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 in numerical-analysis problems, commutativity of multiplication is relied on to rearrange an expression of the form “ a * X * b + c * X * d ” into “( a * b + c * d )* X .” Equations with such expressions 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: “ a * X * b + c * X * d ” cannot be rearranged into “( a * b + c * d )* X .” Equations with such expressions correspond to path problems described by linear context-free languages (LCFLs). In this article, 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.

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 ``c*X + X*d'' 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: ``c*X + X*d'' 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.

Current middlebox (MB) management mechanisms are clumsy and unsuitable for taking full advantage of new MB deployment models and diverse MB functionality. Instead, we advocate for mechanisms that help exercise unified control over the key factors influencing MB operations. Our goal is to realize a software-defined MB networking framework to simplify management of complex, diverse functionalities and engender rich deployments. We discuss the major challenges that arise---representing, manipulating, and knowledgeably controlling MB state---and we present initial thoughts on the appropriate abstractions and interfaces to address them.