justincoslor (justincoslor) wrote,
justincoslor
justincoslor

Book 1 of Possibility Thinking Explorations in Logic and Thought

This is an unfinished work and I disclaim all liability.
----------------------
Book I:
Patterns In Contexts
----------------------
8/25/2005 Stuff that occurred to me while going through some of my
old journal entries (about eight pages worth).
8/22/2005 Numbers & Patterns Across Contexts
5/1/2005 Property grouping axioms in cross-domain relations
8/24/2005 Properties
8/24/2005 Patterns In Contexts: Neural Nets As Priority Systems
10/8/1999 Patterns In Contexts: a computational model for representing
information metaphorically through abductive reasoning
8/5/2005 Augmenting Ideas: Generating New Perspectives on Information
8/6/2005 Epistemology Systems
5/23/2005 Some definitions for Patterns In Contexts Theory
8/2/2005 Cliff Partitions
5/23/2005 Object-oriented processing
5/27/2005 Simulated Models and Utility Axioms
5/22/2005 Operation Spaces continued - Tomographic Data Structures
5/21/2005 Operation Spaces: Grids V.S. Networks
5/14/2005 Key axioms and branch axioms in pattern collections
11/7/2004 Hypothetical Relation Highlighting in Undefined Data Sets
9/23/2004 Am I reinventing the wheel?
8/20/2004 Programming
8/4/2005 Programming Languages
9/22/2004 Patterns In Contexts Cognition Kernel
6/13/2004 Complexity
6/5/2004 Linker Patterns
5/31/2004 Patterns In Contexts Cognition
9/7/2000 Knowledge Mining
7/13/2005 Pattern Occurances
7/5/2005 ePIC Goal Representation
6/3/2005 Cross-Domain Relations in Analogical Relations
6/3/2005 Patterns In Context and Question Asking Systems for
Object-Oriented Programming
5/30/2005 Complexity Progressions
9/7/2004 Metaphoric Operations on Patterns Across Contexts
8/23/2004 Information Theory Quotes
12/25/2004 Metaphoric Operations
1/9/2005 Visual Dictionaries and Axiomatic Abductive Simulation
8/7/2005 Patterns In Contexts: 3D Engine
1/9/2005 Graphical Representation and Visual Heuristics
7/20/2005 Creativity & Understanding
7/17/2005 Concepts
7/8/2005 Measurement Systems
7/2/2005 Re-contextualized Patterns
6/25/2005 Observing patterns and differences
6/24/2005 Patterns Matching
6/20/2005 Remote-Controlled Contexts Via Pre-Processor Switchboards
6/12/2005 Definitions
6/5/2005 Geometric Abstractions
6/4/2005 Index of Topics
6/4/2005 Abstraction
6/3/2005 Analogical Recursions
5/25/2005 Implicit V.S. Explicit Knowledge
5/16/2005 Analogy, Metaphor, and Examples
5/4/2005 Sight
4/25/2003 Computer Vision
5/6/2005 Rules Are Behavioral Expectations
1/7/2005 Categories: Part 1
11/7/2004 Hypothetical Relation Highlighting in Undefined Data Sets
9/12/2004 Some Thoughts on Information Theory
10/22/2004 Some Methods of Proof
8/12/2004 Axiom Notes
6/8/2004 Contexts
6/5/2004 Perception
8/10/2005 Perception -- continued from 6/5/2004....
5/17/2003 A.I. Notes
10/20/2004 Mission Statement
----------------------
Book I:
Patterns In Contexts
----------------------
Copyright 8/25/2005 Justin Coslor
Stuff that occurred to me while going through some of my old journal entries
(about eight pages worth).
Analogies mimic patterns across contexts via cross-domain relations.
That's the basis of Analogical Reasoning. Every pattern in every context
is unique to the properties and axioms of the contexts they exist in.
I've written this book without doing a lick of research or reading
(except where indicated on a few entries), as an experiment to see if I
could generate some new foundations of knowledge and understanding. Some
experts say I succeeded.
A symmetry is an example of an internal algebra. Unique symmetries are
atomic repetitions, and are the simplest form of patterns, distinct from
perceptually apparently random chaos. (I don't believe in ultimate
randomness).
Analogical mimicing results in similar, yet distinctly different patterns.
All truth is but an approximation of a deeper truth. Understanding is
subject to computational complexity of the perceiver and the data forms and
content perceived.
Knowledge is the quest of discovery, and understanding is the growth of
the perceiver. It's how possibilities happen through careful navigation.
There are no dead ends.
Mark fundamental landmark differences in analogically mimiced patterns,
for possible classification category augmentations (for navigation and
data retrieval purposes). Beware of oversimplification of data streams
in order to fit a pattern into a perceptual mold.
Even if my ideas overlap with existing knowledge, they provide a new way
of understanding that knowledge, and that is valuable because my ideas
are not based on mimicry since I haven't studied topics related to them
much (except a college philosophy course and K-13 math). These ideas
exist for the most part in their own context. They can be no doubt eventually
be linked to ideas in other contexts though. Lexicons can often be linked to
external contexts.
Copyright 8/22/2005 Justin Coslor
Numbers & Patterns Across Contexts
Metaphorically speaking, prime numbers are injective and composite numbers
are surjective, when translating functions from one context to another.
Similarly, single-repetition patterns are injective and composite
patterns are surjective, when translating relations from one context to
another. This is an essential part of analogical reasoning.
Copyright 5/1/2005 Justin Coslor
Property grouping axioms in cross-domain relations.
(See diagram.)
1. All variables have properties.
2. All properties are independent of their variable's context(s).
3. All properties have some combination of qualitative relations,
quantitative relations, existential locations, and existential
conditions.
4. Every variable exists within a context and can vary
from context to context.
5. Contexts are composed of networks of patterns, patterns are composed of
networks of variables, and variables are composed of networks of properties.
6. Information can be represented as patterns in contexts, and in that way it
can be represented metaphorically through analogical reasoning and abductive
reasoning. Relations of various kinds, location(s), and condition(s) (apon and
of) exist at all of the various levels, and those are the data access
points.
As writer and owner of this piece of intellectual property I hereby declare it
universally free for use and modification, except I don't condone it's
use in weapon systems or for deception of any kind. This legal agreement
cannot be modified ever, and all modifications of this logic and/or data are
bound by this same agreement. You may sell applications and/or services that
use this logic (or its modifications) but you may not sell the logic itself
and you may not try to prevent others from understanding or using the
logic in any way except if they try to use it for deception or military
weapon applications. Sincerely, Justin Coslor. May 1st, 2005.
