In spite of decades of static-analysis research behind developing precise whole-program analyses, languages that use just-in-time (JIT) compilers suffer from the imprecision of resource-bound analyses local to the scope of compilation. Recent promising approaches bridge this gap by splitting program analysis into two phases: a static phase that identifies interprocedural dependencies across program elements, and a dynamic phase that resolves those dependencies to generate final analysis results. Though this approach is capable of generating precise analysis results without incurring analysis cost in JIT compilers, such "staged analyses'' lack a theoretical backing. In particular, it is unclear if one could transform a general whole-program analysis (that resolves dependencies across all program elements) to a staged one that involves evaluation of statically generated partial results later. Similarly, it would be interesting if one could generate such "partial-result evaluators'' in a way that can also be used to argue about their correctness. In this paper, we propose a novel model of static+dynamic partial analysis that addresses all these points, based on the classic theory of partial evaluation.

We begin by shedding light on the enigmatic idea of partial evaluation as well as the associated notion of Futamura projections to generate specialized program interpreters. We then describe partial analysis as the process of evaluating dependencies across program elements with respect to the statically available parts of a program, resulting into partial results. Next, we devise a strategy (by deriving a novel notion of AM projections from Futamura projections) to statically generate specialized evaluators that can process partial results using dynamic dependencies, at run-time. Later, we use our proposed model to straightforwardly establish the correctness and precision properties of the idea of staging, independent of the program analysis under consideration. We demonstrate the applicability of our model by showcasing examples from non-trivial Java program analyses, implementing the pipeline for one of them, and also discussing future possibilities to extend the same. We believe that our contributions in formulating this theory of partial analysis will significantly extend the usage of existing partial analyzers, as well as promote the design of new ones, for and even beyond Java.