On database query languages for K-relations

The relational model has recently been extended to so-called KK-relations in which tuples are assigned a unique value in a semiring KK. A query language, denoted by RAK+, similar to the classical positive relational algebra, allows for the querying of KK-relations. In this paper, we define more expressive query languages for KK-relations that extend RAK+ with the difference and constant annotations   operations on annotated tuples. The latter are natural extensions of the duplicate elimination operator of the relational algebra on bags. We investigate conditions on semirings under which these operations can be added to RAK+ in a natural way, and establish basic properties of the resulting query languages. Moreover, we show how the provenance semiring of Green et al. can be extended to record provenance of data in the presence of difference and constant annotations. Finally, we investigate the completeness   of RAK+ and extensions thereof in the sense of Bancilhon and Paredaens.