An extension of Newton’s method to ω-continuous semirings

Fixed point equations x = F (x) over ω-continuous semirings are a natural mathematical foundation of interprocedural program analysis. Equations over the semiring of the real numbers can be solved numerically using Newton’s method. We generalize the method to any ω-continuous semiring and show that it converges faster to the least fixed point than the Kleene sequence 0, F (0), F (F (0)),... We prove that the Newton approximants in the semiring of languages coincide with finite-index approximations studied by several authors in the 1960s. Finally, we apply our results to the analysis of stochastic context-free grammars