Copyright 8/24/2005 Justin Coslor
Properties
These are first-level definitions of some useful kinds of properties, any of
which can be networked together to create relations and variables and
patterns and contexts that may exist in physical and/or platonic reality:
----------------------------------
* qualitative identifiers: Categorical names and cross-references.
* qualitative factors: Qualitative pieces of composite patterns.
* quantitative identifiers: Cardinalities (orderings), scalers, and surjective
equalities.
* quantitative factors: Representational methods of measurement of dimension
sets.
* states: Observable distinct configurations that mark and increment step
counts.
* conditions: Dependencies that distinctly configure each state.
* cycle counts: A tally that is increased with each repetition of a process.
* recursions: A self-defined process or network (an internal algebra), or a
function that calls itself.
* repetitions: An algebra, atomic elements that repeat, composite patterns
that repeat, or symmetries.
* activity level: The number of cycles per step (positive, negative, random,
or null).
* step counts: A tally that is increased as conditions of each state
transition is reached.
* location: A place in a memory grid where identifiable data is stored.
* positions: The sequence coordinates of variable in N-dimensional orderings.
* orientations: The perspective that data maps are observed from: This may be
contextual, or spatially framed position maps, and perspectives may even
have translation conditions of their own.
------------------------
As you can see, activity level is just one kind of property, and priority
systems such as neural nets can be based on that property. Other kinds
of systems can be based on other properties.
------------------------
* Relations are the juxtaposition of infrastructures, which result in an
output.
Copyright 8/24/2005 Justin Coslor
Patterns In Contexts: Neural Nets As Priority Systems
Neural nets are essentially priority systems for allocating and
de-allocating priorities of networked elements such as variables on a grid.
Each network can be considered a context, and can be said to be a network of
patterns composed of variables and relations. If the patterns are
functions, then the priority of each pattern determines the level of activity
(cycles per step) of each pattern's function(s).
Some priority level results in a random level of activity, and the other
priority levels result in either positive levels of activity, negative
(reverse) levels of activity, and an undefined priority setting results
in no activity.
Example:
Context: ABCDEFG * R1R2R3R4R5R6R7R8 == a network of patterns (see diagram).
Where each pattern == (a variable)(a relation Rn), and the level of activity
of each pattern is:
(undefined, -3, -2, -1, random, 1, 2. 3)(a variable)(a relation Rn),
and the activity level determines how many cycles per step that the
relation Rn operates on the variable, and the pattern it is linked
to. These values could be anything, this is just an example.
Activity level is one type of property of the variable.
***Variables are composed of networks of properties.
***Patterns are composed of networks of variables and relations.
***Contexts are composed of networks of patterns and relations to other
contexts. Properties can be things like qualitative and quantitative
identifiers and factors, states, conditions, cycle counts, recursions
and repetitions, activity level (cycles per step), step counts, locations,
positions, orientations, etc.
Patterns in Contexts:
a computational model for representing information metaphorically
through abductive reasoning.
All ideas herein are Copyright by Justin Coslor on their respective dates.
These notes are a progression of the concepts in the order they occured to me.
10/8/99 Invent a digital method or circuit (fast physical algorithm) for
mimicking a pattern. Do it by modeling the pattern's relationships
metaphorically (as a metaphor of existing memory or experience).
If circular reasoning is involved, what is the least experience or innate
memory required to start the circular reasoning engine?
Topic: Circular reasoning engines (in logic and computation).
11/19/1999 The definitive nature of knowledge. This occured to me as a
pseudo-sophomore at Carnegie Mellon University in Pittsburgh, PA USA.
(It's a first draft so please forgive it's sketchyness.)
All knowledge=information, which can be represented as metaphors. Metaphors
are applied to specific contexts and general contexts=multiple contexts.
A. All knowledge is metaphors applied to >= 1 context.
B. A metaphor is a set of associations (links, patterns) that can or is
applied to a context.
A single context...A single specific context...general/nonspecific contexts.
C. A context is a set of restrictions (restrictions on information,
associations, links, patterns, sometimes even contexts).
Therefore the statement A is equivalent to this statement:
"Each piece of knowledge is a set of associations that can be applied to >= 1
set(s) of restrictions."
The use of this I had in mind is to make a computer software that could
understand and manipulate (and maybe even apply) metaphors. Many other ideas
occurred to me today too, possibly due to doing yogi breathing and meditation
exercises and taking vitamins since my health had been suffering.
10/31/2003 Epistemology Framework for Artificial Intelligence "Patterns in
Contexts" continued...
* A pattern is a collection of symmetries, where each partition section of
data of every symmetry in the collection corresponds to another partition
section of data in that collection, or sometimes corresponds to a piece
or pieces of data in another collection (or other collections) which may or
may not be part of a similar symmetry in that other collection.
If data has recognizable features, it is a pattern. Repetition is what
makes a symmetry, and is what makes a pattern's features recognizable.
Unique partition sections of data are the atomic elements that a pattern's
features are composed of. A symmetry is a type of repetition,
but a repetition isn't always a symmetry (see metaphor definition below).
* A context is a map of patterns within (thus bounded by) either a set or
stream of data in which other patterns are ignored or are not apparent.
Or a context bounded by a larger pattern than the map itself
(which itself is a pattern that may or may not be part of the larger pattern),
such as the ordinal of the map or a pattern larger than the boundaries
of the scope. There can be many parallel streams, waves, or sets of data
in, traveling through, across, or around the self-updating mapping of
patterns which is chosen to be the context.
Sometimes the corrresponding partitions of data that make up a repetition
are translated by some pattern with each iteration, such as in a methaphor.
Yet similarity remains apparent (identifyable by some means).
Again, I believe that information is patterns in contexts, and that
information is metaphoric in nature. Tip: If confused by this write-up
of my premise, try reading the sentence in reverse order then back through
again.
11/24/2003
Information is a symphony of symbolism and symmetry.
12/23/2003
Information, by it's very nature, is a division. Yet it strives to become
whole again, and at the very least, to become balanced.
