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Sense Theory
(Part 4)
Sense Antiderivative
[P-S Standard]
Egger Mielberg
egger.mielberg@gmail.com
07.03.2020
Abstract.
Like each neuron of the human brain may be connected to up to 10,000
other neurons, passing signals to each other via as many as 1,000 trillion
synaptic connections, in Sense Theory there is a possibility for connecting
over 1,000 trillion heterogeneous objects. An object in Sense Theory is like
a neuron in the human brain. Properties of the object are like dendrites of
the neuron. Changing object in the process of addition or deletion of its
properties is like forming a new knowledge in the process of synaptic
connections of two or more neurons. In Sense Theory, we introduced a
mechanism for determining possible semantic relationships between
objects by connecting-disconnecting different properties. This mechanism
is Sense Integral.
In this article, we describe one of the instruments, sense antiderivative, that
sheds light on the nature of forming new knowledge in the field of Artificial
Intelligence.
1. Introduction
In traditional mathematics, the antiderivative of a function of a single
variable, for example, measures the area under the curve of the function. In
Sense Theory, the antiderivative of a sense function [2] determines a
possible new knowledge (or describing current one more deeply) by
addition or deletion of the properties. It also clearly shows sense
associations between properties of different objects.
2. Problem
Unlike traditional integral calculus where infinitesimals used, in Sense
Theory, we operate sets (finite or infinite) of possible properties of No-
Sense Set (Object No-Sense Set) for zero-object ('s).
For a sense function defined on , a sense integral of the function
will determine the following three possible cases:
1. , for any subset ( ) or set
(
) (I)
2. , for
any or
(II)
3. , for
any or
(III)
In practice, it is crucial to define conditions on which (I), (II) and (III) met. It
will extremely help understand a new knowledge formation.
3. Solution
For (I) we have the following (
),
In practice, we can also have the following case,
The graph above shows that there is a set of for which the sense
integral of will always be constant and equal to the set.
Theorem (Existence of Integral Set):
“For any sense function defined on arbitrary set of , there is at least a
single set of (n>1):
, where j > 1”
For j = 3,
,
,
,
where .
Thus, we may rewrite (I) as follows
, (A)
for any and
where is
The integral in (A) is a sense integral defined on set of objects or
definite sense integral.
Proof.
The proof of the theorem deduces from Definition 5 [1], Theorem
(Surjection of Function) [1] and Definition 8 [1].
For (II) we have the following (
),
And, expressing through sense integral,
(B)
The integral in (B) is a sense integral undefined on set of (
) or
indefinite sense integral.
For (III) we have the following ( ,
For
,
where .
It is obvious that as as
is .
Thus,
, (C)
and
(D)
The integral in (C) and in (D) as well is a sense integral diverging on set of
and set of
, respectfully.
Theorem (Convergent Integral):
“To be convergent, necessary and sufficient, an integrand of the sense
integral must be ”
Proof.
The proof of the theorem deduces from Definition 4 [1], Axiom (Semantic
Derivative on disunion) [2], Axiom (Absence of Derivative) [2], Axiom of
Constancy [2] and, the rules of semantic normalization [2].
Sense Integral on disunion
Let's to be defined on the set of or . Then for any ()
defined on ( ), where M < N and M N, semantic integral (
) on disunion is
or
where .
Properties:
1.
2.
3.
where , according to Axiom
(Sense Limit of Derivative) [2]
4.
5.
6. ,
if , however there is a situation when
7. ,
if ,
Sense Integral on union
Let's to be defined on the set of or . Then for any ()
defined on ( ), semantic integral ( ) on union is
or
Properties:
1.
2. ,
if
3. ,
where
4. , if
5.
6. , if
however there is a situation when
7. ,
if ,
Sense Integral on property (‘s)
Disunion
Let's to be defined on the set of or . Then for any ()
defined on ( ), where M<N and MN, semantic integral ()
on on disunion is
or
,
where – i-property of ,
.
Properties:
Identical to Sense Integral on disunion provided that:
1, 2, 3.
4.
5.
6. ,
7. , ,
Union
Let's to be defined on the set of or . Then for any ()
defined on ( ), semantic integral ( ) on on union is
or
where – i-property of ,
.
Properties:
Identical to Sense Integral on union provided that:
1.
2. , ,
3.
4.
5.
6.
7. ,
Sense Integral on n properties has the same above-mentioned properties
and denoted as:
and
Sense Integral on object
Disunion
Let's to be defined on the set of or . Then for any ()
defined on ( ), where M < N and M N, semantic integral (
) on object on disunion is
or
The properties of sense integral on an object are identical to the properties
of sense integral on disunion and have a place if and only if any sense
derivative on is exist.
Union
Let's to be defined on the set of or . Then for any ()
defined on ( ), semantic integral ( ) on on union is
or
The properties of sense integral on an object are identical to the properties
of sense integral on union and have a place if and only if any sense
derivative on is exist.
8. Conclusion
In this article, we presented the instrument for dynamic formation of a new
knowledge, Sense Integral. It allows finding semantic connections between
a pair of different objects and between billions of objects of different nature
as well. One of the crucial features of Sense Integral is the capability for
determining latent knowledge of an object.
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”, 2018,
https://vixra.org/pdf/1905.0105v1.pdf
[2] E. Mielberg, “Sense Theory. Sense Function, Part 2”, 2018,
https://vixra.org/pdf/1907.0527v1.pdf
[3] E. Mielberg, “Sense Theory. Sense Derivative, Part 3”, 2018,
https://medium.com/@EggerMielberg/sense-theory-derivative-4d6fddd3b4f1
[4] G. Strang, “Calculus”, 2020, MIT,
https://ocw.mit.edu/ans7870/resources/Strang/Edited/Calculus/Calculus.pdf
[5] L. Loomis, S. Sternberg, “Advanced Calculus”, Harvard University,
2020, http://people.math.harvard.edu/~shlomo/docs/Advanced_Calculus.pdf
[6] M. Bittinger, D. Ellenbogen, S. Surgent, “Calculus And Its Applications”,
Santa ANA College, 2012,
https://www.sac.edu/FacultyStaff/HomePages/MajidKashi/PDF/MATH_150/Bus_
Calculus.pdf
[7] G. Williamson, “Tutoring Integral Calculus”, University of California,
2012, http://cseweb.ucsd.edu/~gill/TopDownCalcSite/Resources/TutorCalc.pdf