Semantic Rate-Distortion Theory: Deductive Compression and Closure Fidelity

2026-04-13

AI Summary

This work addresses the limitation of classical rate–distortion theory in capturing semantic fidelity due to its neglect of the logical structure inherent in knowledge bases. The authors propose a semantic fidelity criterion based on deductive closure and introduce the notion of an irreducible core to enable semantic compression while preserving logical equivalence, thereby significantly reducing communication rates. By extracting the core via a fixed-order deletion procedure and leveraging Datalog logic, a semantic source–channel separation theorem, and a strengthened Fano inequality, they prove that the zero-distortion semantic rate is strictly lower than the classical entropy rate and that the semantic rate–distortion function depends solely on the irreducible core. Empirical validation on a Datalog instance with 24,000 facts reveals phenomena such as cost-free transmission of redundant states, semantic leverage effects, and semantic bottlenecks in broadcast scenarios.

Technology Category

• Knowledge Representation and Reasoning: Description Logics
• Data Mining & Knowledge Management: Data Compression
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

Shannon's rate-distortion theory treats source symbols as unstructured labels. When the source is a knowledge base equipped with a logical proof system, a natural fidelity criterion is closure fidelity: a reconstruction is acceptable if it preserves the deductive closure of the original. This paper develops a rate-distortion theory under this criterion. Central to the theory is the irredundant core-a canonical generating set extracted by a fixed-order deletion procedure, from which the full deductive closure can be rederived. We prove that the zero-distortion semantic rate equals a quantity that is strictly below the classical entropy rate whenever the knowledge base contains redundant states. More generally, the full semantic rate-distortion function depends only on the core; redundant states are invisible to both rate and distortion. We derive a semantic source-channel separation theorem showing a semantic leverage phenomenon: under closure fidelity, the required source rate is reduced by an asymptotic leverage factor greater than one, allowing the same knowledge base to be communicated with proportionally fewer channel uses-not by violating Shannon capacity, but because redundant states become free. We also prove a strengthened Fano inequality that exploits core structure. For heterogeneous multi-agent communication, an overlap decomposition gives necessary and sufficient conditions for closure-reliable transmission and identifies a semantic bottleneck in broadcast settings that persists even over noiseless channels. All results are verified on Datalog instances with up to 24,000 base facts.

Problem

• semantic rate-distortion
• closure fidelity
• deductive closure
• knowledge base
• irredundant core

Innovation

• semantic rate-distortion
• closure fidelity
• irredundant core
• semantic leverage
• deductive compression