We define \emph{webs} to be the collections of producers and consumers (\emph{e.g.}, functions and calls) in a program that are constrained: in higher-order languages, multiple functions can flow to the same call, all of which must agree on an interface (\emph{e.g.}, calling convention). We argue that webs are fundamentally the \emph{unit of transformation}: a change to one member requires changes across the entire web. We introduce a web-centric intermediate language that exposes webs as annotations, and describe web-based (that is, flow-directed) transformations guided by these annotations. As they affect all members of a web, these transformations are interprocedural, operating over entire modules. Through the lens of webs we reframe and generalize a collection of transformations from the literature, including dead-parameter elimination, uncurrying, and defunctionalization, as well as describe novel transformations. We contrast this approach with rewriting strategies that rely on inlining and cascading rewrites.

Webs are an over-approximation of the semantic function-call relationship produced by control-flow analyses (CFA). This information is inherently independent from the transformations; more precise analyses permit more transformations. A limitation of precise analyses is that the transformations may not maintain well-typedness, as the type system is a less-precise static analysis. Our solution is a simple and lightweight typed-based analysis that causes the flow-directed transformations to preserve well-typedness, making flow-directed, type-preserving transformations easily accessible in many compilers. This analysis builds on unification, distinguishing types that \emph{look} the same from types that have to \emph{be} the same. Our experiments show that while our analysis is theoretically less precise, in practice its precision is similar to CFAs.