The Derivation Penalty in Premise-Erasure Caching: Capacity, Strong Converse, and Dispersion Dichotomy

2026-03-01

AI Summary

This study investigates the information-theoretic efficiency and performance limits of caching strategies in derivation-based inference engines under independent erasure of premises. By constructing an information-theoretic framework, it compares universal coding-based caching with fact caching constrained by logical derivations, analyzing their capacity, convergence, and dispersion. The work establishes several theoretical contributions: it reveals the “derivation penalty” phenomenon, introduces a structured caching rigidity theorem, and develops new results including capacity separation, a strong converse theorem, and a dichotomy in dispersion behavior. Leveraging Datalog architectures, a strong converse proof via KL-divergence rates, Bahadur–Rao prefactor analysis, and phase diagram modeling, the paper demonstrates that the derivation penalty ratio converges to the inverse of the erasure rate, identifies exponentially sharper phase transitions in balanced merging architectures, and fully characterizes an eight-region phase diagram.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Semantics and Knowledge: Provenance, trust, security and privacy, and ethical issues in managing semantic data

Abstract

We introduce an information-theoretic framework for caching in derivation-based reasoning engines under independent premise erasure. Two decoder models are compared: a coded scheme using an arbitrary bit-string cache with a general-purpose decoder, and a derivation-constrained scheme where the cache consists of logical facts and the decoder must produce a valid proof. Four coding theorems are established. The first proves that each derivation step carries a universal per-step information content determined by the base size. The second reveals an exponential capacity separation between linear-chain and balanced-merge Datalog architectures at equal depth. The third identifies a critical access frequency separating the regimes where caching and on-demand derivation are optimal. The fourth determines the minimum derivation-constrained cache under erasure, decomposing query information into reliable cache and noisy channel capacity. The central result is the derivation penalty: the ratio of the derivation-constrained cache to the coded cache converges to the reciprocal of the erasure rate, universally across query counts, overlap structures, and reliability targets. This penalty originates from a structural caching rigidity theorem showing that only cache facts within the target query's derivation DAG contribute to resilience, precluding cross-coordinate error correction. Beyond capacity, we prove a strong converse at the KL-divergence rate with Bahadur--Rao prefactors, a dispersion dichotomy (positive coded dispersion versus zero derivation-constrained dispersion), and a complete eight-regime phase diagram. The architecture-dependent depth-to-dependency mapping yields exponentially sharper phase transitions for the merge architecture. All results transfer across synonymous representations.

Problem

• derivation penalty
• premise erasure
• caching
• reasoning engines
• information-theoretic framework

Innovation

• derivation penalty
• premise-erasure caching
• information-theoretic framework
• dispersion dichotomy
• strong converse