Semiring Semantics of First-Order Logic: Locality Theorems and Gaifman Normal Forms

Semiring semantics of first-order logic generalizes classical Boolean semantics by allowing truth values from a commutative semiring. This raises the question of how classical model-theoretic properties, such as locality theorems, extend to semiring semantics. The paper studies the generalization of Hanf's and Gaifman's locality theorems, showing that Hanf's theorem holds for all semirings, but Gaifman's theorem only holds for certain semirings like min-max and lattice semirings.