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Content uploaded by Egger Mielberg

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All content in this area was uploaded by Egger Mielberg on Mar 04, 2021

Content may be subject to copyright.

Sense Theory

(part 1)

[P-S Standard]

Egger Mielberg

egger.mielberg@gmail.com

22.04.2019

Abstract.

Cognitive processes of human brain are strongly tied to such a well-

known part of the brain as cortex. All psychological, logical or illogical

solutions made by a human being are the result of the cortex. Thus,

the maximum approximation of mathematical theory to the processes

of the cortex can become a good trampoline to the creation of a self-

learning intellectual system, a Real Artificial Intelligence.

We propose a new concept of mathematical theory which gives a

possibility to form, find and separate senses of two or more objects of

different nature. The theory encompasses the knowledge of

cybernetics, linguistics, neurobiology, and classical mathematics.

The Sense Theory is not a part of traditional mathematics as we

know it now, it is a new paradigm of how we can formalize complex

cognitive processes of the human brain.

1. Introduction

While the definition of artificial intelligence is unclear so far, we believe that

cognitive characteristic is the main and first step anyone who creates AI

should start from. This choice has one strong reason. Humans have five

traditionally recognized senses, sight (vision), hearing (audition), taste

(gustation), smell (olfaction), and touch (somatosensation). All the senses

generate data that the brain needs to perceive and comprehend. Thus,

mechanisms of data processing are crucial for such an important human

act as decision making. That is why we consider the fundamentally different

test in comparison with the Turing test.

Test of Three Persons.

The test consists of the following steps:

1. Three persons, Rick, Nick and Dick are taken. By arbitrary choice,

one of the persons is substituted by a computer program (digital

machine).

2. Three persons exchange their dates of birth and start joint

conversation during the next 30 minutes.

3. A text with three arbitrary dates and numbers is exposed to the

persons for reading.

4. Three persons start a conversation about the text during the next 30

minutes.

5. Each person is asked, “Who is the machine?” with one required

sentence of answer explanation.

6. After the exposition of all answers for the persons, they are secondly

asked, “Who wants to change the answer?”.

7. The answers of the persons are fixed and calculated. If the machine

(Rick, Nick or Dick) was not chosen by other two persons

simultaneously, the test is passed.

Remark: During the test, each person (man) can make notices.

Graphically it can be shown as follows:

The passed test means only that the machine is capable of thinking

intellectually. For the level of human thinking it needs to pass the test of N

persons. We called the test of three persons as a test of first order. And the

test of N persons as a test of second order. Importance of intellectual level

for the machine can be compared with a process of approximation. In our

context, the figure inside the machine circle has as many sides as the

number of persons.

So, an intellectual excellence of the machine can be reached as soon as

the internal machine’s figure becomes the circle with minimum error.

2. Problem

Classical mathematics, namely its basis, classical mathematical logic is not

capable of operating with such an object as a sense. In other words, with

the qualitative properties of the object. However, it is crucial for building a

self-learning system of any kind.

In mathematics, as we know it now, there are a number of direct proofs of

the foregoing statement. We will briefly describe here only two of them

which are the basic ones according to the author of this article.

The law of the excluded middle (third).

By simple words, according to this law, if statement A is true then

statement which is opposite to the statement A is always false.

For example:

Statement A - "Bob is a stupid man".

Statement B - "Bob is a smart man".

Statement C - "Bob is a good man".

Statement D - “Bob is a bad man”.

So, if someone states that Bob is a stupid man, according to the law of the

excluded middle Bob cannot be simultaneously a smart man. At first sight,

it seems logical but as soon as we list the characteristics (properties) of a

stupid man as well as a smart one we will necessarily bump into a

contradiction. As a matter of fact, part of the properties of the smart (stupid)

man can be the same one of any other (not stupid nor smart) man. In terms

of mathematical logic, we have:

or

where without fail.

Further, if someone states state that Bob is a stupid but good man, we

have:

and

or

But in the practical realization of any intellectual system, we frequently

meet the situation when a man has several properties simultaneously or

property that does not have direct opposite value. For example, the

following expression cannot be firmly established or refuted:

or

where S – statement “Bob is a shapely man”, G – statement “Bob is an

elegant man”. Thus, the classical mathematical logic is good only for

homogeneous objects that do not have qualitative properties.

Gödel’s incompleteness theorems.

Gödel’s theorems are a good example of the absence of a clear and single

definition of what "negation operation" is all about. In classical logic, it is

primarily used in the context of two possible values, "true" or "false". In this

way, only propositions that can be evaluated by two states are possible for

operation and analysis. Therefore, the negation operation is good if and

only if the outcome of any proposition can take two opposite forms.

One of the Gödel’s theorem says that we cannot derive two formulas

simultaneously, where .

