10 Jun
2024
10 Jun
'24
6:48 p.m.
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-typ…
• 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-ide…
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-ide…
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
11 Jun
11 Jun
6 p.m.
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-pai…
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
12 Jun
12 Jun
6:24 p.m.
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-pai…
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-cac…
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
13 Jun
13 Jun
7:36 p.m.
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-cac…
Figure 3. Parse Graph and Traverse String
•
https://inquiryintoinquiry.files.wordpress.com/2022/06/theme-exposition-par…
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
14 Jun
14 Jun
4:54 p.m.
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
16 Jun
16 Jun
11:12 p.m.
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
17 Jun
17 Jun
4:04 p.m.
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-interpre…
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-interpret…
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
19 Jun
19 Jun
3:48 p.m.
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…
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…
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-interpretati…
Regards,
Jon
cc: https://www.academia.edu/community/leMa3v
cc: https://mathstodon.xyz/@Inquiry/110039613333677509
20 Jun
20 Jun
7:20 p.m.
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-…
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-…
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
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Jon Awbrey