💻 computer science

Non-negative Rational Semantic Numeration Systems

This paper introduces positive rational Semantic Numeration Systems, defines their carry and remainder operations for cardinal semantic operators, analyzes their dynamical properties through examples, and proposes a preliminary framework for partial integer Semantic Numeration Systems.

Original authors: Alexander Chunikhin

Published 2026-05-01

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Alexander Chunikhin

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a giant, magical kitchen where you are constantly mixing, splitting, and combining ingredients. In traditional math, we usually think of numbers like whole apples: you have 3 apples, you give away 1, you have 2 left. You can't really have "half an apple" in a whole-number system without breaking the rules of that specific game.

This paper introduces a new way of playing that game, called Semantic Numeration Systems (SNS), but with a twist: it allows for rational numbers (fractions like 1/2, 3/4, or 2.5) to flow through the system naturally, without forcing them to be whole numbers first.

Here is a simple breakdown of the paper's main ideas using everyday analogies:

1. The Old Way vs. The New Way

Traditional Math (The Static Photo): Usually, when we write a number in a system (like base-10), we treat it as a finished picture. It's just a static list of digits.
This Paper's Approach (The Movie): The author treats numbers as a dynamic process. Instead of just looking at the final result, we watch the "movie" of how the number transforms step-by-step. It's like watching water flow through pipes rather than just looking at a bucket of water at the end.

2. The Building Blocks: "Abstract Entities"

Think of the system as a network of buckets (called Cardinal Abstract Entities).

Each bucket holds a certain amount of "stuff" (a number).
The buckets are connected by pipes (called Cardinal Semantic Operators).
The goal is to move "stuff" from one bucket to another according to specific rules.

3. The Rules of the Flow (The Operators)

The paper defines four main types of "pipes" or rules for moving the stuff. In this new system, the rules allow for fractions to be the "carrying" amount, not just whole numbers.

The Linear Pipe (L-Operator): Imagine a bucket with a hole at the bottom. You pour water in. The rule says, "For every 10 units you have, 1 unit flows out to the next bucket." In this new system, if you have 15 units, 1.5 units flow out. It's a direct, one-to-one transfer.
The Distribution Pipe (D-Operator): Imagine a bucket that splits its flow into two different pipes. If you have 20 units, the rule might say, "Send 1.2 units to Bucket A and 0.8 units to Bucket B." The total flow is still based on the fraction of what you started with.
The Fusion Pipe (F-Operator): Imagine two buckets pouring into a single funnel. The rule is: "The amount that flows out is limited by the smallest amount available in either bucket." If Bucket A has enough for 5 units of flow, but Bucket B only has enough for 3, the system only moves 3 units. The "remainder" (the extra 2 units in Bucket A) stays behind.
The Multi-Pipe (M-Operator): A combination of the above, where multiple buckets feed into multiple other buckets simultaneously, all governed by these fractional rules.

4. The "Carry" Without the Floor

In normal math, if you divide 7 by 3, you get 2 with a remainder of 1. You usually throw away the "1/3" part or keep it as a remainder.

The Innovation: In this new system, the "carry" is exactly 7/3 (2.333...). There is no "floor" function (no rounding down). The system accepts the fraction as a valid, moving part of the process. This means the "remainder" can also be a fraction, and the system keeps flowing perfectly without losing precision.

5. A Dynamic System (The Movie in Motion)

The author shows that this isn't just a static calculation; it's a dynamic system.

Imagine a row of buckets. You pour water into the first one.
Step 1: Water flows to the second and third buckets based on the rules.
Step 2: The water in the second and third buckets now flows to the next ones.
The paper provides a specific example (Example 2) where a starting amount of "33" flows through a complex network of 7 buckets. After just three steps, the water has settled into a new distribution. The math proves that the system is stable and predictable, even with all these fractions floating around.

6. The "Negative" Experiment (A Glimpse of the Future)

The paper briefly touches on a more advanced idea: what if we allowed the pipes to have negative coefficients?

Imagine a pipe that doesn't just add water to a bucket, but removes water from it.
The author proposes a "Proposition": Even if we use negative numbers to move things around, the final amount in every bucket must stay positive (you can't have negative water in a bucket).
This is presented as a "first step" toward a more complex system that could handle whole numbers (including negatives) in this dynamic way, but the paper admits this is still a work in progress.

Summary

This paper proposes a new way to think about numbers. Instead of treating them as static whole numbers, it treats them as fluid, fractional flows moving through a network of connected containers. By removing the rule that forces us to round down to whole numbers, the system can model continuous, real-world processes more accurately, step-by-step, using a set of logical "pipes" that handle fractions naturally.

1. Problem Statement

Traditional numeration systems (positional systems) are generally viewed as static representations of numbers where the focus is on the final expansion. Furthermore, traditional systems typically rely on integer bases and integer arithmetic, often utilizing the floor function ( $\lfloor \cdot \rfloor$ ) to define carries and remainders.

The author identifies two primary limitations in existing frameworks:

Static vs. Dynamic: Traditional systems treat number representation as a completed state, whereas the author argues for viewing it as a dynamic process of transformations within a state space.
Integer vs. Rational Constraints: Existing "Semantic Numeration Systems" (SNS) have been defined primarily over non-negative integers ( $N_0$ ). There is a lack of formalism for systems where the base, parameters, and resulting values are non-negative rational numbers ( $Q^+$ ), which is necessary for modeling real-world processes involving "parts" and "pieces" rather than just discrete "whole portions."

