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Semiring provenance is a successful approach, originating in database theory, to provide detailed information on the combinations of atomic facts that are responsible for the result of a query. In particular, interpretations in general provenance semirings of polynomials or formal power series give precise descriptions of the successful evaluation strategies or 'proof trees' for the query, and by evaluating these polynomials or power series in specific application semirings, one can extract practical information for instance about the confidence of a query or the cost of its evaluation.While provenance analysis in databases has, for a long time, been largely confined to negationfree query languages such as conjunctive queries, positive relational algebra, Datalog, and several others, a recent approach extends this to model checking problems for logics with full negation. Algebraically this relies on new quotient semirings of dual-indeterminate polynomials or power series, and it has intimate connections with a provenance analysis of finite and infinite games. So far, this new approach has been developed mainly for first-order logic (FO) and for the positive fragment of least fixed-point logic (posLFP). What has remained open is an adequate treatment for fixed-point calculi that admit arbitrary interleavings of least and greatest fixed points such as full LFP or the modal µ-calculus, but also temporal logics such as CTL.The common approach for dealing with least fixed point inductions, as in Datalog or posLFP, is based on ω-continuous semirings and Kleene's Fixed Point Theorem. It turns out that this is not sufficient for arbitrary fixed points. We show that an adequate framework for the provenance analysis of full fixed-point logics is provided by semirings that are (1) fully continuous, (2) absorptive, and (3) chain-positive. Full continuity guarantees that provenance values of least and greatest fixedpoints are well-defined. Absorptive semirings provide a symmetry between least and greatest fixedpoint computations and make sure that provenance values of greatest fixed points are informative. Finally, chain-positivity is not a necessary requirement in all applications, but it is responsible for having truth-preserving interpretations, which give non-zero values to all true formulae.We further identify semirings of generalized absorptive polynomials S ∞ [X] and prove universality properties that make them the most general appropriate semirings for LFP. We illustrate the power of provenance interpretations in these semirings, by relating them to provenance values of plays and strategies in the associated model-checking games. Specifically we prove that the provenance value of an LFP-formula gives precise information on the evaluation strategies in these games.

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