Hierarchical Number System (HNS) Overview

• HNS is a family of domain-specific formalisms that encode hierarchical structures across numeral systems, biological taxonomies, algebraic systems, and visual representations.
• It encompasses diverse methods such as numerical-rank schemes in biology, distance-sensitive tree representations, and recursively compressed binary representations that offer efficiency gains.
• HNS extends to hypercomplex number systems and OILU symbolic numeration, providing unique solutions for typification, optimized arithmetic, and energy-efficient display techniques.

Hierarchical Number System (HNS) is not a single standardized concept across the arXiv literature. The label is applied to several structurally different constructions: a numerical-rank scheme for biological nomenclature, a distance-sensitive tree-based numeral system, a recursively compressed tree representation for natural numbers, and, in algebraic papers, hypercomplex number systems; a further related usage appears in a hierarchical, line-based decimal symbolic called OILU (Shipunov, 2017, Rofa, 2013, Tarau, 2013, Kalinovsky et al., 2014, Boiarinova et al., 2021, Mostefai et al., 2021). This suggests that HNS is best treated as a family of domain-specific formalisms linked by explicit hierarchy, rather than as a unique mathematical object.

1. Terminological scope and disambiguation

The cited literature uses the same abbreviation for different technical purposes. In some cases the hierarchy is taxonomic, in others it is combinatorial, recursive, algebraic, or visual.

Usage in the literature	Core construct	Representative source
Numerical ranks	Numerical prefixes encode taxonomic rank from species to kingdom	(Shipunov, 2017)
Distance-sensitive HNS	Rooted symmetric tree; vertex value depends on place values and distance from the root	(Rofa, 2013)
Hereditarily binary numbers	Recursively run-length encoded bijective base-2 trees	(Tarau, 2013)
Hypercomplex HNS	Finite-dimensional algebras, including generalized quaternion-like systems	(Kalinovsky et al., 2014, Boiarinova et al., 2021)
OILU symbolic	Line-based, hierarchical / pyramidal decimal numeration and marker encoding	(Mostefai et al., 2021)

A common misconception is that “Hierarchical Number System” necessarily denotes a positional numeral system. The literature does not support that restriction. In (Shipunov, 2017), the hierarchy encodes biological rank; in (Kalinovsky et al., 2014) and (Boiarinova et al., 2021), HNS denotes hypercomplex number systems rather than numeral representations; and in (Mostefai et al., 2021), hierarchy is primarily visual and multi-facet rather than arithmetically positional.

2. Numerical ranks as an HNS for biological nomenclature

In biological nomenclature, HNS refers to the proposal called “numerical ranks,” introduced as a very simple Hierarchical Number System for naming taxa above the genus/species level (Shipunov, 2017). The proposal replaces base name plus rank-specific postfixes with a numerical prefix that directly encodes rank. The principal ranks are assigned integers from lowest to highest:

• $1 =$ species
• $2 =$ genus
• $3 =$ family
• $4 =$ order
• $5 =$ class
• $6 =$ phylum
• $7 =$ kingdom

Intermediate ranks are represented by decimal modifiers:

• $.2 =$ super-
• $.5 =$ infra-
• $.8 =$ sub-

The general format is

$2 =$0

The paper gives explicit examples such as 7. Araneus = Kingdom Animalia, 3. Brassica = Family Cruciferae, 4. Agaricus = Ordo Agaricales, 4.5. Homo = Infraclass Theria, and 5.8. Scolopendra = Subphylum Myriapoda (Shipunov, 2017). It also presents more literal renderings such as 7. Animalia, 4. Agaricales, 3. Cruciferae, 4.5. Theria, and 5.8. Myriapoda.

The motivation is explicitly nomenclatural. The proposal identifies four problems in conventional higher-taxon naming: non-uniformity across codes of nomenclature, long and hard-to-read names, obscuring of the base name, and awkward handling of intermediate ranks (Shipunov, 2017). The numerical system is presented as simpler, more straightforward, and easy to extend. Because the prefix leaves the underlying taxon name unchanged, the base name remains visible. Because decimal modifiers encode super-, infra-, and sub-categories, intermediate ranks do not require ad hoc ending systems.

A further feature is the link to typification. The proposal states that all numerical-rank names require typification and expresses the hope that such a system would facilitate the creation of typified names for higher-level taxonomic groups (Shipunov, 2017). The same paper notes a LaTeX implementation via the classif2 package, cited as Shipunov, 2008, CTAN. The proposal is therefore both a nomenclatural reform and a typographic formalization.

