Semiring semantics relies on the idea to evaluate logical statements not just by true or false, but by values in some commutative semiring. In this context, the standard semantics re-appears as the special case when the Boolean semiring is used. Valuations in other semirings provide additional information, beyond the truth or falsity of a statement, such as costs, confidence scores, access levels, or the number of successful evaluation strategies. Further, semirings of polynomials or formal power series permit us to track which atomic facts are used (and how often) to establish the truth of a sentence in a given structure. Semiring semantics originated in the provenance analysis for (positive) database query languages, but in the last years it has been systematically extended to many logical systems, including first-order logic and fixed-point logic. This raises the question to what extent the standard model-theoretic and algorithmic results and techniques of classical logic (and specifically finite model theory) extend to semiring semantics, and how such extensions depend on the choice of the underlying semiring.


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