A Naı̈ve Evaluation of Queries over Incomplete Databases

framework for powerset semantics. We now cast the powerset semantics in our general relation-based framework, which enables us to establish when naı̈ve evaluation works for it. For a set D of database objects and a set C of complete objects, we have a pairR “ pRval,Rsemq of relations withRval Ď Dˆ2 C andRsem Ď 2 ˆC. The first relation corresponds to applying multiple valuations (e.g., relating D with sets th1pDq, . . . , hnpDqu). The second relation, in our example, is RY “ tpX , Xq | X “ Ť X u. The semantics given by R is again the composition of two relations: D P rrDss R iff DpRval ̋RsemqD. The basic conditions on these relations are essentially the same as we used before for non-powerset semantics except that we need to deal with relations between C and 2. Let idl Ď Cˆ 2 C contain precisely all pairs pc, tcuq and idr Ď 2 C ˆ C contain precisely all pairs ptcu, cq for c P C. We say that a semantics rr ss R is given by R if both relations are total, relation Rval equals idl when restricted to C, relation Rsem contains idr, and D P rrDss R iffDpRval ̋RsemqD . Previously we just used identity instead of idl and idr. We say thatRsem is transitive ifRsem ̋ idl ̋Rsem Ď Rsem. Note thatRY is transitive. Now we have an analog of Proposition 4.2. PROPOSITION 10.2. A pair R “ pRval,Rsemq gives rise to a fair database domain if Rsem is transitive. PROOF. We prove a more general necessary and sufficient condition for fairness: LEMMA 10.3. A powerset semantics given by R “ pRval,Rsemq gives rise to a fair database domain iff Rval ̋ Rsem ̋ idl ̋ Rsem Ď Rval ̋ Rsem. In particular if Rsem is transitive then the database domain is fair. PROOF. Assume first thatRval ̋Rsem ̋idl ̋Rsem Ď Rval ̋Rsem, and take an arbitrary x P D and c P C. We have (1) c P rrcss R . Indeed we know pc, tcuq P Rval and ptcu, cq P Rsem, then c P rrcssR. (2) c P rrxss R implies rrcss R Ď rrxss R . Indeed if c P rrxss R there exists y Ď C such that px, yq P Rval and py, cq P Rsem. Moreover if c P rrcss R then pc, cq P idl ̋ Rsem (because Rval is idl when restricted to C). Hence px, cq P Rval ̋Rsem ̋ idl ̋Rsem. This implies px, c q P Rval ̋Rsem, and therefore c P rrxss R . ACM Transactions on Database Systems, Vol. V, No. N, Article A, Publication date: January YYYY.