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

Ehrenfeucht-Fra\"iss\'e games provide a fundamental method for proving elementary equivalence (and equivalence up to a certain quantifier rank) of relational structures. We investigate the soundness and completeness of this method in the more general context of semiring semantics. Motivated originally by provenance analysis of database queries, semiring semantics evaluates logical statements not just by true or false, but by values in some commutative semiring; this can provide much more detailed information, for instance concerning the combinations of atomic facts that imply the truth of a statement, or practical information about evaluation costs, confidence scores, access levels or the number of successful evaluation strategies. There is a wide variety of different semirings that are relevant for provenance analysis, and the applicability of classical logical methods in semiring semantics may strongly depend on the algebraic properties of the underlying semiring. While Ehrenfeucht-Fra\"iss\'e games are sound and complete for logical equivalences in classical semantics, and thus on the Boolean semiring, this is in general not the case for other semirings. We provide a detailed analysis of the soundness and completeness of model comparison games on specific semirings, not just for classical Ehrenfeucht-Fra\"iss\'e games but also for other variants based on bijections or counting. Finally we propose a new kind of games, called homomorphism games, which are based on the fact that there exist certain rather simple semiring interpretations that are locally very different and can be separated even in a one-move game, but which can be proved to be elementarily equivalent via separating sets of homomorphisms into the Boolean semiring. We prove that these homomorphism games provide a sound and complete method for logical equivalences on finite and infinite lattice semirings.