# Fully local and efficient evaluation of alternating fixed points

## Abstract

We introduce Partitioned Dependency Graphs (PDGs), an abstract framework for the specification and evaluation of arbitrarily nested alternating fixed points. The generality of PDGs subsumes that of similarly proposed models of nested fixed-point computation such as Boolean graphs, Boolean equation systems, and the propositional modal mu-calculus. Our main result is *an efficient local algorithm for evaluating PDG fixed points*. Our algorithm, which we call LAFP, combines the simplicity of previously proposed induction-based algorithms (such as Winskel's tableau method for *v*-calculus model checking) with the efficiency of semantics-based algorithms (such as the bit-vector method of Cleaveland, Klein, and Steffen for the equational Μ-calculus). In particular, LAFP is simply specified, we provide a completely rigorous proof of its correctness, and the number of fixed-point iterations required by the algorithm is asymptotically the same as that of the best existing global algorithms. Moreover, preliminary experimental results demonstrate that LAFP performs extremely well in practice. To our knowledge, this makes LAFP the first efficient local algorithm for computing fixed points of arbitrary alternation depth to appear in the literature.

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