# On Desirable Semantics of Functional Dependencies over Databases with Incomplete Information

## TL;DR

This paper explores the semantics of functional dependencies in databases with incomplete information, proposing desirable properties and showing their feasibility in certain models while highlighting limitations in more general settings.

## Contribution

It introduces a set of desirable properties for FDs over incomplete databases and demonstrates their realization in vague tables, revealing trade-offs in more complex models.

## Key findings

- FDs with all desirable properties exist for vague tables
- No FD notion can satisfy all properties in disjunctive tables
- Formalizes trade-off between model generality and FD properties

## Abstract

Codd's relational model describes just one possible world. To better cope with incomplete information, extended database models allow several possible worlds. Vague tables are one such convenient extended model where attributes accept sets of possible values (e.g., the manager is either Jill or Bob). However, conceptual database design in such cases remains an open problem. In particular, there is no canonical definition of functional dependencies (FDs) over possible worlds (e.g., each employee has just one manager). We identify several desirable properties that the semantics of such FDs should meet including Armstrong's axioms, the independence from irrelevant attributes, seamless satisfaction and implied by strong satisfaction. We show that we can define FDs such that they have all our desirable properties over vague tables. However, we also show that no notion of FD can satisfy all our desirable properties over a more general model (disjunctive tables). Our work formalizes a trade-off between having a general model and having well-behaved FDs.