Rate-Distortion Theory for Deductive Sources under Closure Fidelity

2026-04-17

AI Summary

This work investigates lossy compression of knowledge base statements under the constraint of preserving deductive closure semantics. Treating the knowledge base as a structured information source within a deductive system, the authors formulate a rate–distortion framework wherein fidelity is measured by deductive closure preservation rather than conventional symbol-level accuracy. By introducing decoders with bounded inference depth and leveraging conditional entropy analysis alongside deductive system modeling, they demonstrate that redundant statements become invisible in the rate–distortion function. The main contributions include a proof that the minimum rate at zero distortion equals the product of the mass of the irreducible core set and its conditional entropy, and the further result that the complete rate–distortion function is entirely determined by this core set.

Technology Category

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

Application Category

• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Content-based information diffusion

Abstract

We study lossy compression of a finite statement source generated in a fixed deductive environment. The source symbols are statements in a knowledge base endowed with a proof system, and reconstruction fidelity is measured by preservation of deductive closure rather than by symbolwise equality. This induces, once the proof system and canonical order are fixed, a decomposition of the source into an irredundant core and redundant stored consequences. Under a natural disjointness condition on zero-distortion reconstruction sets, we show that the minimum zero-distortion rate equals the source mass of the core times the entropy of the source conditioned on that core. For reconstruction alphabets contained in the deductive closure of the source knowledge base, we further prove that the full rate-distortion function depends only on the core, so redundant states are invisible to both rate and distortion. When the decoder is limited to a bounded number of inference steps, we obtain an exact fixed depth rate-delay-distortion characterization. Under an additional order-robustness assumption identifying the chosen core with the order-free essential set, this characterization interpolates between classical symbolwise compression and unconstrained deductive compression. These results formulate deductive compression as a structured source coding problem and quantify how shared inference structure changes the fundamental limits of communication.

Problem

• Rate-Distortion Theory
• Deductive Sources
• Closure Fidelity
• Lossy Compression
• Source Coding

Innovation

• deductive compression
• rate-distortion theory
• deductive closure
• irredundant core
• structured source coding