Newton’s Method for ω-Continuous Semirings

Abstract

Fixed point equations ${\bf\it X} = {\bf\it f}({\bf\it X})$ over?-continuous semirings are a natural mathematicalfoundation of interprocedural program analysis. Generic algorithmsfor solving these equations are based on Kleene's theorem, whichstates that the sequence ${\bf{0}}, {\bf\it f}({\bf{0}}), {\bf\itf}({\bf\it f}({\bf{0}})), \ldots$ converges to the least fixedpoint. However, this approach is often inefficient. We report onrecent work in which we extend Newton's method, the well-knowntechnique from numerical mathematics, to arbitraryω-continuous semirings, and analyze its convergencespeed in the real semiring.