Proof-Valid Caching under Premise Erasures: Local Structural Limits and Shared-Workload Gains

2026-08-12

AI Summary

This study addresses the problem of reliable query recovery in semantically transparent caching systems under premise-independent erasures, ensuring that cached objects remain logical consequences of the premise base. The work proposes a logic-derivation-structure-based caching mechanism, introducing strategies of local path interception and shared semantic modules, and establishes the Query Local Projection Theorem and the Residual Leaf Node Law. It presents, for the first time, an exact reliability criterion for semantic caching and proves the optimality of shared modules under homogeneous cost assumptions. By integrating deterministic canonical witnesses, semantic module partitioning, MDS erasure codes, and Datalog modeling, the theoretical analysis shows that cache selection in derivation DAGs of depth two is NP-complete, that MDS-based caching enables recovery of leaf payloads from the loss of at most one packet, and precisely quantifies the relationship among caching overhead, erasure rate, and module cost.

Abstract

We study reliable query recovery under independent premise erasures in semantically transparent caching systems, where every cached object must be a logical consequence of the premise base. Recovery succeeds only when the query remains derivable from surviving premises and the cache. Under a deterministic canonical-witness regime, we prove a query-local projection theorem and an exact residual-leaf law: recovery fails exactly when an erased base leaf retains a cache-free path to the query. Single-query design becomes weighted partial path interception. For shared workloads, we introduce semantic modules and derive exact reliability laws under joint and maximal-error criteria. The shared-module cache is exactly optimal under exact module routing and homogeneous costs, whereas optimal selection in general derivation DAGs is NP-complete at depth two. Against a coded benchmark recovering workload-relevant leaf payloads, MDS parity caching is optimal up to one packet. Leaf-only transparency incurs a first-order overhead inversely proportional to the erasure rate; shared modules multiply that inverse-erasure-rate scaling by the module-to-leaf cost ratio divided by the number of protected leaves. A Datalog instance and Monte Carlo checks illustrate the theory. For derivation-structured content, the results provide exact stochastic-erasure counterparts of function-correcting storage and an exact distributional quantification of maximal recoverability.

Problem

• semantic caching
• premise erasures
• query recovery
• logical consequence
• reliability

Innovation

• semantic caching
• premise erasures
• query recovery
• derivation DAGs
• MDS parity caching