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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