4/7/2004 Category Theory: Abductive context changing using identified
metaphoric patterns. Some dimension additions for alternating or specializing
the application of a pattern or set of patterns:
- Location
- Relative rate, relative timeline framework
- Newly recognized relations found under sequential and parallelly recursive
brute force and intuitively adaptive experimental logic search strategies --
Yields hypothetical considerations which can be temporally prioritized and
recursively checked and updated from state to state and organized
intelligently by current 1. depth, 2. branch size, 3. branch cardinality
(alpha-numeric, etc), 4. task growth rate, and 5. average task completion rate
(for scaling computability).
When you figure out why a variable is a variable in a particular way, that
understanding becomes a new relation to consider, which in effect and affect
either increases or decreases the dimensionality of the variable's context.
Some dimensions that are added usually increase task completion rate (such as
specialization) other dimensions that ar added usually increase task growth
rate (such as broadening the context or broadening the number of class
categories to consider). Generalization can in some cases merge categories,
classes, and/or contexts, or blur them for simplicity, and can increase or
decrease completion rate.
Generalization is useful for experimentation.
All truth is but an approximation of a deeper truth.
A pattern is like a function, and a context is like a field.
Each has relations, variables (when thought of metaphorically),
and often the potential for variations and unconsidered variables of
dimensionality.
5/15/2004
A working definition of the mystery of consciousness might be ascribed to the
interplay between 1. perspective, 2. priorities, 3. intentions, and
4. awareness; all of which depend on the flexibility, state, and mechanisms of
belief held by the subject.
5/16/2004
My data symmetry section analysis technique for perception through patterns in
contexts may be able to play a key role in automating axiom and theorem
discovery for any given context (i.e. contexts such as the integers, the
reals, wavefield analysis, map data, behavioral intention charts,
language/speech modeling and representation, transform sequences, etc.).
Any pattern discovered within a particular context can be applied to any of
the known axioms and theorems of that context, and patterns that are
discovered can sometimes be related to undiscovered axioms in that context.
Anytime an axiom or theorem is discovered in a context, the entire context is
redefined (as well as its subcontexts), and in doing so, its scope is
narrowed. Choose -> Search -> Experiment -> Classify -> Test -> Prove.
Choose/define context -> search for patterns -> searchfor patterns that relate
discovered patterns -> postulate a classification for each discovered
relation.
For each relation, if a classification category does not exist that
closely matches the relation, then further experimentation, context choosing
(add and/or subtract context dimensions), and pattern searching must be done,
starting with the characteristics of all partially matched categories, until
an accurate or exact classification or category definition can be derived.
After the relation's category is realized, search for more examples of that
relation and derrive a proof of it. If the relation can be proven to be
applicable to all patterns in a given context and all subsets of that context,
it can be said to be an axiom of that context.
8/3/2004
Context can be thought of as a network as well as a shell that encompasses
abstract nodes. The context of a set is merely its powerset, that is, until
relations ar applied. I don't believe in randomness, but I do believe that
some contexts are larger or deeper than the scope of our perceptions.
8/4/2004
A context can also be thought of as a network of patterns, or even the network
of relations that tlinks patterns. But when relations are applied to a
context, it becomes an organism. An organism that is capable of translation
(metaphoric operations), modification (adaptationn), division
(duplication/reproduction/partitioning, and/or growth and association with
other contexts.
8/5/2004
There are patterns, and they exist within and between/across contexts, and
there are relations that act as reasoning engines that operate on the systems
of patterns and contexts.
Patterns can have analogue distortion, digital distortion, or metaphoric
distortion. Contexts can be approximations of larger contexts, and
elaborations or extensions of smaller contets or extensions of other contexts
in general There can be relatively unique (somewhat unique, minimal
commonality) patterns and contexts. Note: the word "commonality" is based on
the greek root "monality", which is the "commonality" of the prime numbers.
Each prime number is a "co-monality." This can be visualized in terms of
geometry, to some extent. Every prime number is balanced, and is symmetrical,
and contains a unique number of dimensions, which are also unique kinds
of dimensions. Patterns and contexts and relations can also be symmetry pieces
of other patterns and contexts and relations, regardless of whether or not
they are distorted in any given state or piece or part or linkage.
8/30/2004
Every context is founded on its own set of axioms and theorems, and adding an
axiom to or from a context's foundation fundamentally changes the context
profoundly, yet some structures may remain un-affected.
(*Note many of these notes may become invalid or ridiculous as you read more,
so mental filtering may be necessary.)
8/23/2004
This is a quote from my journal.
""Metaphor" is a relational model of recursion, where the circular
reasoning (in recursive definitions & recursive functions) cross-relates
the elements of definitions & functions from multiple (or different)
contexts. That is why cross-domain relations are so crucial to the metaphoric
representation of knowledge and knowledge systems (logics)."
8/26/2004
"I also believe that information is metaphoric in nature (has algebraic
interconnectivity), and that it can be represented as a composition of
patterns in contexts, where the contexts themselves can be patterns, and the
atomic elements of each pattern are composed of symmetry sections
(partitiopns of data, where each partition is part of a local or dislocated
repetition (a symmetry, and algebra)). And it is only through the repetition
of a data section that part of a pattern can become recognizable from
apparently random white noise. Randomness and white noise are probably
patterns that are larger than the scope of our perceptions, so the data
appears random.
And I say that metaphors can be represented geometrically because all of
the prime numbers (the balance points in the universe) are symmetrical when
represented geometrically, and it is likely through primarily symmetrical
sensory and cognitive structures that our minds can interpret information.
And I think of metaphors not as A=B, but more like the similarity of the
juxtaposition of A's elements in the context of B, and B's elements in
the context of A, in terms of general systems theory.
I equate truth with workable patterns that become more and more refined
and defined as they get used. I believe that all truth that we are capable
of perceiving is but a small approximation of the whole truth. And that
the truth/patterns that we are capable of using is often subject to perception
within varying contexts. But there seem to exist connections between
information none-the-less, through whatever means. Possibly since (in my
opinion) everything came from oneness)."
9/13/2004
Scope & Context -> Boundaries and Restrictions/Limitations
Class -> Purpose
Type -> Syntax
Pattern Definitions -> Semantics
Data Element Groups -> Configurations (Data Maps & Dependencies)
9/22/2004 Patterns in Contexts Cognition Kernal
Database -> Metabase -> Context Rotator -> Experiment Application Field
Expandable
Adaptable
Translatable
Summarizable subjectively/objectively
======>
Metaphoric Linkers
Patterns Toolkit
Augmentation Socket Parameters
Analysis Scope Dimensionality
of (Input/Internal perspective "eyes")
Geometry & quantitative & qualitative properties of simultaneous interrupts
and their instantaneous functional interrelations and interactions
across multivariate sequence states (such as time & symmetry equivalencies).