The core problem addressed is the theoretical extension of Semantic Numeration Systems to the domain of non-negative rational numbers ( $SNS(Q^+)$ ) and the initial exploration of extending this to negative integers ( $SNS(q^-)$ ).

2. Methodology

The paper employs a theoretical framework based on Cardinal Abstract Entities (CÆ) and Cardinal Semantic Operators (CSO). The methodology involves:

Redefining the Carry Mechanism: Unlike traditional systems that use the floor function to calculate carries ( $p_i = \lfloor \#_i / n_i \rfloor$ ), the proposed $SNS(Q^+)$ defines the carry as a direct division without flooring:
$p_i = \#_i / n_i$
Consequently, the remainder is calculated as:
$rem \#_i = \#_i - p_i \cdot n_i$
In this rational system, the remainder is not necessarily zero unless the division is exact, but the system allows for rational remainders and carries.
Operator Basis Expansion: The author defines four fundamental cardinal semantic operators adapted for rational values ( $Q^+$ ):
1. L-operator (Linear): Transforms a single entity to another.
2. D-operator (Distribution): Distributes a carry from one entity to multiple targets.
3. F-operator (Fusion): Combines carries from multiple entities into a single target, using a min function to determine the common carry ( $p_{common} = \min(p_i, p_j)$ ).
4. M-operator (Multi): A generalization combining fusion and distribution.
Dynamical Systems Formulation: The paper models the Semantic Numeration System as a discrete dynamical system.
- State Vector: The system state at step $k$ is represented by a vector $|\mathbf{\#}(k)\rangle$ containing the cardinals of all abstract entities.
- State Equation: The evolution of the system is governed by matrix equations involving:
  - $\mathbf{N}$ : The diagonal radix operator.
  - $\mathbf{R}_T$ : The transposed conversion operator (coefficients).
  - $\mathbf{\Lambda}$ : The common carry operator (handling min logic for fused operators).
- The general state equation is derived as:
  $|\mathbf{\#}(k+1)\rangle = |\mathbf{\#}(k)\rangle + (\mathbf{R}_T - \mathbf{N})\mathbf{\Lambda}\mathbf{N}^{-1}|\mathbf{\#}(k)\rangle$
Topological Flexibility: The methodology allows for heterogeneous and anisotropic topologies (non-linear connections), distinguishing it from the linear topology of traditional positional systems.

3. Key Contributions

Introduction of $SNS(Q^+)$ : The paper formally introduces a new class of Semantic Numeration Systems where all parameters (bases, conversion coefficients) and values (cardinals, carries, remainders) are non-negative rational numbers.
Rational Carry Definition: A novel definition of the "common carry" that operates without the floor function, enabling continuous, rational-valued transformations.
Unified Dynamic Framework: The formulation of the State Equation for Semantic Numeration Systems, treating them as dynamic systems evolving in a multidimensional state space. This unifies various operator types (L, D, F, M) under a single matrix-based mathematical model.
Proposal for $SNS(q^-)$ : A preliminary conceptual framework for systems allowing negative transformants (and thus negative intermediate values) while maintaining non-negative final cardinals, serving as a bridge toward full integer ( $Z$ ) and rational ( $Q$ ) systems.

4. Results

Analytical Examples: The paper provides a complex numerical example (Example 2) involving a stationary system with 7 entities and 4 different operator types (2M2, L, D2, 2F).
- The system starts with integer inputs ( $33_i, 21_j$ ).
- Through the application of the rational state equations, the system evolves through 3 steps, producing rational values (e.g., $27/4, 189/160$ ) at intermediate stages.
- The example demonstrates that the system converges and that the dynamics are fully determined by the configuration matrix and the initial state.
Illustrative $SNS(q^-)$ Case: Example 3 demonstrates a system where a negative conversion coefficient ( $r = -1$ ) is used. This results in a negative transformant ( $q = -3$ ), which is subtracted from a target entity. The result shows that the system can handle negative flows while ensuring the final cardinal remains non-negative ( $8_s$ ), validating the proposition for adjustable cardinal systems.

5. Significance

Modeling Real-World Processes: By moving from integers to rationals, $SNS(Q^+)$ allows for the modeling of processes that involve fractions, continuous flows, and partial quantities, which integer-based systems cannot adequately represent.
Paradigm Shift: The paper shifts the perspective of numeration from a static "representation" to a dynamic "transformation process." This is significant for fields requiring the modeling of complex state changes, such as biochemistry (given the author's affiliation), information theory, and complex system dynamics.
Foundational Work for Future Extensions: While $SNS(Q^+)$ is established, the paper highlights the non-trivial challenges in extending this to full rational ( $Q$ ) and integer ( $Z$ ) systems (specifically handling negative numbers and the definition of common carries in those contexts). This sets the stage for future research into "whole number" semantic systems.
Topological Diversity: The framework supports complex, non-linear topologies, offering a more flexible alternative to the rigid linear structure of traditional positional notation.

In conclusion, Chunikhin successfully extends the theory of Semantic Numeration Systems into the rational domain, providing a rigorous mathematical foundation (state equations and operator definitions) for dynamic, rational-valued numeration systems, while outlining the path toward more complex integer-based semantic systems.