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Content uploaded by Egger Mielberg
Author content
All content in this area was uploaded by Egger Mielberg on Mar 04, 2021
Content may be subject to copyright.
Sense Theory
(part 2)
Sense Function
[P-S Standard]
Egger Mielberg
egger.mielberg@gmail.com
25.07.2019
Abstract.
The Sense Theory is not a part of traditional mathematics. It is a new
paradigm of how we can formalize complex cognitive processes of
the human brain. The basis of the theory is a sense function which
determines sense conformity between a set of objects or/and events
and a single subject (described object/event).
The sense function has a series of unique properties that can help
find associative connections between trillions different-type
objects/events.
By the function, we can investigate a whole process of forming a
single sense of big data set of different business or scientific events.
1. Introduction
It has been far away in the past when a first mathematical function was
formulated. Since antiquity, people used quantitative analysis in their
everyday life. After hundreds of years, a great number of different
mathematical sets and spaces were postulated. But none of them was not
focused on such qualitative characteristic as a sense. In this article, we
describe a new sense-centered function which is the main element of
Sense Space (see next articles).
2. Problem
In the traditional mathematical theory, a function is a relation that
associates each element “x” of a set “X”, the domain of the function, to a
single element “y” of another set “Y”, the codomain of the function. In many
cases, set X, as well as set Y, are numeric. Such function was originally
focused on how the elements of one numeric set depends on varying
elements of another set.
In the advent of AI concept, many complicated and unsolvable tasks were
formulated. However, neither differential calculus nor other mathematical
methods helped to even approximate to a solution of the tasks. The reason
for that is the codomain of the function operates elements of numeric
nature whereas AI operates senses.
Below we introduce a new-paradigm function which is sense-centered.
3. Solution
As in classical mathematics, we need to define a "coordinate" system for
operating with such an abstract object like a sense unit. Unlike rules of
setting a point on the line, in Sense Theory, we need to set three
conditions:
1. sense units (SU) -
units
The table shows how many variants of the sense units can be chosen for a
procedure of sense approximation.
S
O(N)
S
C
2. initial set (IS) is a set of Non-Sense Sets, Object Non-
Sense Sets, Complete Sense Sets and Incomplete Sense Sets which is
used for first iteration of the sense approximation.
3. sense direction (SD): subject or event.
The subject of the sense approximation is any element (sub-element) of
Text Set which is a noun with qualitative properties.
The event of the sense approximation is a thing that happens during some
time.
As well as the coordinate method of classical mathematics, in Sense
Theory, there is own method, the method of sense units.
The method of sense units is the algorithm for determining the meaning
(sense direction) of a text set by a chosen element or group of sense units.
For clarity's sake, we need to introduce such notation as levels of the
sense approximation.
Definition 1: The first level of the sense approximation is a result of the first
iteration of the calculating algorithm where the Text Set is the Initial Set.
Given:
SU – chosen variant of the sense units.
IS – chosen Text Set for the sense approximation.
SD – chosen “subject” or “event”.
Definition 2: The second or N level of the sense approximation is a result of
the second (N’s) iteration of the calculating algorithm where the Text Set is
the first (N-1) level result.
Given:
SU – chosen variant of the sense units.
IS – first (N-1) level result.
SD – chosen “subject” or “event”.
Definition 3: Sense function is a sense conformity between No-Sense
Set (Object No-Sense Set) and zero object .
Notation: or or , where .
Like in traditional mathematics, in Sense Theory we introduce such
notations as to the domain and codomain of the function. In our case, the
domain of is a sense sequence. The codomain of is always one of the
set of . For example, in case of we have
the following functional expression:
or
Specifying a function ():
1. Analytical method [1]:
or
2. Graphical method.
This method shows the depth of the object description as well as the level
of correlation with other objects with the same set of No-Sense Set
elements.
Fig.1
Definition 4: The function that has N No-Sense Set (Object No-Sense
Set) elements and determines only one zero object is a -complete
function.
Definition 5: The function that has N No-Sense Set (Object No-Sense
Set) elements and determines more than one zero object is a -partial
function.
It is obvious that in many practical cases the function graph will be looking
like this:
Fig.2
3. Table method.
This method allows for defining all possible combinations of No-Sense Set
elements that relates to one of Object Set elements.
