Abstract
It is known that there is no finite set of inference rules that is complete for embedded multivalued dependencies. This paper considers the problem, called membership problem, of deciding whether a given embedded multivalued dependency X ↠ V on U′ (or functional dependency X → V) is implied by a given set D of functional and embedded multivalued dependencies. A new inference rule for embedded multivalued dependencies is presented and the following results about membership problems are shown. Let U be a set of attributes and let W n ⊊…⊊ W 0 ⊊ U . Let D = F ∪ E ∪ E 1 ∪…∪ E n , where 1. (i) F is a set of functional dependencies Y → Z satisfying Y ⊆ W 0 or Z ∩ W 0=∅, 2. (ii) E is a set of embedded multivalued dependencies Y ↠ Z on T satisfying at least one of Y ⊆ W 0 ⊆ T, Z ∩ W 0 = ∅ and ( T − Y ∪ Z)∩ W 0=∅ and 3. (iii) each E υ 1 ⩽ i ⩽ n, is a set of embedded multivalued dependencies on W. 1. (1) If X, V ⊆ U′ ⊆ W 0 then the membership problem is solaable, that is, it is decidable whether X ↠ V on U′ (or X → V) is implied by D. 2. (2) If X, V ⊆ U′ ⊆ W n then the membership problem cah be solved in O(| D|·| V− X|) time, where | D| is the size of the description of D. 3. (3) If X ⊆ W n and X, V ⊆ U′ ⊆ W 0 then the membership problem can be solved in O(| D| 2·| W n − X|· Π n t=1 (| W 1− W 1+ l |+1)) time.
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