Semiring Provenance in the Infinite

Semiring provenance evaluates database queries or logical statements not just by true or false but by values in some commutative semiring. This permits to track which combinations of atomic facts are responsible for the truth of a statement, and to derive further information, for instance concerning costs, confidence scores, number of proof trees, or access levels to protected data. The focus of this approach, proposed and developed to a large extent by Val Tannen and his collaborators, has first been on (positive) database query languages, but has later been extended, again in collaboration with Val, to a systematic semiring semantics for first-order logic (and other logical systems), as well as to a method for the strategy analysis of games. So far, semiring provenance has been studied for finite structures. To extend the semiring provenance approach for first-order logic to infinite domains, the semirings need to be equipped with addition and multiplication operators over infinite collections of values. This needs solid algebraic foundations, and we study here the necessary and desirable properties of semirings with infinitary operations to provide a well-defined and informative provenance analysis over infinite domains. We show that, with suitable definitions for such infinitary semiring, large parts of the theory of semiring provenance can be succesfully generalised to infinite structures.