But what exactly does “” mean? Suppose we have the following

series of values:

Thus, all the above values are true. Then, “” should mean situation

when the values are false. In other words, is undefined for . For

example, if we take the following simple formula , ,

then . In case of “” means opposite value,

. Thus, we have two formulas and two sets. It clearly shows that

Gödel’s theorems as well as classical logic (its operators) are primarily

focused on Boolean domain. In other words, it works only when a bijective

function is defined.

In the context of the Sense Theory as well as any practical realization of a

semantical live algorithm, there are more than two states for an object. For

example, the object “device” can have more than one qualitative properties

such as "plastic", "thin", etc. But in the context of sense, it is undefined if it

does not have a single property. In practice, the Sense Theory operates

multivalued functions.

Resuming above-said and what can be derived from it, all logical operators

of the classical logic are primarily suiting to bijective sets. But it is

absolutely not suited to the nature of cognitive processes as well as the

Sense Theory.

3. Solution

At the core of the theory lies an object which has a qualitative property (‘s).

The object can be the nature of any kind. For example, a word “device”

presents a template of some element with no relationship to any categorical

context. As soon as we prefix the word “medicine” to the word “device”, the

corresponding context becomes evident. In this case, the word “medicine”

is the qualitative property of the word “device”.

,

where – object,

index – quantity of object properties.

In case of prefixing more different-in-sense-words, we get the following

notation:

or

,

where – total quantity of object properties.

In terms of linguistics, the property of any object can be any part of speech.

The object that does not have any properties is called zero object:

or

device

A Sense Set (SS) of the objects is a plurality thought of as a sense unit.

Let us consider the following several objects:

,

,

,

,

In the next step, we will consider the all above-mentioned objects as

properties:

This set of properties forms a No-Sense Set (NS):

As it was said earlier, an object has a qualitative property (‘s). But some of

them can be a zero ones with no properties at all.

Now, if we start to select a zero object iteratively, with high probability we

will end up with the object “aircraft” or “airplane”.

,

and

or

where S – Sense Set.

Unlike zero object, Sense Set cannot be empty as it is a result of “inclusion”

of two elements, zero object and No-Sense Set.

Definition 1: S is a Sense Set if and only if the following expression is true:

where N,K = {1,2,3,…n}, K N, K,N – finite numbers.

Definition 2:

where N – finite number.

Definition 3: is a zero or empty object if and only if the following

expression is true:

Definition 4:

where N – finite number of objects.

Definition 5:

Definition 6:

where

Definition 7:

Definition 8:

Definition 9: Object A semantically connects to Object B if the following

expression is true:

“Semantic connection” (SC) – is measured by percent. The following

formula is used:

where – number of similar properties of both objects,

– number of properties of largest object.

In order to formulate the first Axiom of the Sense Theory, we need to enter

such definitions as “sense sequence” and “sense limit”.

Definition 10: The set A of elements is a sense sequence if

and only if there is at least a single zero object that satisfies the

following expression:

Definition 11: The sense limit of the set A is the zero object of a Sense Set.

In other words, an object the properties of which are the elements of the set

A is a sense limit of that set.

It has the following notation:

where .

For the subset of the set A, we can have two outcomes:

or

The Axiom of Object Limit:

“Every object of Sense Set consists of two parts, zero object and sense

sequence, where the first one is a sense limit of the second one.”

The following two expressions are equivalent:

The Axiom of Object Equality:

Object is equal to Object if the following expression is true:

or

The Axiom of Set Equality:

The Sense Set is equal to the Sense Set if the following expression is

true:

or

The Axiom of Semantic Union (left-to-right):

1. For any two No-Sense Sets,

and there is such the properties of which are

both, the properties of and :

where K = M + L.

2. For any two Objects, and , there is such that the

following two expressions are true:

3. For any two Sense Sets, and there is such that the

following two expressions are true:

The Axiom of Semantic Subset:

Any Object can be only one of two types of subsets for any Sense Set

:

Subset of first order:

where K = N.

Subset of second order:

where K = N or K N.

The Axiom of Set of Subsets:

“There are at least N+1 subsets for any .”

Theorem (Existence of Set).

“The Sense Set () is defined if and only if there is a sense limit of

.”

Proof.

For any given there are two elements, and by definition. Now,

presume that there is no sense limit of . In symbolic notation, it

presents the following:

and

The latter expression contradicts the definition of Sense Set. The theorem

is proven.

Theorem (Existence of Subsets).

“There is at least one subset for and N+1 subsets for .”

Proof.

Further,

The theorem is proven.

4. Conclusion

In this article, we presented the new “mathematical” theory with own

signature. Unlike classical mathematical or intuitionistic logic, the Sense

Logic which is the basis for the Sense Theory can drastically improve

understanding methods and possible algorithms in the creation of human-

like AI.

We hope that our decent work will help other AI researchers in their life

endeavors.

To be continued.

Appendix

“Semantic Intersection” is commutative for all operands.

Definition 9.

Associativity (“inclusion”):

Associativity (“semantic union”):

Associativity (“semantic disunion”):

References

More than 100 books and articles.