Complexity of Logics with Semiring Semantics

We study the expressive power and computational properties of first-order logic and its extensions under the semiring semantics originating from the seminal work of Green, Karvounarakis, and Tannen. While semiring semantics is currently extensively used, e.g., in the study of provenance in database theory and description logic, a comprehensive computational analysis of these logics acting over general semirings is still lacking. We analyse expressivity, and complexity of model-checking of first-order formulas in this framework, providing characterizations in terms of generalized Blum–Shub–Smale machines over semirings. We also show a variant of Fagin's theorem, i.e., a logical characterization of nondeterministic polynomial time over semirings using a version of existential second-order logic. We further generalize Cook's theorem for the semiring framework and show that propositional satisfiability in the semiring semantics is complete for this notion of NP, and that the true existential first-order theory of the semiring is complete for its Boolean fragment.