Fully Local and E cient Evaluation of Alternating Fixed Points ? ( Extended

We introduce Partitioned Dependency Graphs (PDGs), an abstract framework for the speci cation and evaluation of arbitrarily nested alternating xed points. The generality of PDGs subsumes that of similarly proposed models of nested xed-point computation such as Boolean graphs, Boolean equation systems, and the propositional modal mu-calculus. Our main result is an e cient local algorithm for evaluating PDG xed points. Our algorithm, which we call LAFP, combines the simplicity of previously proposed induction-based algorithms (such as Winskel's tableau method for -calculus model checking) with the e ciency of semantics-based algorithms (such as the bit-vector method of Cleaveland, Klein, and Ste en for the equational -calculus). In particular, LAFP is simply speci ed, we provide a completely rigorous proof of its correctness, and the number of xed-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 rst e cient local algorithm for computing xed points of arbitrary alternation depth to appear in the literature.