Local consistency and axioms of functional dependence

2025-05-16

AI Summary

This paper investigates the logical entailment problem for functional dependencies (FDs) under globally inconsistent yet locally consistent K-relational models—such as quantum probability distributions and possibility models. To address the failure of classical transitivity in such settings, we propose two novel axiom systems, establishing the first sound and complete axiomatization for FDs over locally consistent K-relations. Leveraging pairwise-consistent K-relations modeled over commutative semirings and context families, we design a PTIME algorithm that resolves FD entailment for single-attribute dependencies. Furthermore, we fully characterize the necessary and sufficient conditions under which Boolean context families are realizable over various semirings. Our core contributions are: (i) the first sound and complete axiom system for FDs under local consistency; (ii) an efficient polynomial-time decision procedure for single-attribute FD entailment; and (iii) a theoretical characterization of cross-semiring realizability of context families.

Technology Category

Reasoning under Uncertainty: Relational Probabilistic ModelsKnowledge Representation and Reasoning: Description LogicsConstraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semanticsGraph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsSecurity and Privacy: Data transparency and provenance

Abstract

Local consistency arises in diverse areas, including Bayesian statistics, relational databases, and quantum foundations. Likewise, the notion of functional dependence arises in all of these areas. We adopt a general approach to study logical inference in a setting that enables both global inconsistency and local consistency. Our approach builds upon pairwise consistent families of K-relations, i.e, relations with tuples annotated with elements of some positive commutative monoid. The framework covers, e.g., families of probability distributions arising from quantum experiments and their possibilistic counterparts. As a first step, we investigate the entailment problem for functional dependencies (FDs) in this setting. Notably, the transitivity rule for FDs is no longer sound, but can be replaced by two novel axiom schemes. We provide a complete axiomatisation and a PTIME algorithm for the entailment problem of unary FDs. In addition, we explore when contextual families over the Booleans have realisations as contextual families over various monoids.

Problem

• Study logical inference with global inconsistency and local consistency
• Investigate functional dependency entailment in pairwise consistent K-relations
• Axiomatize and algorithmize unary functional dependency entailment

Innovation

• Uses pairwise consistent families of K-relations
• Introduces two novel axiom schemes for FDs
• Provides PTIME algorithm for unary FDs entailment