https://www.academia.edu/community/LGJ0vb

cc: https://mathstodon.xyz/@Inquiry/110039613333677509

• https://inquiryintoinquiry.com/2024/06/13/theme-one-program-exposition-4-b/

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 probably won't 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.

Resources —

Theme One Program • Overview

• https://oeis.org/wiki/Theme_One_Program_%E2%80%A2_Overview

Theme One Program • Exposition

• https://oeis.org/wiki/Theme_One_Program_%E2%80%A2_Exposition

Theme One Program • User Guide

• https://www.academia.edu/5211369/Theme_One_Program_User_Guide

Survey of Theme One Program

• https://inquiryintoinquiry.com/2024/02/26/survey-of-theme-one-program-6/

Regards,

Jon

cc: https://mathstodon.xyz/@Inquiry/110039613333677509

Jun 14, 2024, 10:56:07 AMJun 14

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

Theme One Program • Exposition 5

• https://inquiryintoinquiry.com/2024/06/14/theme-one-program-exposition-5-b/

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 that on hold and fast forward to

the logical substance.

Resources —

Theme One Program • Overview

• https://oeis.org/wiki/Theme_One_Program_%E2%80%A2_Overview

Theme One Program • Exposition

• https://oeis.org/wiki/Theme_One_Program_%E2%80%A2_Exposition

Theme One Program • User Guide

• https://www.academia.edu/5211369/Theme_One_Program_User_Guide

Survey of Theme One Program

• https://inquiryintoinquiry.com/2024/02/26/survey-of-theme-one-program-6/

Regards,

Jon

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

cc: https://mathstodon.xyz/@Inquiry/110039613333677509

• https://inquiryintoinquiry.com/2024/06/14/theme-one-program-exposition-5-b/

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 that on hold and fast forward to

the logical substance.

Resources —

Theme One Program • Overview

• https://oeis.org/wiki/Theme_One_Program_%E2%80%A2_Overview

Theme One Program • Exposition

• https://oeis.org/wiki/Theme_One_Program_%E2%80%A2_Exposition

Theme One Program • User Guide

• https://www.academia.edu/5211369/Theme_One_Program_User_Guide

Survey of Theme One Program

• https://inquiryintoinquiry.com/2024/02/26/survey-of-theme-one-program-6/

Regards,

Jon

cc: https://mathstodon.xyz/@Inquiry/110039613333677509

Jun 16, 2024, 5:12:26 PMJun 16

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

Theme One Program • Exposition 6

• https://inquiryintoinquiry.com/2024/06/16/theme-one-program-exposition-6-b/

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/2024/06/09/theme-one-program-exposition-1-b/

[2] https://inquiryintoinquiry.com/2024/06/10/theme-one-program-exposition-2-b/

[3] https://inquiryintoinquiry.com/2024/06/12/theme-one-program-exposition-3-b/

[4] https://inquiryintoinquiry.com/2024/06/13/theme-one-program-exposition-4-b/

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/2024/06/14/theme-one-program-exposition-5-b/

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.

Resources —

Theme One Program • Overview

• https://oeis.org/wiki/Theme_One_Program_%E2%80%A2_Overview

Theme One Program • Exposition

• https://oeis.org/wiki/Theme_One_Program_%E2%80%A2_Exposition

Theme One Program • User Guide

• https://www.academia.edu/5211369/Theme_One_Program_User_Guide

Survey of Theme One Program

• https://inquiryintoinquiry.com/2024/02/26/survey-of-theme-one-program-6/

Regards,

Jon

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

cc: https://mathstodon.xyz/@Inquiry/110039613333677509

• https://inquiryintoinquiry.com/2024/06/16/theme-one-program-exposition-6-b/

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/2024/06/09/theme-one-program-exposition-1-b/

[2] https://inquiryintoinquiry.com/2024/06/10/theme-one-program-exposition-2-b/

[3] https://inquiryintoinquiry.com/2024/06/12/theme-one-program-exposition-3-b/

[4] https://inquiryintoinquiry.com/2024/06/13/theme-one-program-exposition-4-b/

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/2024/06/14/theme-one-program-exposition-5-b/

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.