Sense determinant and limit.
Function
is called sense determinant. It determines a rule of sense
conformity between No-Sense Set (Object No-Sense Set) elements and a
single zero object of Object Set.
Zero object in the following expression:
is called sense limit. It is a maximum achieved level of the sense
approximation.
Unlike traditional mathematics, in Sense Theory, there is no one-to-one
inverse function. Otherwise, from the following formulas:
or
we would get the following result:
or
and
or
The result is a direct contradiction to Definition 1 [1] and the main definition
of the sense function.
Theorem (Equivalence of functions).
“Two functions
and
are equivalent if and only if the following
expression is met:
“.
Proof.
The theorem is proofed by using two Axioms, “The Axiom of Object Limit”
and “The Axiom of Object Equality”.
Theorem (Nonexistence of composition functions).
“There is no composition function
as the following expression does
not make a sense:
”.
Proof.
The proof of the theorem deduces from the main definition of and
Definition 3 [1].
Properties of :
is strictly surjective.
Theorem (Surjection of function).
“Each element of has semantic connection to at least one sense
unit of “.
Proof.
The proof of the theorem deduces from Definition 3 and Definition 11 [1].
Definition 6: In the case when the number of elements tends to N,
the number of semantically connected zero objects tends to 1 or 0. In this
case, is called an semantically increasing function.
Definition 7: In the case when the number of elements tends to 1,
the number of semantically connected zero objects tends to N. In this case,
is called an semantically decreasing function.
Definition 8: is defined if and only if there is a sense limit:
or
Definition 9: is not defined if and only if there is no sense limit:
or
4. Conclusion
In this article, we presented the new “sense-focused” function. Unlike
classical mathematical function, the sense-based codomain of our function
is purely qualitative. It helps design different semantical models for
purposes of any kind.
We hope that our decent work will help other AI researchers in their life
endeavors.
To be continued.
References
[1] E. Mielberg, “Sense Theory”, Part 1, 2019,
http://vixra.org/abs/1905.0105
[2] E. Titchmarsh, “The Theory of Functions”, 1939,
https://archive.org/details/TheTheoryOfFunctions/page/n5
[3] S. Sternberg, “Theory of functions of a real variable”, 2005,
http://www.math.harvard.edu/~shlomo/docs/Real_Variables.pdf
[4] J. Harkness, F. Morley, “The Theory of Functions”, 1898,
https://www.ams.org/journals/bull/1899-06-02/S0002-9904-1899-00671-
2/S0002-9904-1899-00671-2.pdf
[5] R. Remmert, “Theory of Complex Function”, 1991,
https://www.matem.unam.mx/~hector/Remmert-TheoryCpxFtns.pdf
[6] I. Natanson, “Theory of Functions of a Real Variable”,
https://epdf.pub/theory-of-functions-of-a-real-
variable345a969e7942ecf12b11e3b8055dd80725702.html
[7] C. Pinter, “A Book of Set Theory”, 2014,
http://matematicas.uis.edu.co/adrialba/sites/default/files/SetTheoryDover-
%20Charles%20C%20Pinter.pdf
[8] K. Kunen, “Set Theory”, 1980,
https://pdfs.semanticscholar.org/8929/ab7afdb220d582e9880b098c23082d
a8bc0c.pdf
[9] R. Andr´e, “Axioms and Set Theory”, 2014,
http://www.math.uwaterloo.ca/~randre/1aaset_theory_140613.pdf
[10] H. Schwichtenberg, “Mathematical Logic”, 2004,
http://www.mathematik.uni-
muenchen.de/~schwicht/lectures/logic/ws03/ml.pdf
[11] J. Aspnes, “Notes on mathematical logic”, 2010,
http://www.cs.yale.edu/homes/aspnes/pinewiki/attachments/MathematicalL
ogic/mathematical-logic.pdf
[12] M. Abramowitz, I. Stegun, “Handbook of Mathematical Functions with
Formulas, Graphs and Mathematical Tables”, 1972,
http://people.math.sfu.ca/~cbm/aands/abramowitz_and_stegun.pdf
[13] J. Nicholas, J. Hunter, J. Hargreaves, “Functions and Their Graphs”,
