Kashirin I.Yu.

Definition of Hierarchical Numbers

Kashirin I.

Hierarchical numbers are numbers of the form [s] a₀ . a₁ . a₂ . ... . a_i . ... .a_n ,

where a_i are positive integers from the set N = {0, 1, 2, 3...}. s is the symbol of the "+" or "-" sign,

a positive sign may not be specified.

For example, hierarchical numbers may look like this:

0.0.12.48.0 or -2.33.0.0.4 .

The symbol denoting the set of hierarchical numbers is H.

Applied definition . Hierarchical numbers are a branch of mathematical number theory that allows you to adequately operate on the main characteristics of taxonomies. In the theory of knowledge representation of artificial intelligence, hierarchical numbers allow:

− to determine the semantic similarity of concepts;

− calculate indexes of hyperonyms and hyponyms;

− receive regular knowledge structures;

− verify trees of generic and causal classifications.

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)

 Let N be a set of positive integers with elements ni ϵ N, let there also be a highlighted character ".".

The set A = N ᴗ "." ᴗ L is defined as an alphabet with integers n, where "ᴗ" is the operation of combining sets, and L is an empty character.

Then the grammar is:

ĥ → L,

ĥ → h,

ĥ → - h,

h → < n >,

h → < n > . <h>

describes a set of hierarchical numbers H with elements ĥ.

The rule  ĥ → - h   allows for the full range of hierarchical numbers, including the middle negative elements of the number:

0.10.-1.-1.127 

 These numbers are already used in one way or another in the practice of classification or addressing, for example: a universal decimal code or the IP address of a computer on a global network. However, the introduction of

an algebraic system of hierarchical numbers makes it possible to perform operations with them similar to formal arithmetic, and to isolate binary relations for their comparison and analysis of non-trivial properties of operations and relations.

Consider the algebra of binary hierarchical numbers.

Let B be a set of numbers with elements {0, 1}, n ϵ B (n = 0 or n = 1), let there also be a highlighted character ".".

The set A = B ᴗ "." ᴗ  L is defined as an alphabet with integers from B, where "ᴗ" is an operation combinations of sets, and L is an empty character. Then the grammar is:

ĥ → L, ĥ → h, ĥ → -h,

h → < n >, h → < n > . <h>

describes a set of binary hierarchical numbers H with elements of h.

Examples of binary hierarchical numbers can be given: 0.1.0.0.1 or 1.0.-1.0 .

Binary hierarchical numbers 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, this can be represented by a tree spreading in a positive or negative direction (Figure 1):

 Figure 1 – An image of algebraic operations in the form of trees

However, using negative elements can make the meaning of operations more complicated. For example, the generation of the "+" trace may look like this:

0.1 + 1.1 = 0.1.1.1, 0 + 0.1 = 0.0.1

However, the example 0.0.-1 + 1.1 = 0.0.-1.1.1 indicates the presence of more complex tree travel routes using not only descent but also local ascents. This application of hierarchical numbers will be discussed further with specific examples.

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

0.1.1.1-- = 0.1.1, 0.1.0-- = 0.1, 0.-1.-1 -- = 0.-1.-1 .

In graphical interpretation, the number can be considered the absolute index of any vertex, i.e. starting from the top of the tree 0 or relative, displaying the path through the tree from one of any vertices to other vertices up and down.

The absolute index always starts with the character 0.

When solving practical problems, only the positive part of the algebra of binary hierarchical numbers can be considered. In this case, operations claiming to receive a negative index will have a result of 0.

One more rather popular operation can be cited, namely, the calculation of the most common vertex, which is interpreted as a search for a common ancestor of two argument vertices: 

 Figure 2 – Another example of hierarchy

This generalization/multiplication operation is commutative, i.e. a º b = b º a.

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

The important operation is "Ù" as the calculation of the path from the vertex given by the first argument to the vertex given by the second argument. For the previous figure, examples of such calculations could be given:

 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 relation of equality of the lengths of two hierarchical numbers.

  However, such a calculation leads to an unnecessarily complex result.

 Note that the common ancestor for 0.1.1.1 and 0.1.0 is 0.1, from which both 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 “Ù” turns out 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 Ù 1.1.1.0

After considering the semantics of the above operations, we can define a universal arithmetic algebra of hierarchical 

 numbers H:

                                                 H = < H, Ω >, Ω = {+, -, --, º , Ù, ⊕},

where Ω is the signature of the algebra, i.e. many operations.

  All operations are binary, except for "--", which is unary. The meaning of the operation “⊕” will be discussed later, 

 using relevant examples.

 The considered algebra can be supplemented to the algebraic system 

H = < H, Ω, R > by introducing a set of relations R = {< , >, @ , = },

where the relations “a > b” and “b < a”, respectively, “the number a is more complex numbers b" and "number b is 

 shorter than number a."

A brief example of a taxonomy is shown in the figure.

 Figure 3 – Brief taxonomy example

More examples on this site in articles about icf taxonomies.