*Every set state is but an approximation of the possible combinatorial
translations.
11/20/2004 Epistemology thoughts on Metaphor Abduction
Metaphors hide cross-domain relations between generalized nouns,
adjectives, and systems within a semi-subjective context of perspective.
The descriptive mappings of metaphors and multi-layered metaphoric operations
are generallymore foundational than their analogical counterparts, as the
metaphoric objects and relational context is generalized (from set, type, and
categorical specifics), which simplifies the computational complexity of the
models' qualitative factors, and provides new bases for consideration and
re-application of data, relations, and knowledge. Metaphor generation provides
the architectural basis and objective of considering newe relations and data
experimentatiopn for deriving and arriving at new models of understanding.
data --> context unknown
patterns --> hypothetical contexts
relations --> categorical context parsing
metaphoric relations --> cross-domain functions across contexts
specific knowledge --> contextual scope focusing/narrowing
analogies --> applies metaphoric relations to different examples of specific
knowledge for partial transitivity
new knowledge --> modifies existing contexts to incorporate new axioms.
4/6/2005 Prioritization and choice in decision systems
(Part of a reasoning engine.)
----------------------
New action (such as prioritization or actual action)
^---^
Evaluaction ->criteria
^---^
outcome
^---^
choice
^---^
initiative factor(s)
^---^
prioritization
^---^
evaluation-->criteria
^---^
possibilities
----------------
4/24/2005
Inventing industries with patterns in contexts
What the world needs more of in order to support the ever rising
population levels, is more industries. An entire industry can be created
simply by developing a new kind of alagorithm, or an algorithm that creates a
niche for people to fill with services or products.
**********************
An algorithm can be developed by applying an axiom to a new context.
**********************
This may require forming or describing a new context or kind of context, with
intentions and expectations and attributes or properties in mind, as axoms are
chosen and adapted to make that possible. Theorems can then be derived from
those axioms, that are specific to that context, and when possible, they can
be metaphorically related to theorems in other contexts. This is the basis for
the patterns in contexts model for creating new information. It relates
directly to abductive reasoning, analogical reasoning, and cross-domain
relations.
Axioms depend on which dimensions they can exist in and apply to.
For they are the links that connect different dimensions, parts of dimensions,
and sets of dimensions, with the goal of unique lowest-terms representation.
Usually they incorporate at least some implicit knowledge or material
structure in their model.
Copyright 5/1/2005 Justin Coslor
Property grouping axioms in cross-domain relations.
(See diagram.)
1. All variables have properties.
2. All properties are independent of their variable's context(s).
3. All properties have some combination of qualitative relations, quantitative
relations, existential locations, and existential conditions.
4. Every variable exists within a context and can vary from context to
context.
5. Contexts are composed of networks of patterns, patterns are composed of
networks of variables, and variables are composed of networks of properties.
6. Information can be represented as patterns in contexts, and in that way it
can be represented metaphoricly through analogical reasoning and abductive
reasoning. Relations of various kinds, location(s), and condition(s) (apon and
of) exist at all of the various levels, and those are the data access points.
As writer and owner of this piece of intellectual property I hereby declare it
universally free for use and modification, except I don't condone it's use in
weapon systems or for deception of any kind. This legal agreement cannot be
modified ever, and all modifications of this logic and/or data are bound by
this same agreement. You may sell applications and/or services that use this
logic (or its modifications) but you may not sell the logic itself and you may
not try to prevent others from understanding or using the logic in any way
except if they try to use it for deception or military weapon applications.
Sincerely, Justin Coslor. May 1st, 2005.
Copyright 8/5/2005 Justin Coslor
Augmenting Ideas: Generating New Perspectives on Information
Today in the 61C Cafe I was talking to my friend Jason Bacasa telling him
about how I come up with ideas. Besides keeping an ever growing network of
questions in the back of my mind, I take a topic or generate a topic by
combining keywords, and then think about how that topic is typically
represented, then I try to epistemologically dissect that representation and
then rebuild the content using different, if not more foundational
contextualization of those concepts. Then I go off on a tangent exploring the
most interesting parts by associating other concepts, patterns, contexts,
and operations to the new representation of the concepts in the original
topic.
It is often very valuable to have alternative representations of ideas
and concepts and topics because each representation can yield a useful
perception. If there is any word sense ambiguity, or use of metaphor,
then each alternative representation can yield many perceptions, each of
which could uncover previously unseen or unconsidered aspects of the topics,
ideas, and concepts. So in the end, exploring and mapping out alternative
representations of concepts, ideas, and topics is a way to augment their
knowledge base, by generating new perspectives on the information, which
can generate entirely new contexts, which can generate entirely new
knowledge bases, by treating all information metaphorically. People are
currently very good at metaphoric interpretation and analogical reasoning.
Computer programs currently are not. It's the next step towards
computational methods of abductive (round-about scenic-route) reasoning.
Anyway, Jason said I should make a program that does what I do, i.e.:
a program that recontextualizes information from different perspectives
of association, sort of like a choose-your-own-adventure story, but more like
a choose-your-own-perspective program. Like a computer program that generates
alternative representations of ideas, topics, and concepts. Or even more
generally, a computer program that generates alternative representations of
patterns (thoughtforms) in a variety of contexts (settings).
Copyright 8/6/2005 Justin Coslor
Epistemology Systems
Categories, and complete dictionaries as foundations. Quantified objects
(and systems) can be juxtaposed into relations that balance alternative
representations of objects and systems via a structural or syntactic
methodology that acts as a transformation into some of the possible
alternative representations of the quantified objects and systems.
Algebras as alternative representations of information. Algebras can
rename, or point to representations of information, as well as interconnect
and dissect informational objects and systems. All objects and systems
are named.
Simulations, recontextualizations, and "polymachines" as alternative
models of systems.
Proof is contextual, in other words: proof is dependent on perspective and
representation. In much larger contexts than the original context in which
something was proven, most "proof" becomes incomplete or uncompatible, and
sometimes even false if more foundational epistemological structures are
found to have been overlooked. Proof is complete, logically consistent
introspection of perceptions of concepts.
