Past exam of the computer science course of the University of Oxford / 2015 / Databases / 1 / c

Assume the two database queries have compatible output database attributes. ThenFor a finite input database, the result of the relational difference is a subset of the finite result of $Q_1$, so it is finite and introduces no new values. For the usual domain independence of a database query, let $D$ be the input database and let $A\subseteq A^{\prime }$ be two ambient value domains containing its data and the query constants. Database query safety givesand thereforeAlthough a domain relational calculus expression for the relational difference uses negation, its positive membership requirement in $Q_1$ supplies the necessary range restriction. Negation is not by itself a reason for failure of database query safety.

Take a unary database relation $R$ and the positive domain relational calculus queryFor the finite input $R=\{0\}$, every pair $(0,d)$ and $(d,0)$ belongs to $Q$, for every value $d$ in the ambient domain. On an infinite ambient domain the answer is infinite. Even if one compares only finite ambient domains, adding a fresh value $d$ changes the answer while leaving $R$ fixed, violating domain independence of a database query.

The logical disjunction leaves $y$ unrestricted in its first branch and $x$ unrestricted in its second branch. Thus both variables occur in relational atoms somewhere in the formula, yet database query safety still fails. No negation or implication symbol occurs.