Syntactic Type Soundness Results for the Region Calculus

Abstract

The region calculus of Tofte and Talpin is a polymorphically typed lambda calculus with annotations that make memory allocation and deallocation explicit. It is intended as an intermediate language for implementing Hindley-Milner typed functional languages such as ML without traditional trace-based garbage collection. Static region and effect inference can be used to annotate a statically typed ML program with memory management primitives. Soundness of the calculus with respect to the region and effect system is crucial to guarantee safe deallocation of regions, i.e., deallocation should only take place for objects which are provably dead. The original soundness proof by Tofte and Talpin requires a complex co-inductive safety relation. In this paper, we present two small-step operational semantics for the region calculus and prove their type soundness with respect to the region and effect system. Following the standard syntactic approach of Wright, Felleisen, and Harper, we obtain simple inductive proofs. The first semantics is store-less. It is simple and elegant and gives rise to perspicuous proofs. The second semantics provides a store-based model for the region calculus. Albeit slightly more complicated, its additional expressiveness allows to model operations on references with destructive update. A pure fragment of both small-step semantics is then proven equivalent to the original big-step operational approach of Tofte and Talpin. This leads to an alternative soundness proof for their evaluation-style formulation.

A Retrospective on Region-Based Memory Management

“…Tofte and Talpin's quite involved soundness proof for the region type system [57,58] has sparked a number of investigations of alternative and simpler techniques for proving type safety and soundness results for the region calculus [5,11,12,31,66].…”

Region-Based Memory Management for a Dynamically-Typed Language

“…The expressions given above includes those for representing run-time values (ranged over by v): They have been borrowed from the formalization of Calcagno et al [4]. An expression n at ρ stores an integer n in region ρ and returns (a pointer to) the integer.…”

“…We define an operational semantics of our target language, following the formalization of Calcagno et al [4].…”

“…As in [4], function applications are carried out by using substitutions, so that the identity of each pointer is lost. (For example, we cannot tell whether or not two occurrences of 1 ρ point to the same location.)…”

“…These theorems together imply that a well-typed, closed expression never accesses a deallocated region. Our proof is based on the syntactic type soundness proof of Calcagno et al [4], and extends it to handle union/recursive types and polymorphism.…”