Any given proof is only applicable to specific axiom sets. I.E. a proof
based on one axiom set may be incomplete or uncompatible or even false in a
context composed of a different set of axioms. Therefore concepts must be
analogically translated into other contexts and their translations must be
formed concurrently with their proof validity in their new context, as a
best-fit categorical search procedure. The proof is a complete, concise
system, so the proof in it's new context can be considered to be a
polymachine, since it is an alternative representation of that system. A
polymachine is a set of cross-domain relations that operate on
analogically-matched patterns from an original context to a new context, and
represents an alternative form of a system in a different context.
Polymachines are created by inductive, deductive, or (in the case of
analogically translated proofs) abductive reasoning. Cross-domain relations
are relations that analogically match the domain of a relation in one context
to the domain of a relation in another context whose range approximates the
same infrastructure and quantitative parameters while leaving the qualitative
parameters categorically open-ended; they are a form of analogical reasoning.
Input Devices->Internal model buffer->Association and repetition
filter->Analysis/comparison engine->Perceptions on experience->Algebraic
Conceptualization->Character sets and dictionaries, or number systems and
axiom sets -> statements, arguments, inquiries, propositions, implications,
operations, filtrations, combinations, exegesis, dissertation, assignments,
contextualizations, templates, associations, compositions, dissections,
introspections, modifications, adaptations, introductions, translations,
transformations, distortion, refinement, recontextualization, proof,
mapping, search, buffering, sorting, indexing, encoding, decoding,
regulation, pattern formulation, trans-substantiation (joke), frollick.
Copyright 5/23/2005 Justin Coslor
Some definitions for patterns in contexts theory
Metaphoric objects are informational objects defined by their
relational properties. In relational contexts, sub-contexts of each
property are independent of the application context. Qualitative factors
are computed by mapping and defining a lexicon of their properties.
Qualitative factors are reflective and algebraic usually. Quantitative factors
are computed by counting and performing materialistic operations on them,
and mapping them in that way. Quantitative factors are materialistic and
geometric usually.
Copyright 8/2/2005 to 8/3/2005 Justin Coslor
Cliff Partitions
Cliff partitions are perceptual references that distinguish deeply layered
patterns from surface patterns, much like a cliff wall bordering the ocean.
In the ocean, every couple of feet down an ocean wall is a new layer, much
like how layers of pixel groups can be laid out on a visual canvas, with some
stacked up several layers high on an edge.
Cliff partitions are essential markers of where a topology has a steep
slope that may or may not be an overhanging awning above a hidden hollow or
cave. In topology, cliff partitions are useful for analyzing the depth
perception of a view.
In linguistics, cliff partitions may indicate a sentence that is placed in
the wrong order, or it may indicate a sudden change of topic, or a jump from
one perspective of a context to a deeper or more superficial depth of
perspective of that same context. Cliff partitions in linguistics may also
indicate the boundaries of a given context, where one context ends and another
begins. Cliff partitions are only conceptual perceptual references in
linguistic domains, as writers and speakers linearly paint a nonlinear
picture with their words.
Copyright 5/23/2005 Justin Coslor
Object-oriented processing
Grids (a.k.a. manifolds), networks, and gridded networks all can house
patterns in contexts of information data sectors as the representation of
knowledge (knowledge is information that contains meaning). Grids, networks,
and gridded networks are materialistic operation spaces for knowledge
representation, whereas the notion of "patterns in contexts" are the Platonic
operation spaces that form the meaning behind the scenes on the materialistic
operation spaces. Identifying the representation of knowledge in an operation
space as "patterns in contexts" and specifying the details allows us to work
with the information in an object-oriented manner.
Copyright 5/27/2005 Justin Coslor
Simulated Models and Utility Axioms
If a network or grid is composed of N elements, then it is capable of
simulating every possible permutation of those elements by forming internal
networks and sub-networks (& grids). Grid networks allow for an infinite
number of combinations to be simulated though, but only some simulations are
of any use. Maybe there are utility axioms that can be defined to tell us what
classes of models contain useful representations. It seems like some factors
that might determine whether or not a model is useful would be:
1. Compatibility with existing useful models.
2. Novel representation or novel perspective.
3. Incorporation of new information.
4. Novel capability.
5. Ability to link two or more other models together.
6. Ability to prune other models.
There may be many more factors directly related to evaluating the worthiness
of a model. Simulation allows for recontextualization of models and problems
and systems.
Copyright 5/22/2005 Justin Coslor
Operation Spaces continued - Tomographic Data Structures
In the gridded network system, as described previously, a multidimensional
array is built between selection of nodes in a network, where elements of this
array can be used to build internal networks between the primary node anchors
of the array, or between other nodes in other networks --as in cross-domain
relations. This process can repeat to an infinite depth, in the order of
network node to array anchor to array node to tomographic network to
cross-domain relation network to array grid, cyclically. This is a way of
creating tomographic data structures of an infinite depth and of infinite
permutations, due to the potential for infinite depth, all without adding any
extra primary nodes. Every array element and every node represents a relation
to or between their anchors or parent nodes.
Copyright 5/21/2005 Justin Coslor
Operation Spaces: Grids V.S. Networks
Rows and columns and layers are dimensions of a grid, but dimensions can
also be parts of an N-dimensional array. Each of the dimensional intersections
form a unique partition that relates or is categorized by it's parent sets'
position along their own sequences. So in this way, elemenets on a grid
(i.e. in an array) come from multiple parents, wheras elemenets in a network
can often come from only one parent (an injective branch). However, in some
networks, such as where a planar geometry can exist by the interconnection of
more than two nodes, multiple parents can be a grid of subspaces between the
nodes on the plane that they make, and in those subspaces multivariable
position and quantized quality relations can be said to exist, that are
anchored to multiple origin points (each vertex be treated as an origin,
and angles between them only serve to define the partitioning of the planar
grid). I'll call this kind of transformation of a network "a subspace grid
of vertices". Maybe this combination of a grid network can enhance the
operation space by making any nodes on a network able to be related to
eachother, in grid format, between particular data sections on the subspace
grid as well as between other primary nodes.
The other kind of operation space is the Swiss cheese like structure that
surrounds a subspace geometrized grid transformed network. The inner edge of
that space is where one context ends and other contexts may begin to exist.
Copyright 5/14/2005 Justin Coslor
Key axioms and branch axioms in pattern collections.
Patterns are composed of smaller parts, with the smallest parts being
repetitions of unique elements in which no sub-patterns are apparent; also,
these smallest parts exist and their repetitions make them algebraicly
recognizable due to certain axioms, which act as fundamental truths
(self-evident assumptions) for which no proof is said to be needed.
