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A Mathematical Framework of Semantic Communication based on Category Theory

Shuheng Hua, Yao Sun, Kairong Ma, Dusit Niyato, Muhammad Ali Imran

TL;DR

This work introduces a category-theoretic mathematical framework for semantic communication (SemCom) that models semantic entities within a semantic probability space and defines semantic entropy $H_s(E)$. By proving $H_s(E)$ can be reduced via knowledge bases (KBs) that capture semantic dependencies, the paper derives a KB-enhanced semantic channel capacity $C_s$ and demonstrates entropy reduction at both entity and message levels. The framework is validated numerically through KB-aided semantic coding (based on Fano coding), showing improved coding efficiency and lower entropy than traditional approaches, particularly for distributions with strong semantic dependencies (e.g., Zipf). Overall, the approach provides a rigorous foundation for designing and analyzing SemCom systems with KB integration and offers guidance for optimizing KB depth and semantic representations in practical wireless settings.

Abstract

While semantic communication (SemCom) has recently demonstrated great potential to enhance transmission efficiency and reliability by leveraging machine learning (ML) and knowledge base (KB), there is a lack of mathematical modeling to rigorously characterize SemCom system and quantify the performance gain obtained from ML and KB. In this paper, we develop a mathematical framework for SemCom based on category theory, rigorously modeling the concepts of semantic entities and semantic probability space. Within this framework, we introduce the semantic entropy to quantify the uncertainty of semantic entities. We theoretically prove that semantic entropy can be effectively reduced by exploiting KBs, which capture semantic dependencies. Within the formulated semantic space, semantic entities can be combined according to the required semantic ambiguity, and the combined entities can be encoded based on semantic dependencies obtained from KB. Then, we derive semantic channel capacity modeling, which incorporates the mutual information obtained in KB to accurately measure the transmission efficiency of SemCom. Numerical simulations validate the effectiveness of the proposed framework, showing that SemCom with KB integration outperforms traditional communication in both entropy reduction and coding efficiency.

A Mathematical Framework of Semantic Communication based on Category Theory

TL;DR

This work introduces a category-theoretic mathematical framework for semantic communication (SemCom) that models semantic entities within a semantic probability space and defines semantic entropy . By proving can be reduced via knowledge bases (KBs) that capture semantic dependencies, the paper derives a KB-enhanced semantic channel capacity and demonstrates entropy reduction at both entity and message levels. The framework is validated numerically through KB-aided semantic coding (based on Fano coding), showing improved coding efficiency and lower entropy than traditional approaches, particularly for distributions with strong semantic dependencies (e.g., Zipf). Overall, the approach provides a rigorous foundation for designing and analyzing SemCom systems with KB integration and offers guidance for optimizing KB depth and semantic representations in practical wireless settings.

Abstract

While semantic communication (SemCom) has recently demonstrated great potential to enhance transmission efficiency and reliability by leveraging machine learning (ML) and knowledge base (KB), there is a lack of mathematical modeling to rigorously characterize SemCom system and quantify the performance gain obtained from ML and KB. In this paper, we develop a mathematical framework for SemCom based on category theory, rigorously modeling the concepts of semantic entities and semantic probability space. Within this framework, we introduce the semantic entropy to quantify the uncertainty of semantic entities. We theoretically prove that semantic entropy can be effectively reduced by exploiting KBs, which capture semantic dependencies. Within the formulated semantic space, semantic entities can be combined according to the required semantic ambiguity, and the combined entities can be encoded based on semantic dependencies obtained from KB. Then, we derive semantic channel capacity modeling, which incorporates the mutual information obtained in KB to accurately measure the transmission efficiency of SemCom. Numerical simulations validate the effectiveness of the proposed framework, showing that SemCom with KB integration outperforms traditional communication in both entropy reduction and coding efficiency.

Paper Structure

This paper contains 20 sections, 3 theorems, 52 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Background and Related Works
  3. Motivation and Challenges
  4. Contributions and Organization
  5. Semantic Entity & Semantic Probability Space
  6. Semantic Probability Space
  7. Category Space
  8. S-category Space
  9. Semantic Probability Space
  10. Categorizing Entropy
  11. Entropy Reduction at Entity Level
  12. Combine Synonyms without Semantic Ambiguity
  13. Scaling Semantic Attributes with Ambiguity
  14. Entropy Reduction at Message Level
  15. Message Entropy
  16. ...and 5 more sections

Key Result

Proposition 1

For a given discrete information source $\mathcal{B}$ and a semantic probability space with the set of entities $E$, if each element $b_i\in\mathcal{B}$ uniquely corresponds to an entity $e_i\in E$, the categorizing entropy of entities is equal to classical entropy of the source, i.e.,

Figures (10)

  • Figure 1: The location of entity in scs
  • Figure 2: Different perspectives for locating an entity
  • Figure 3: Combine synonyms without semantic ambiguity
  • Figure 4: Scaling semantic attributes with ambiguity
  • Figure 5: Coding efficiency vs. Number of attributes
  • ...and 5 more figures

Theorems & Definitions (9)

  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 1
  • Definition 4
  • Proposition 2
  • Proposition 3
  • Remark 1
  • Remark 2