Extending relational algebra with similarities

In this paper we propose various extensions to the relational model to support similarity-based querying. We build upon the K-relation model, where tuples are assigned values from an arbitrary semiring K, and its associated positive relational algebra RA(K)(+). We consider a recently proposed extension to RA(K)(+) using a monus operation on the semiring to support negative queries, and show how, surprisingly, it fails for important 'fuzzy' semirings. Instead, we suggest using a negation operator. We also consider the identities satisfied by the relational algebra RA(K)(+) We show that moving from a semiring to a particular form of lattice (a De Morgan frame) yields a relational algebra that satisfies all the classical (positive) relational algebra identities. We claim that to support real-world similarity queries realistically, one must move from tuple-level annotations to attribute-level annotations. We show in detail how our De Morgan frame-based model can be extended to support attribute-level annotations and give worked examples of similarity queries in this setting.