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We develop a model-checking algorithm for a logic that permits propositions to be defined using greatest and least fixed points of mutually recursive systems of equations. This logic is as expressive as the alternation-free fragment of the modal mu-calculus identified by Emerson and Lei, and it may therefore be used to encode a number of temporal logics and behavioral preorders. Our algorithm determines whether a process satisfies a formula in time proportional to the product of the sizes of the process and the formula; this improves on the best known algorithm for similar fixed-point logics.