Incomplete relations are relations which contain null values, whose meaning is “value is at present unknown”. Such relations give rise to two types of functional dependency (FD). The first type, called the strong FD (SFD), is satisfied in an incomplete relation if for all possible worlds of this relation the FD is satisfied in the standard way. The second type, called the weak FD (WFD), is satisfied in an incomplete relation if there exists a possible world of this relation in which the FD is satisfied in the standard way. We exhibit a sound and complete axiom system for both strong and weak FDs, which takes into account the interaction between SFDs and WFDs. An interesting feature of the combined axiom system is that it is not k-ary for any natural number k ⩾ 0. We show that the combined implication problem for SFDs and WFDs can be solved in time polynomial in the size of the input set of FDs. Finally, we show that Armstrong relations exist for SFDs and WFDs.
