Kashirin I.Yu.

The idea of hierarchical numbers

Binary hierarchical numbers
Kashirin I.Yu.

The simplest description of the theory of hierarchical numbers

for knowledge representation models.

It can be used to calculate the semantic proximity of words and sentences in a natural language. 

The idea of hierarchical numbers was first proposed in 2020 in an article

 I.Yu.Kashirin. Hierarchical numbers for designing artificial intelligence ICF taxonomies/2020. № 71.  P.71-82 (rus)

They are numerical indices of the vertices of two binary trees: positive and negative with one common vertex 0.

The generation of a vertex to the left of 0 is performed by the binary operation 0+0 = 0.0, the generation of a vertex to the right is performed by the binary operation      0+1 = 0.1.

The generation of negative vertices is performed by the "-" operation, respectively:
0-0 = -0.0, 0-1 = -0.1.

Graphically, it looks like this:

  The operation of generating the "+" trace may be more complicated:

0.1 + 1.1 = 0.1.1.1
0 + 0.1 = 0.0.1

The reverse operation of generation, removal of the terminal vertex "--", is unary:

0.1.1.1-- = 0.1.1
0.1.0-- = 0.1

To consider a number as an absolute or relative index of the vertex of a binary tree is determined by a person solving an applied problem using hierarchical numbers. The absolute index always starts with the character 0.

Only the positive part of the algebraic system of binary hierarchical numbers can be considered. In this case, operations claiming to receive a negative index will have a result of 0.

Calculating the most common vertex is interpreted as a search for their common ancestor:

  0.1.1.1 º 0.1.0 = 0.1.0 º 0.1.1.1 = 0.1

This generalization/multiplication operation is commutative. The operation can also be designated as "*".

Multiplying a positive number by a negative number is always 0.

The concept of the inverse element (number) can be described as follows.

What can be said about the numbers 0.1.1.0 and 1.0.0.1 ?
They are obtained by the complete inversion of atomic elements 0 and 1 in all digits.

Multiplying such numbers will always give a zero result. Obviously, these numbers cannot be considered absolute (counted from the root of the tree).

 
However, the access paths to the terminal vertices for these numbers are completely opposite: if we consider "1" to be a positive choice

and "0" to be a negative one, then any descent in the tree one level lower for the first number will be the opposite of the choice determined by the second number.

When the first number says "yes", the second one definitely says "no" and vice versa.

 
If we always talk only about absolute numbers, then the first digit "0" can be omitted.

An important operation is "Ù" as the calculation of the path from the vertex specified by the first argument to the vertex specified by the second argument.

Examples of such calculations can be given for the previous figure:

0.1.0 Ù 0.1.1.1 = 0.1.0. 0.1. 0.1.1. 0.1.1.1
      0.1.1.1 Ù 0.1.0 = 0.1.1.1 0.1.1. 0.1 0.1.0

0.1.0. 0.1. 0.1.1. 0.1.1.1 @ 0.1.1.1 0.1.1. 0.1 0.1.0

Here " @ " is the ratio of the equality of the lengths of two hierarchical numbers.

However, the common ancestor for 0.1.1.1 and 0.1.0 is 0.1, which is where these numbers begin.

As a result, when calculating the operation Ù, these fragments are omitted for all vertices of the path from the first vertex to the second.

This is necessary to get an idea of the complexity of the path from the first vertex to the second, regardless of the depth of the tree.

Then the correct operation is like this:

0.1.0 Ù 0.1.1.1 = [0.1.]0. [0.1.] [ 0.1.]1. [ 0.1].1.1 = 0.1.1.1

0.1.1.1 Ù 0.1.0 = [0.1.]1.1 [0.1.]1. [0.1] [0.1.]0 = 1.1.1.0

0.1.1.1 @ 1.1.1.0

What is considered a right or left descendant? This is in the application of numbers in practice. First, the left descendant is written, then the right one.

Orderliness needs to be specifically specified.

(Static Dynamics) (Beautiful Ugly) (Good Bad) (Small Big)

(Simple Complex) (Delicious Tasteless) (Friend Enemy)

(Cause Consequence) (Yesterday Tomorrow)

To be continued ...