Refunctionalization of abstract abstract machines: bridging the gap between abstract abstract machines and abstract definitional interpreters (functional pearl)

Abstract

machines is a systematic methodology for constructing sound static analyses for higherorder languages, by deriving small-step abstract abstract machines (AAMs) that perform abstract interpretation from abstract machines that perform concrete evaluation. Darais et al. apply the same underlying idea to monadic definitional interpreters, and obtain monadic abstract definitional interpreters (ADIs) that perform abstract interpretation in big-step style using monads. Yet, the relation between small-step abstract abstract machines and big-step abstract definitional interpreters is not well studied. In this paper, we explain their functional correspondence and demonstrate how to systematically transform small-step abstract abstract machines into big-step abstract definitional interpreters. Building on known semantic interderivation techniques from the concrete evaluation setting, the transformations include linearization, lightweight fusion, disentanglement, refunctionalization, and the left inverse of the CPS transform. Linearization expresses nondeterministic choice through first-order data types, after which refunctionalization transforms the first-order data types that represent continuations into higher-order functions. The refunctionalized AAM is an abstract interpreter written in continuation-passing style (CPS) with two layers of continuations, which can be converted back to direct style with delimited control operators. Based on the known correspondence between delimited control and monads, we demonstrate that the explicit use of monads in abstract definitional interpreters is optional. All transformations properly handle the collecting semantics and nondeterminism of abstract interpretation. Remarkably, we reveal how precise call/return matching in control-flow analysis can be obtained by refunctionalizing a small-step abstract abstract machine with proper caching.

Liberate Abstract Garbage Collection from the Stack by Decomposing the Heap

“…Without a continuation component, an Eval PD state is an evaluation configuration and an Apply PD state is an evaluation result. Except for the presence of the time component, State PD exhibits precisely the configuration and result shapes one finds in many stack-precise CFAs [17,8,1,18].…”

“…Formulating our semantics in big-step style offers two advantages to our setting: First, we can readily express them by big-step definitional interpreters at which point we can apply systematic abstraction techniques [1,18] to obtain corresponding CFAs exhibiting perfect stack precision. Second, they emphasize the availability of the configuration store at the delivery point of the evaluation result; this availability is crucial to our ability to shift to a compositional treatment of the store.…”

Staged abstract interpreters: fast and modular whole-program analysis via meta-programming

“…The caching algorithm is also called a co-inductive caching or truncated depth-first evaluation [Rosendahl 2013]. It has been used in other abstract interpreters or fixed-point computation [Darais et al 2017;Rosendahl 2013;Wei et al 2018]. The idea is to use an in cache and an out cache during the depth-first evaluation.…”

“…The unstaged abstract interpreter in this paper follows the abstracting definitional interpreters (ADI) approach [Darais et al 2017]. Similar to ADI, reconstructing big-step abstract interpreters with delimited continuations was shown by Wei et al [2018]. Keidel et al [2018] presented a modular, arrow-based abstract interpreter that allows soundness proofs to be constructed compositionally.…”

“…and sensitivity analyses [Darais et al 2015;Gilray et al 2016a,b]. It has also been applied to various small-step abstract machines Might 2010, 2012;Sergey et al 2013] and big-step definitional interpreters [Darais et al 2017;Keidel et al 2018;Wei et al 2018]. Based on the idea of AAM, more pragmatically, several implementation strategies in high-level functional programming languages have emerged.…”