3. Distance-sensitive tree representations

A mathematically different HNS appears in the theory of number representation by rooted symmetric trees (Rofa, 2013). Here the defining idea is that numbers are represented by all vertices of a rooted symmetric tree, and the representation of a vertex depends on its distance from the root. The system is therefore explicitly distance sensitive.

For a radix sequence

$2 =$1

the place value set $2 =$2 is defined through

$2 =$3

with $2 =$4, and equivalently by the recursion

$2 =$5

For example, when $2 =$6,

$2 =$7

so

$2 =$8

Vertices are encoded by tuples $2 =$9 corresponding to positions in a rooted symmetric tree with degree sequence $3 =$0. The value map is

$3 =$1

The extra $3 =$2 is the feature that makes the representation depend on distance from the root (Rofa, 2013). A consequence is that leading zeros are no longer value-neutral: the paper explicitly contrasts $3 =$3 and $3 =$4, which differ in this framework.

The system preserves a place-value structure and radix constraints $3 =$5, but generalizes ordinary positional notation in three ways. First, numbers correspond to all non-root vertices, not just leaves. Second, the same coefficient pattern can yield different values at different depths. Third, the range of represented numbers is contiguous: $3 =$6 and the range of $3 =$7 is exactly

$3 =$8

The inverse conversion is performed by the Distance Sensitive Division Algorithm, which produces the unique representation $3 =$9 (Rofa, 2013). Standard decimal and binary systems appear as special cases when the radix sequence is constant; the paper names these the distance decimal number system and distance binary number system. In the distance binary example with $4 =$0, the place value set is

$4 =$1

and there are 14 distinct distance binary numbers with at most 3 digits, compared to only 8 ordinary binary numbers of at most 3 digits (Rofa, 2013). Arithmetic is also developed, using recursive relations such as

$4 =$2

with examples including $4 =$3, $4 =$4, $4 =$5, and $4 =$6.

4. Hereditarily binary natural numbers

A second numeral-theoretic use of hierarchy appears in hereditarily binary natural numbers, a tree-based representation obtained by recursively run-length encoding bijective base-2 digits (Tarau, 2013). The starting operators are

$4 =$7

with $4 =$8 represented by the empty sequence $4 =$9. The paper gives the basic examples

$5 =$0

together with

$5 =$1

The recursive representation is specified by the datatype $.8 =$7 where E is zero and V and W encode alternating run-length blocks. The interpretation map $5 =$2 is given by

$5 =$3

$5 =$4

$5 =$5

$5 =$6

$5 =$7

The inverse map $5 =$8 is defined recursively by parity: $5 =$9 The paper states that $6 =$0 and $6 =$1 on their domains (Tarau, 2013).

The key conceptual distinction is between bit-size and structural complexity. The paper defines bitsize(t) as the total number of applications of $6 =$2 and $6 =$3, and tsize(t) as the number of tree nodes excluding the root. It proves

$6 =$4

This permits algorithms whose cost is governed by tree structure rather than by raw digit length. Implemented operations include successor, predecessor, doubling, halving-like hf, exponentiation of 2, addition, subtraction, comparison, multiplication, squaring, exponentiation, left and right shift by a power of 2, division and remainder, and an integer square root in the appendix (Tarau, 2013).

The complexity claims are explicit. Successor, predecessor, doubling, halving, exponentiation by 2, and similar operations are described as “practically constant time” under the structural model; addition, subtraction, multiplication, and comparison can be much faster than standard binary algorithms on highly regular numbers; in the best case, complexity can collapse by super-exponential factors; and in the worst case the methods remain within a constant factor of traditional arithmetic because $6 =$5 (Tarau, 2013). Illustrative examples include $6 =$6, whose tsize values are only $6 =$7, and the largest known Mersenne prime at the time, $6 =$8, represented with structural complexity 22 and a DAG with only 7 shared nodes.

5. HNS as hypercomplex number systems

In algebraic literature, HNS often denotes hypercomplex number systems rather than hierarchical numeral systems (Kalinovsky et al., 2014, Boiarinova et al., 2021). One important line of work studies 4-dimensional non-commutative HNS obtained by the non-commutative Grassmann–Clifford procedure of doubling 2-dimensional systems and relating them to generalized quaternions (Kalinovsky et al., 2014).

A generalized quaternion is written as

$6 =$9

with conjugation

$7 =$0

and norm

$7 =$1

The doubling construction uses the basis

$7 =$2

with the key noncommutativity

$7 =$3

From the 2-dimensional systems $7 =$4, $7 =$5, and $7 =$6, the paper derives six isomorphism classes: $7 =$7, $7 =$8, $7 =$9, $.2 =$0, $.2 =$1, and $.2 =$2 (Kalinovsky et al., 2014).