This being said, we can say that all patterns that are unique in some
manner must contain at least some unique axioms, and if we look at a
collection of basic patterns and determine what is unique about each one
and what is in common between them, and then figure out how those
similarities and differences ar ordered on an axiomatic level, we may discover
key axioms and branch axioms which can be represented in a nodal network
graph.
The value of this is that we can then understand, at the most basic level,
what makes a pattern exist, what makes a pattern recognizable and similar to
other patterns, and what makes a pattern unique.
We can use that understanding to select axioms suitable to generate a set
of patterns with a measurable degree of flexibility/adaptability, to use
in constructing a system of perception, similar to a painter mixing paints on
an artists pallete, while he mixes concepts in his mind's eye.
Copyright 11/7/2004 Justin Coslor
Hypothetical Relation Highlighting in Undefined Data Sets:
If categorical names have been assigned to finite elements in a domain,
the rest of the data in the set can be hypothetically considered to be
relations or parts of relations (on those elements and elements not in that
buffered data set). Or they may be elements of categories you don't yet
recognize or know of yet.
9/23/2004 Justin Coslor
Am I reinventing the wheel?
Today while studying a diagrammatic map on "Can Computers Think?" that
Seth Casana gave me I learned of work that has already been done in Artificial
Intelligence that is very similar to some of the concepts that I came up with
on my own.
For instance, there has been work done in the area of making computer
software that can understand "analogies". That is very similar to my concept
of "metaphoric operations". Also, in 1989 in seems, Keith Holyoak and
Paul Thagard created ACME, which is a connectionist network that discovers
"cross domain analogical mappings." That soundsd just like my concept of
"cross domain relations for alternative route mathematics", that I have
written about prior to reading anything about it, and I came up with it all on
my own earlier this year. Here are some Analogy Systems:
Copycat - Douglas Hofstadter and Melanie Mitchell 1995.
SME - Brian Falkenhaimer, K. Forbus, and D. Gentner, 1990.
ACME - Keith Holyoak and Paul Thagard, 1989.
8/20/2004 Justin Coslor
Programming
In the preface to the introductory computer programming book
"The Little Lisper" second edition ISBN 0-574-21955-2 it says: (that in LISP)
"the primary programming activity is the creation of (potentially) recursive
definitions." Now to me, that sounds like the main task (and goal) is to map
out and/or define patterns that are either finite or infinite and to put them
into a relational context that is capable of transforming incoming data
patterns by relating them to stored data patterns, so that the output can be
1. represented, 2. stored, and 3. used/manipulated. I believe this because
nothing is more recursive than a pattern (nothing is less recursive than a
pattern as well, except that which is totally random). Patterns always exist
within a context or contexts, otherwise they are not recognizable and look
like random garbage (see Godel's Theorems). On page vii it also says that
"LISP is the medium of choice for people who enjoy free style and flexibility.
LISP was initially conceived as a theoretical vehicle for recursion theory and
for symbolic algebra." (and likely Lambda Calculus & the EMACS environment for
Artificial Intelligence)... LISP syntax looks very similar to my old nonlinear
style of thought notation, with its parenthesis within parenthesis (which was
good for scaling depth on tangents and concept descriptions).
Copyright 8/4/2005 Justin Coslor
Programming Languages
"Programming languages are formal languages that have been designed
to express computations." - How to Think Like a Computer Scientist -
Java Edition
In other words, programming languages are mappings of balanced processes.
The flow of any kind of process can be mapped, if not only approximated by a
systematic contextualization of patterns and relations involved in the
process. Every system is like a state machine in motion, where the elements
and operators are encapsulated by their interconnectivity via
contextualization, which is a form of perspective of finite scope.
Formal languages have fully defined axioms, and are consistent and
complete in the mechanics of their methodology. But what is the methodology of
mappings of balanced processes in general? The universality concept applies to
them: they are consistent and complete because they are balanced about a tight
contextualization, where the interconnectivity of the process's elements acts
like a fulcrum (when thought of quantitatively), with no element left
unconnected. That's why patterns in any context can be transformed through
operations into different patterns, so long as there is a method of
representing both sets of patterns. The balance comes from having multiple
methods of representing each state of the elements in the process. The mapping
comes from being able to contextualize the processes, which is only possible
if the processes have finite scope, and are completely defined (thus
interconnected), and must be systematic (thus logically consistent) in
order to be precisely mappable with regularity throughout their states of
operation.
Copyright 9/22/2004 Justin Coslor
Patterns In Context Cognition Kernel
[Database]-> [Metabase]-> [Context Rotator]-> [Experiment Application Field]
-----------------------
The following are *a. Subjectively and *b. Objectively
1. Expandable,
2. Adaptable,
3. Translatable, &
4. Summarizable:
------------------------
Metaphoric Linkers
------------------------
Pattern Toolkit
------------------------
Augmentation Socket Parameters
------------------------
*Considerations:
-----------------
I. Analysis
II. Scope Dimensionality (of input/internal perspective "eyes")
III. Geometry & Quantitative & Qualitative properties of simultaneous
interrupts and their instantaneous functional interrelations and interactions
across multivariate sequence states (such as time & symmetry equivalences).
**Every set state is but an approximation of the possible combinatorial
translations.
Copyright 6/13/2004 Justin Coslor
Complexity
Commercial or proprietary software is surjective or injective, but free
open-source software is bijective.
Part of the FRDCSA Tutorial (Free Research Database Cluster Study and
Apply) on frdcsa.org says a blurb from an ACM paper about measuring the power
of a set of axioms in order to measure the information contained within the
set of theorems that can be deduced from those axioms. It says that one can
only get out of a axiom sets what one puts in. The paper says something like:
"If a set of theorems constitutes t bits of unique information, and the set of
axioms that the theorems are based on contains less than t bits of unique
information, then it is impossible to deduce those theorems from that set of
axioms."
My friend Andrew J. Dougherty of FRDCSA says that to understand the
general necessity of having more software, simply replace "theorems" with
"problems", and "axioms" with "programs", and "deduce" with "solve" in the
previous statement. Doing that we get: "If a set of problems constitutes t
bits of unique information, and a set of programs contains less than t bits of
unique information, then it is impossible to solve these problems using just
that set of programs. By "problems", I think he means "explicitly defined
problems", because an explicitly defined problem is a program that has yet to
be executed. Abduction may be necessary to define all of the elements and
operators of a problem in the process of turning a problem into a program.
