Semantic Rate-Distortion Theory: Deductive Compression and Closure Fidelity

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.