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Local consistency and axioms of functional dependence

Timon Barlag, Miika Hannula, Juha Kontinen, Nina Pardal, Jonni Virtema

TL;DR

This work studies local consistency in families of $K$-relations under potential global inconsistency, introducing contextual $K$-families and a realisability framework that unifies relational, probabilistic, and quantum-style data. It proves a complete realisability characterisation for cancellative monoids on chordless-cycle contexts via edge cycle covers in the overlap projection graph, and provides a complete axiomatisation for unary functional dependencies in the contextual setting, showing that transitivity fails while cycle and chain rules govern derivations. A polynomial-time procedure is developed to decide unary FD derivability from reflexivity, cycle, and contextual chain rules, establishing tractable entailment in this setting. The results illuminate how local consistency shapes logical inference about dependencies, with implications for quantum foundations, databases, and probabilistic models that lack a global joint distribution. The paper also connects realisability to constructions over $ ext{R}_{ ext{ge}0}$ and $ ext{N}$ and outlines several open questions for broader context sets and non-unary FDs.

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.

Local consistency and axioms of functional dependence

TL;DR

This work studies local consistency in families of -relations under potential global inconsistency, introducing contextual -families and a realisability framework that unifies relational, probabilistic, and quantum-style data. It proves a complete realisability characterisation for cancellative monoids on chordless-cycle contexts via edge cycle covers in the overlap projection graph, and provides a complete axiomatisation for unary functional dependencies in the contextual setting, showing that transitivity fails while cycle and chain rules govern derivations. A polynomial-time procedure is developed to decide unary FD derivability from reflexivity, cycle, and contextual chain rules, establishing tractable entailment in this setting. The results illuminate how local consistency shapes logical inference about dependencies, with implications for quantum foundations, databases, and probabilistic models that lack a global joint distribution. The paper also connects realisability to constructions over and and outlines several open questions for broader context sets and non-unary FDs.

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.
Paper Structure (12 sections, 19 theorems, 10 equations, 6 figures, 1 algorithm)

This paper contains 12 sections, 19 theorems, 10 equations, 6 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $K$-relations and Contextual Families
  4. $K$-realisability of Contextual Families
  5. Contextual $\mathbb{R}_{\geq 0}\xspace$-realisability
  6. Entailment of Dependence
  7. Classical setting
  8. FDs in the Contextual Setting
  9. Unary FDs with Binary Contexts
  10. Unary FDs with Ternary Contexts
  11. Complexity of Derivation
  12. Conclusion

Key Result

Proposition 2

If $\mathcal{R}$ is a contextual $K$-family, then $\mathrm{Supp}(\mathcal{R})$ is a contextual family.

Figures (6)

  • Figure 1: Teachers, courses, and students.
  • Figure 2: Example of an Overlap Projection Graph
  • Figure 3: Non-realisable contextual family.
  • Figure 4: Counterexample for the falsification of the cycle rule.
  • Figure 5: Relations arising from Items \ref{['item:fd1']}, \ref{['item:c1']}, and \ref{['item:c2']}.
  • ...and 1 more figures

Theorems & Definitions (37)

  • Definition 1: Contextual $K$-families
  • Proposition 2
  • Proposition 3
  • Example 4
  • Example 5
  • Lemma 6
  • Example 7
  • Lemma 7
  • proof
  • Theorem 8
  • ...and 27 more