The evaluation of SPARQL algebra queries on various kinds of annotated RDF graphs can be seen as a particular case of the evaluation of these queries on RDF graphs annotated with elements of so-called  spm-semirings . Spm-semirings extend semirings, used for representing the provenance of positive relational algebra queries on annotated relational data, with a new operator to capture the semantics of the non-monotone SPARQL operators. Furthermore, spm-semiring-based annotations ensure that desired SPARQL query equivalences hold when querying annotated RDF. In this work, in addition to introducing spm-semirings, we study their properties and provide an alternative characterization of these structures in terms of semirings with an embedded boolean algebra (or seba-structure for short). This characterization allows us to construct spm-semirings and identify a universal object in the class of spm-semirings. Finally, we show that this universal object provides a provenance representation of poly-sized overhead and can be used to evaluate SPARQL queries on arbitrary spm-semiring-annotated RDF graphs.
['Floris Geerts', 'Thomas Unger', 'Grigoris Karvounarakis', 'Irini Fundulaki', 'Vassilis Christophides']
Algebraic Structures for Capturing the Provenance of SPARQL Queries