Abstract: Methods, systems, and apparatus, including computer programs encoded on computer storage media, for evaluating recursive predicates and exiversal predicates. One of the methods includes receiving an expression having an exiversal quantifier that specifies a condition x and a condition y. The expression is transformed into a second expression having multiple recursive calls, the second expression having a first non-negated term with a first recursive call in the condition y, and a second negated term with a second recursive call in the condition y. The second expression is evaluated using evaluation predicates that include a single delta predicate that replaces only the recursive call in the first non-negated term to identify data elements in the relation having at least one attribute satisfying the condition x and the condition y, and, for each attribute of the data element satisfying the condition x, the attribute also satisfies the condition y.

Abstract: Methods, systems, and apparatus, including computer programs encoded on computer storage media, for computing transitive closures of relations. One of the methods includes initializing the transitive closure F of an initial iteration with the tuples in another relation f. New first tuples and new second tuples are iteratively computed until no new first tuples are generated, including: generating new first tuples on each iteration by matching destination elements of tuples in the auxiliary relation of a previous iteration with source elements of tuples in the auxiliary relation of the previous iteration, generating second tuples on each iteration by matching destination elements of the new first tuples with source elements of tuples in F of the previous iteration, and adding the new first tuples and the new second tuples to F of a current iteration.

Abstract: Methods, systems, and apparatus, including computer programs encoded on computer storage media, for evaluating recursive predicates and exiversal predicates. One of the methods includes receiving an expression having an exiversal quantifier that specifies a condition x and a condition y. The expression is transformed into a second expression having multiple recursive calls, the second expression having a first non-negated term with a first recursive call in the condition y, and a second negated term with a second recursive call in the condition y. The second expression is evaluated using evaluation predicates that include a single delta predicate that replaces only the recursive call in the first non-negated term to identify data elements in the relation having at least one attribute satisfying the condition x and the condition y, and, for each attribute of the data element satisfying the condition x, the attribute also satisfies the condition y.