Axiomatization of approximate exclusion
Matilda Häggblom
TL;DR
This paper axiomatizes approximate exclusion atoms in team semantics by defining $x|_p y$ as satisfiable when a subteam of size at most $p$ times the original team yields the usual exclusion atom $x|y$. It proves a complete axiomatization and a polynomial-time decision procedure for finite implication problems, with a completeness result for the usual exclusion atoms and an extension to the approximate case for $p<\tfrac{1}{2}$. The results formalize how exclusion-like constraints can tolerate bounded error and establish a rigorous basis for reasoning about exclusion dependencies in databases, including potential extensions to first-order and propositional logics. Overall, the work provides foundational tools for reasoning about approximate exclusion constraints and suggests directions for broader applicability and future research.
Abstract
We define and axiomatize approximate exclusion atoms in the team semantic setting. A team is a set of assignments, which can be seen as a mathematical model of a uni-relational database, and we say that an approximate exclusion atom is satisfied in a team if the corresponding usual exclusion atom is satisfied in a large enough subteam. We consider the implication problem for approximate exclusion atoms and show that it is axiomatizable for consequences with a degree of approximation that is not too large. We prove the completeness theorem for usual exclusion atoms, which is currently missing from the literature, and generalize it to approximate exclusion atoms. We also provide a polynomial time algorithm for the implication problems. The results also apply to exclusion dependencies in database theory.