On fixed point equations over commutative semirings

Abstract

Fixed point equations x = f (x) over ω-continuous semirings can be seen as the mathematical foundation of interprocedural program analysis. The sequence 0, f (0), f2(0),... converges to the least fixed point μf. The convergence can be accelerated if the underlying semiring is commutative. We show that accelerations in the literature, namely Newton's method for the arithmetic semiring [4] and an acceleration for commutative Kleene algebras due to Hopkins and Kozen [5], are instances of a general algorithm for arbitrary commutative w-continuous semirings. In a second contribution, we improve the Ο(3n) bound of [5] and show that their acceleration reaches μf after n iterations, where n is the number of equations. Finally, we apply the Hopkins-Kozen acceleration to itself and study the resulting hierarchy of increasingly fast accelerations. © Springer-Verlag Berlin Heidelberg 2007.