I say, replace "theorems" with "context", and "axioms" with "patterns",
and "solve" with "create". This yields: "If a set of contexts constitutes t
bits of unique information, and the set of patterns that the contexts are
based on contains less than t bits of unique information, then it is
impossible to create those contexts from that set of patterns."
Copyright ?/5/2004 Justin Coslor
This is part of my method of knowledge representation for my
epistemological representation of artificial intelligence through Patterns in
Contexts. Contexts come from patterns that are combined. There can be patterns
noticed in the cross-examination of different contexts, but those "patterns"
are elements of a greater scope of context than any of the contexts being
cross-examined, that is to say, when those cross-context patterns are not
noticable when only examining any one of those contexts in relation to itself.
This method of knowledge representation may hopefully prove to be useful in
the abductive search for new axioms within and across representable contexts.
A context is represented by its systems of patterns (a.k.a. it's system of
axioms).
Copyright 6/5/2004 Justin Coslor
New patterns can be discovered by experimenting with data sets: analyzing
them in relation to metaphoric operations on other data sets. Metaphoric
operations are operations that translate, juggle, predict/locate, and/or
transform specified elements across specified contexts.
Copyright 6/7/2004 Justin Coslor
New metaphors can be discovered by combining axioms that come from
multiple number sets, orderings, and/or algebras. Metaphors are esoteric
relations. The application of a metaphoric operation on a data set sometimes
results in the discovery of new axioms through the new perspective's set of
relations brought about by the application of esoteric relations.
Metaphoric perception is all about cross-domain relations. This is because
the application of metaphors brings about both:
1. relations between the range of the metaphor and the range of all applicable
operations (operations of applications) of the data set, and
2. new cross-domain relations (new domain perspectives) for both the
operations of potentially all applications of the data set; and sometimes new
cross-domain relations and new ranges for the system and set of relations
that algebraically defines the metaphor (when applying the unmatched relations
that are not bijective of the operations of applications of the data set)
metaphorically (i.e. algebraicly to the metaphor).
Copyright 6/5/2004 Justin Coslor
Linker patterns
Linker patterns require both an observation buffer (that is at least of
equal size to the sum of the contexts to be linked), and linker patterns
require an operation buffer that is at least as big as the observation buffer
(though far larger is necessary for some observations, even though the amount
of data that ends up in the operation buffer may be far less, in some
instances, than the amount of data filtered out of the sum of the contexts
into the observation buffer).
Data gets filtered out of every applicable context by the linker
pattern's "filter specifications", right into the linker pattern's
observation buffer. Then the linker pattern's set of metaphoric patterns
operates on the observation buffer one at a time or in parallel, but inside
the operation buffer.
The linker pattern contains a set of metaphoric patterns whose elements
are referenced algebraically to the applicable data elements present in the
observation buffer for every possible metaphoric pattern combination present
in the linker pattern innately.
Metaphors which are algebraically a complete set of elements to
applicable/valid data elements are used in the observation buffer, then inside
the operation buffer they perform their calculation (translating, juggling,
and/or transforming of the data section by the metaphor) and the linker
pattern then places the output in an organized form (so it can be referenced
later), and those outputs are placed into a buffer called "the unified
context" of the original contexts. This "unified context" includes the linker
pattern's filter specifications and metaphor set that was used (i.e. the set
that was computable).
Linker patterns can duplicate themselves to divide up the work of
applying their metaphoric pattern sets to the observation buffers' data (and
they update each other with each successful operation).
Each linker pattern is like a mobile set of operators that copies select
groups of contexts and gives birth to unified contexts (which are new
contexts). It is each linker pattern's unique set of filter specifications
that differentiates one linker pattern from another.
New axioms and theorems that are found elsewhere and within each operation
are found and get added to the metaphor set after the valid discovered
patterns are provably generalized. They are placed in all of the linker
patterns.
Linker patterns can also update each other's set of metaphoric patterns by
sharing ones the other doesn't have, and copying new ones from the other.
The observation buffer performs general quantifier type matching.
Copyright 5/31/2004 Justin Coslor
Patterns In Context Cognition
A context is any specified number set, ordering, or system of numbers that
is representative of something (symbolic).
Take the desired outcome (the goal) and break it down into unique aspects.
Treat each aspect as an element of a context that contains it, or as an
element of several contexts that contain it. each element/aspect may have its
own unique context at first. We will be striving to find the pattern or
patterns that link all aspects of the goal into one context.
A "linker pattern" can be a linker of the contexts that each of the
elements of our goal exist within. Such a pattern links contexts together by
assigning a system of translating, juggling, predicting/locating, and/or
transforming the specified elements across their specified contexts.
This "linker pattern" is metaphoric, and can act as the "unified context" in
which we will search for the aspects of our goal, as well as search for
alternate routes to each of these aspects (for optimization).
After this experimental search has completed and an optimal cross-domain
relation search for shorter routes to each aspect has been completed, we will
have generated the optimal route map to our goal.
Cross-domain relations can also be thought of as possible associations, or
simply as patterns. They can very explicitly depict ambiguous relationships,
such as when they are used with graph theory. Cross-domain relations are
a little bit like surjective and bijective networks in logic but where two
domains lead to the same range in a number set, even when the domains come
from different contexts. They can also be thought of as alternative routes.
Cross-domain relations can be searched for that relate aspects of our goal
that are also aspects of goals that have different unified contexts than our
goal. It's important to mark the optimal routes out of the cross-domain
relations, but keep the other relations (possible routes) for use in future
goal structures. By linking multiple goals in this manner, we expand our
network of understanding.
Copyright 9/7/2000 Justin Coslor
Knowledge Mining
Maybe amassing huge intelligent databases that can draw conclusions and
make abstractions and predictions towards goals that can recognize & ask for
specific data it needs to output one or more units of truth, which could help
demystify fields of study and help major breakthroughs occur, if not by simply
abstracting and relating so much specific data and general patterns in so many
areas; to help bring everything to one's fingertips. A massively parallel
search and correlation engine:
The computer has to be able to understand a goal enough to figure out how
to better understand that goal, so that it can design it's own searches
(determine its own search criteria), and know what a conclusion would look
like and would require to be complete enough to make an abstraction.
