Recently, Esparza et al. generalized Newton’s method – a numerical-analysis algorithm for finding roots of real-valued functions – to a method for finding fixed-points of systems of equations over semirings. Their method provides a new way to solve interprocedural dataflow-analysis problems. As in its real-valued counterpart, each iteration of their method solves a simpler ``linearized'' problem.

One of the reasons this advance is exciting is that some numerical analysts have claimed that "all effective and fast iterative [numerical] methods are forms (perhaps very disguised) of Newton’s method.'' However, there is an important difference between the dataflow-analysis and numerical-analysis contexts: when Newton’s method is used on numerical-analysis problems, multiplicative commutativity is relied on to rearrange expressions of the form “cX + Xd” into “(c+d) * X.” Such equations correspond to path problems described by regular languages. In contrast, when Newton’s method is used for interprocedural dataflow analysis, the ``multiplication'' operation involves function composition, and hence is non-commutative: “cX + Xd” cannot be rearranged into “(c+d) * X.” Such equations correspond to path problems described by linear context-free languages (LCFLs).

In this paper, we present an improved technique for solving the LCFL sub-problems produced during successive rounds of Newton’s method. Our method applies to predicate abstraction, on which most of today’s software model checkers rely.