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Ehrenfeucht-Fra\"iss\'e games provide a fundamental method for provingelementary equivalence (and equivalence up to a certain quantifier rank) ofrelational structures. We investigate the soundness and completeness of thismethod in the more general context of semiring semantics. Motivated originallyby provenance analysis of database queries, semiring semantics evaluateslogical statements not just by true or false, but by values in some commutativesemiring; this can provide much more detailed information, for instanceconcerning the combinations of atomic facts that imply the truth of astatement, or practical information about evaluation costs, confidence scores,access levels or the number of successful evaluation strategies. There is awide variety of different semirings that are relevant for provenance analysis,and the applicability of classical logical methods in semiring semantics maystrongly depend on the algebraic properties of the underlying semiring.  While Ehrenfeucht-Fra\"iss\'e games are sound and complete for logicalequivalences in classical semantics, and thus on the Boolean semiring, this isin general not the case for other semirings. We provide a detailed analysis ofthe soundness and completeness of model comparison games on specific semirings,not just for classical Ehrenfeucht-Fra\"iss\'e games but also for othervariants based on bijections or counting.  Finally we propose a new kind of games, called homomorphism games, which arebased on the fact that there exist certain rather simple semiringinterpretations that are locally very different and can be separated even in aone-move game, but which can be proved to be elementarily equivalent viaseparating sets of homomorphisms into the Boolean semiring. We prove that thesehomomorphism games provide a sound and complete method for logical equivalenceson finite and infinite lattice semirings.

Ehrenfeucht-Fra\"iss\'e Games in Semiring Semantics