Logical Approaches to Non-deterministic Polynomial Time over Semirings

We provide a logical characterization of non-deterministic polynomial time defined by BSS machines over semirings via existential second-order logic interpreted in the semiring semantics developed by Grädel and Tannen. Furthermore, we show that, similarly to the classical setting, the satisfiability problem of propositional logic in the semiring semantics is the canonical complete problem for this version of NP. Eventually, we prove that the true existential first-order theory of the semiring is a complete problem for the so-called Boolean part of this version of NP.