Inclusion with repetitions and Boolean constants -- implication problems revisited
Matilda Häggblom
TL;DR
This paper investigates the implication problem for inclusion dependencies in three natural extensions: repetition-enabled inclusions that express equalities, and inclusion with Boolean constants in both standard and Boolean-value settings. It provides a new two-value completeness proof for repetition-free inclusions and extends it to inclusion with repetitions, establishing complete axiomatizations and PSPACE-completeness. It then introduces a complete axiomatization for inclusion with Boolean constants, proves that no $k$-ary system suffices, and confirms PSPACE-completeness for the extended variants. By connecting these results to team semantics, the work lays groundwork for broader logics and demonstrates the robustness of these dependency classes across database theory and logical frameworks.
Abstract
Inclusion dependencies form one of the most widely used dependency classes. We extend existing results on the axiomatization and computational complexity of their implication problem to two extended variants. We present an alternative completeness proof for standard inclusion dependencies and extend it to inclusion dependencies with repetitions that can express equalities between attributes. The proof uses only two values, enabling us to work in the Boolean setting. Furthermore, we study inclusion dependencies with Boolean constants, provide a complete axiomatization and show that no such system is k-ary. Additionally, the decision problems for both extended versions remain PSPACE-complete. The extended inclusion dependencies examined are common in team semantics, which serves as the formal framework for the results.