a certain set of attributes is a candidate key. For example, if we had considered only r1 then we would have concluded that {A,B} is a candidate key, which is incorrect. However, we might be able to conclude from such a relation that a certain set is not a candidate key, because that set does not have the uniqueness property (example {A,D} for r1). Note that the existence of a proper subset of a set that has the uniqueness property cannot in general be used as evidence that the superset is not a candidate key. In particular, note that in the case of an empty relation, every subset of the heading has the uniqueness property, including the empty set. Determining candidate keys The set of all candidate keys can be computed e.g. from the set of functional dependencies. To this end we need to define the attribute closure for an attribute set . The set contains all attributes that are functionally implied by . It is quite simple to find a single candidate key. We start with a set of attributes and try to remove successively each attribute. If after removing an attribute the attribute closure stays the same, then this attribute is not necessary and we can remove it permanently. We call the result . If is the set of all attributes, then is a candidate key. Actually we can detect every candidate key with this procedure by simply trying every possible order of removing attributes. However there are many more permutations of attributes () than subsets (). That is, many attribute orders will lead to the same candidate key. There is a fundamental difficulty for efficient algorithms for candidate key computation: Certain sets of functional dependencies lead to exponentially many candidate keys. Consider the functional dependencies which yields candidate keys: . That is, the best we can expect is an algorithm that is efficient with respect to the number of candidate keys. The following algorithm actually runs in polynomial time in the number of candidate keys and functional dependencies: function find_candidate_keys(A, F) /* A is the set of all attributes and F is the set of functional dependencies */ K[0] := minimize(A); n := 1; /* Number of Keys known so far */ i := 0; /* Currently processed key */ while i < n do for each α → β ∈ F do /* Build a new