What is the criteria of a conclusion? Is a conclusion just one particular
perspective in every situation? How can the perspective be intelligently
shifted and rotated in a search to generate and array of complimentary
conclusions? At what point does the difference in goals generate opposing
conclusions? (i.e. when do conclusions become apparently contradicting when
using the same set of data...) Taking this into account, what difference
in goals produces contradictory conclusions (perceptions) when searching
(parsing) intersecting sets of data?
Input something like a handbook of chemistry and physics, with a goal of
making valid correlations that are not a listed part of that original data
set. Start out with general patterns like input types, leading to language
semantic patterns, leading to patterns of contextual settings, leading to
metaphoric patterns between contexts, *leading to applications of the
generalized raw data to the metaphoric patterns, leading to generalized
predictions of the outcome of the previous step*, parsing the conclusions
listed in the raw data and matching it to the metaphoric mold (the pattern and
logic) that led the contained data elements (or equivalents) to that listed
conclusion. . .in short: enable the software to understand how the data
elements were led to the conclusion listed in the raw data, so that those
patterns (metaphoric molds or logic operations) can be understood enough to be
applied to the raw data in different permutations (ways) to uncover
conclusions of previously unconsidered possibilities. Those patterns and
derivation/discovery methods could also be used as a guide for designing
new patterns built from recombining the old patterns with unique data.
And since unique data almost always is unique due to its being composed of
at least some unique patterns; parsing the old (known) patterns from new
(unknown) patterns might make it easier to clarify what exactly the new
pattern is or at least how it operates (or at the very least, its function).
This is knowledge mining. . .One form of artificial intelligence.
Circular Reasoning:
I aught to look up the dictionary definition of a bunch of the key words
in this.. Hey, why not all of the words? A number could correspond to the
number of words the dictionary definition of each word had to reference
(on every level of the tree of lookup words, each branch pausing when it ends
up at it's own word (a loop)) until the parts of the world applicable to the
context of the base word have been described (mapped)), until an upper limit
has been reached on each word. The highest number out of all of the words will
be the number of words in the applicable dictionary to the context of that
paragraph (no repeats). It will be a complete system of circular reasoning.
A complete system of circular reasoning is where every word in a
dictionary is mapped to at least one other word in that dictionary. Some may
be mapped to every word in that dictionary. A complete system of circular
reasoning is one unit. It is aversion/perspective model of a truth. And
different ones can be combined to build complex systems of truth. Like
mitochondria building cells building structures.
Copyright 7/13/2005 Justin Coslor
Pattern Occurrences
Some patterns are designed or brought about intentionally, and other
patterns are brought about naturally, and others are brought about as an
unintentional consequence of bringing about intentional patterns, such
as in unintentional contexts that are created as a result of layering
patterns, and grouping patterns, and modifying patterns.
Some patterns occur naturally according to certain variable probabilities
specific to their contexts, while others are subject to haphazard creation,
randomness, and free will.
Copyright 7/14/2005 Justin Coslor
Since all patterns are composed of repetitions, and since the repetitions
are what makes the parts recognizable, and since anything that is recognizable
can be considered a pattern, the reference pieces for the parts of each
pattern can be local, as part of the pattern's context, or the reference
pieces can be remote, as part of other contexts that are accessible to the
perception system. The reference pieces are instances of the repetitions that
make the parts recognizable, and are usually cataloged by order of exposure to
them, as well as by associations.
When new patterns are encountered they are either recognized (thus
categorizable), or they are unrecognizable (thus not categorizable) because
their parts and properties are unknown, or they are partially recognized (thus
potentially categorizable and partly referencable). If the pattern is
new and it is recognized, then its parts are already known but are arranged in
a new configuration and with potentially new properties due to the novel
association of its parts.
So basically, once the perception system is exposed to contexts, the
pattern matching/classification system begins its task of dissecting new
patterns into reference pieces, and classifying recognizable patterns into
association contexts and utility contexts, and assigning priority ratings to
everything so that the perception system can decide what to pay attention to.
Priority ratings get constantly updated, and depend on how much bandwidth and
processing power the perception system and reasoning engine have available.
The reasoning engine does all of the heavy calculations, task and priority
assignments, memory management, simulation modeling, and most of the decision
making.
Copyright 7/5/2005 Justin Coslor
ePIC Goal Representation
(ePIC = electronic Patterns In Contexts)
A goal is an abstract construct, and the attainment of a goal is to fill
in all of the details of the goal either:
1. in Platonic Reality (information space), or
2. in physical reality (matter configuration space).
If the details have been filled in in Platonic Reality, then the result is a
simulation. If the details have been filled in in physical reality, then the
result is a working model. A prototype can be a preliminary model or
preliminary simulation.
The abstract construct of a goal is the starting point for changing your
reality in some way. One need only be able to partially perceive of the
abstraction to initiate the existence of the goal, but to fully specify it, a
viable plan needs to be formulated. Usually there are unknown variables in
every abstract goal, and specifying each variable becomes an iterative
process. Often the abstract goal can be stated in the form of a question, and
is the result of the questions that arose from some problem. Often times
further questioning of the problem impetus is necessary to specify the goal
and in doing so, the problem gets solved as the unknowns become decided or
calculated.
Many goals are qualitative/categorical subjective/objective priority
system calculations, that rely on preference, perspective, universal truths,
contextual restrictions, and contextual properties. However, all problems,
goals, and solutions can be represented as patterns in contexts, such as
undecided patterns in partially determined contexts, that evolve through
storing and grouping of categorical, qualitative, and quantitative patterns
across different contexts into an experimental buffer/model space towards the
sufficiently representative construct or construction of networks of
systems of patterns, that satisfy the objectives of the problem and goal,
in the context of the final form of the problem and goal.
The Patterns In Contexts concept is an epistemological language,
which I strongly believe can be used to represent anything, any concept,
and any information of any kind, including first person, second person,
third person information in past present or future tense,
and it adapts well into any other language.
Copyright 6/3/2005 Justin Coslor
Cross-Domain Relations in Analogical Relations
A true cross-domain relation would have two domains that each lead to the
same range. Analogical relations do something very similar to this, however
not quite. In an analogical relation, the relation between the domain and
range of one context is mimicked across a somewhat similar domain and range in
a different context (only some properties need to be similar for the analogy
to be formed, since a barely recognizable similarity needs to exist).
The result is like having generalized an abstraction of the two
domains and the relation, and using that abstraction to perform the
abstracted relation on the second domain in the other context.
--------------------------------\
This is an unfinished work and I disclaim all liability.
--------------------------------
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