Resources —

Theme One Program • Overview

• https://oeis.org/wiki/Theme_One_Program_%E2%80%A2_Overview

Theme One Program • Exposition

• https://oeis.org/wiki/Theme_One_Program_%E2%80%A2_Exposition

Theme One Program • User Guide

• https://www.academia.edu/5211369/Theme_One_Program_User_Guide

Survey of Theme One Program

• https://inquiryintoinquiry.com/2024/02/26/survey-of-theme-one-program-6/

Regards,

Jon

cc: https://mathstodon.xyz/@Inquiry/110039613333677509

Jun 17, 2024, 10:04:43 AMJun 17

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

Theme One Program • Exposition 7.0

• https://inquiryintoinquiry.com/2024/06/17/theme-one-program-exposition-7-b/

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

Resources —

Theme One Program • Overview

• https://oeis.org/wiki/Theme_One_Program_%E2%80%A2_Overview

Theme One Program • Exposition

• https://oeis.org/wiki/Theme_One_Program_%E2%80%A2_Exposition

Theme One Program • User Guide

• https://www.academia.edu/5211369/Theme_One_Program_User_Guide

Survey of Theme One Program

• https://inquiryintoinquiry.com/2024/02/26/survey-of-theme-one-program-6/

Regards,

Jon

cc: https://www.academia.edu/community/Vv80oG

cc: https://mathstodon.xyz/@Inquiry/110039613333677509

• https://inquiryintoinquiry.com/2024/06/17/theme-one-program-exposition-7-b/

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

Resources —

Theme One Program • Overview

• https://oeis.org/wiki/Theme_One_Program_%E2%80%A2_Overview

Theme One Program • Exposition

• https://oeis.org/wiki/Theme_One_Program_%E2%80%A2_Exposition

Theme One Program • User Guide

• https://www.academia.edu/5211369/Theme_One_Program_User_Guide

Survey of Theme One Program

• https://inquiryintoinquiry.com/2024/02/26/survey-of-theme-one-program-6/

Regards,

Jon

cc: https://mathstodon.xyz/@Inquiry/110039613333677509

Jun 18, 2024, 8:30:26 AMJun 18

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

Theme One Program • Exposition 7.1

• https://inquiryintoinquiry.com/2024/06/17/theme-one-program-exposition-7-b/

All,

I'm attaching the Tables for the previous post.

Sending them separately on account of their size.

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/2024/06/17/theme-one-program-exposition-7-b/

All,

I'm attaching the Tables for the previous post.

Sending them separately on account of their size.

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

Jon

Jun 19, 2024, 9:48:28 AMJun 19

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

Theme One Program • Exposition 8

• https://inquiryintoinquiry.com/2024/06/18/theme-one-program-exposition-8-b/

Re: Theme One Program • Exposition 7

• https://inquiryintoinquiry.com/2024/06/17/theme-one-program-exposition-7-b/

Mathematical Structure and Logical Interpretation —

The main things to take away from the previous post are

the following two ideas, one syntactic and one semantic.

• Syntax. The compositional structures of cactus graphs

and cactus expressions are constructed from two kinds

of connective operations.

• Semantics. There are two ways of mapping the compositional

structures of syntax 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, as shown below.

• 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, as shown below.

• 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

cc: https://www.academia.edu/community/leMa3v

cc: https://mathstodon.xyz/@Inquiry/110039613333677509

• https://inquiryintoinquiry.com/2024/06/18/theme-one-program-exposition-8-b/

Re: Theme One Program • Exposition 7

• https://inquiryintoinquiry.com/2024/06/17/theme-one-program-exposition-7-b/

Mathematical Structure and Logical Interpretation —

The main things to take away from the previous post are

the following two ideas, one syntactic and one semantic.

• Syntax. The compositional structures of cactus graphs

and cactus expressions are constructed from two kinds

of connective operations.

• Semantics. There are two ways of mapping the compositional

structures of syntax 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, as shown below.

• 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, as shown below.

• 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

cc: https://www.academia.edu/community/leMa3v

cc: https://mathstodon.xyz/@Inquiry/110039613333677509

Jun 20, 2024, 1:21:01 PMJun 20

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

Theme One Program • Exposition 9

• https://inquiryintoinquiry.com/2024/06/19/theme-one-program-exposition-9-b/

Transformation Rules and Equivalence Classes —

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

and partitioning 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 a transformation which preserves

equivalence classes and reduces the level of

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 just in case

every component cactus joined to a node itself

reduces to a node, as shown below.

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

A “lobe reduction” is permitted just in case

exactly one component cactus listed in a lobe

reduces to an edge, as shown below.

• 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

cc: https://www.academia.edu/community/Vj8Gg0

cc: https://mathstodon.xyz/@Inquiry/110039613333677509

• https://inquiryintoinquiry.com/2024/06/19/theme-one-program-exposition-9-b/

Transformation Rules and Equivalence Classes —

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

and partitioning 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 a transformation which preserves

equivalence classes and reduces the level of

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 just in case

every component cactus joined to a node itself

reduces to a node, as shown below.

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

A “lobe reduction” is permitted just in case

exactly one component cactus listed in a lobe

reduces to an edge, as shown below.

• 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

cc: https://www.academia.edu/community/Vj8Gg0

cc: https://mathstodon.xyz/@Inquiry/110039613333677509

Reply all

Reply to author

Forward

0 new messages

Search

Clear search

Close search

Google apps

Main menu