These systems align with generalized quaternion parameter choices. The paper identifies $.2 =$3 with $.2 =$4, $.2 =$5 with $.2 =$6, $.2 =$7 with $.2 =$8, $.2 =$9 with $.5 =$0, $.5 =$1 with $.5 =$2, and $.5 =$3 with $.5 =$4. Across all six systems, conjugation has the same coordinate form,

$.5 =$5

and the pseudonorm satisfies

$.5 =$6

Zero divisors are characterized by $.5 =$7, with no zero divisors for quaternions $.5 =$8 and explicit vanishing conditions for the other five cases (Kalinovsky et al., 2014).

A separate hypercomplex use of HNS appears in symbolic exponentiation in a fifth-dimensional system (Boiarinova et al., 2021). There, a hypercomplex number is

$.5 =$9

and the multiplication law for the class under study is

$.8 =$0

The exponential $.8 =$1 is constructed as the particular solution of

$.8 =$2

with initial condition

$.8 =$3

For the 5-dimensional HNS $.8 =$4, the characteristic numbers split into one real eigenvalue and two pairs of complex conjugates, and the paper concludes that

$.8 =$5

The symbolic workflow is implemented in the Maple-based HCS package, using commands such as SearchHNS, VizHNS, HNSnumber, inMulti, Eigenvalues, solve, and factor(numer(...))/factor(denom(...)) (Boiarinova et al., 2021). In this usage, HNS designates a finite-dimensional algebraic environment rather than a numeral encoding.

6. Visual hierarchical numeration and OILU symbolic

A further hierarchical usage is the line-based decimal numeration system called OILU symbolic, described as line-based, human-machine readable, hierarchical / pyramidal, capable of representing numbers as multi-facet objects, and useful for number generation, coding, and visual markers (Mostefai et al., 2021). Here the hierarchy is graphical and view-dependent rather than primarily algebraic.

The system uses 10 symbols generated from simple folded-line patterns. The first four symbols are stated as | for digit 1, L for digit 2, u for digit 3, and } for digit 0. The six missing symbols are obtained by successive quarter-turn counterclockwise rotations of two base shapes; the paper says that rotating {L}-type symbols produces the symbols for even digits, while rotating {} symbols produces the symbols for odd digits (Mostefai et al., 2021). The extracted text is degraded, and the complete digit-to-symbol table is therefore not fully reconstructable from the summary alone.

The system’s central structural claim is that numbers can be superimposed in pyramidal form without losing value. For the decimal number 3172, the related numbers observed from different facets are 3172, 9158, 7136, and 5194. The readout direction is explicitly stated to be from the outside to the inside (Mostefai et al., 2021). This makes the representation multi-facet rather than flat.

The most concrete transformation mechanism is digit splitting on standard seven-segment displays. Each digit can be decomposed into two OILU symbols using three strategies: (a) the central horizontal segment is shared by both parts, (b) it belongs to the upper part, or (c) it belongs to the lower part. The decomposition is intended to be fully reversible, and if extra symbols appear that are not valid OILU symbols, they are replaced using strategy (a) because it has no extra symbol and is therefore safer and unambiguous (Mostefai et al., 2021).

The paper claims a display-efficiency advantage of 25 segments versus 49 for classical displayers and “at least 50%” reduction in display energy. It also proposes applications to visual markers for augmented reality and UAV navigation, with markers built from superimposed OILU symbols and intended to be real-time identifiable and highly distinguishable (Mostefai et al., 2021). A plausible interpretation is that OILU is a hierarchical visual numeration and encoding framework rather than a conventional positional arithmetic system in the strict sense.

7. Comparative perspective

Across these literatures, hierarchy plays different technical roles. In biological nomenclature, it is a rank code attached to a taxon name (Shipunov, 2017). In the rooted-tree HNS, hierarchy is literal depth in a symmetric tree and appears algebraically through the distance term $.8 =$6 in the value map (Rofa, 2013). In hereditarily binary numbers, hierarchy is recursive compression of run lengths, and algorithmic performance depends on structural complexity rather than bit-size (Tarau, 2013). In hypercomplex HNS, the term refers to algebraic systems with explicit multiplication, conjugation, norm or pseudonorm, and symbolic computation of functions such as the exponential (Kalinovsky et al., 2014, Boiarinova et al., 2021). In OILU, hierarchy is pyramidal superposition and multi-facet decoding (Mostefai et al., 2021).

The principal source of ambiguity is therefore terminological rather than mathematical. The same acronym spans nomenclatural standardization, nonstandard positional representation, recursive compression, hypercomplex algebra, and visual symbol design. Any technical use of “HNS” requires immediate domain qualification. Without that qualification, the term does not identify a unique formalism in contemporary arXiv usage.