HLLSet Theory: A Unified Framework for Probabilistic Knowledge Representation

This paper introduces HLLSet (HyperLogLog Set), a probabilistic data structure that behaves like a set under all standard operations while containing no explicit elements. Unlike traditional HyperLogLog, which only estimates cardinality, HLLSets support full set operations (union, intersection, difference) through enhanced register structures and provide a principled framework for representing semantic relationships. We establish a category-theoretic foundation for HLLSets, where objects are contextual representations and morphisms are directed similarity relations defined by a dual-threshold system (τ for inclusion tolerance, ρ for exclusion intolerance). We introduce Bell State Similarity (BSS), a directed similarity metric that measures the overlap between probabilistic representations. The framework demonstrates that balanced addition and deletion operations on HLLSet-based representations give rise to a discrete conservation law analogous to Noether’s theorem, providing a principled steering mechanism for AI system evolution. We formalize HLLSets within a categorical framework and establish that HLLSet collections form sheaves over ϵ-isometry categories, with the condition |N| − |D| = 0 serving as a stability criterion that enables self-regulating system dynamics.