Partial Redundancy Elimination for Multi-threaded Programs

Multi-threaded programs have many applications which are widely used such as operating systems. Analyzing multi-threaded programs differs from sequential ones; the main feature is that many threads execute at the same time. The effect of all other running threads must be taken in account. Partial redundancy elimination is among the most powerful compiler optimizations: it performs loop-invariant code motion and common subexpression elimination. We present a type system with optimization component which performs partial redundancy elimination for multi-threaded programs.

💡 Research Summary

The paper “Partial Redundancy Elimination for Multi‑threaded Programs” introduces a novel approach to applying Partial Redundancy Elimination (PRE) in the context of multi‑threaded code. PRE is a well‑known compiler optimization that combines loop‑invariant code motion and common subexpression elimination, but existing formulations have been limited to sequential languages. The authors address this gap by designing a type‑system‑based framework that can safely perform PRE on programs written in a simple imperative language called FWHILE, which includes a fork construct for spawning parallel threads.

The core of the approach is a series of static analyses expressed as typing rules. First, a modified analysis computes, for each program point, the set of variables that have been assigned up to that point. This is a forward “must” analysis whose subtyping relation is reverse set inclusion, ensuring that the set of modified variables only grows along execution paths. Second, the authors define a concurrent‑modified function C that, for any statement, returns the set of variables that might be altered by other concurrently executing threads. For a fork statement, C aggregates the modified sets of all child threads, thereby capturing the potential interference between threads.

Building on these two analyses, the paper presents two classic data‑flow properties adapted to the multi‑threaded setting: anticipability (which expressions are guaranteed to be evaluated before any of their operands are modified on all paths) and partial availability (which expressions have already been computed and remain unmodified on some paths). The anticipability analysis is performed backward, and it explicitly removes any expression that depends on a variable in C, preventing unsafe hoisting across thread boundaries. The partial availability analysis proceeds forward, tracking expressions that are both already computed and not later killed, again respecting the concurrent‑modified set.

The authors integrate these analyses into a type system with an optimization component. Typing judgments have the form Γ ⊢ s : ant, cpav → s', where ant and cpav are the anticipable and partially available expression sets, and s' is the optimized version of statement s. For ordinary constructs (assignments, conditionals, loops) the rules are identical to those of Saabas and Ustaluu’s earlier work on sequential PRE. The new contribution lies in the rule for the fork statement: each thread is typed independently, using its own ant and cpav sets, while the concurrent‑modified function ensures that any expression that could be invalidated by another thread is excluded from hoisting. Consequently, the optimizer can safely move invariant computations out of loops or across thread boundaries without risking incorrect results.

A substantial part of the paper is devoted to soundness proofs. The authors define a semantic relation σ ⊨ m indicating that a state satisfies a modified‑variable set, and they prove that if a statement is typed with a certain m then executing the statement from a state satisfying m yields a state that also satisfies the resulting m. Similar proofs are given for anticipability and partial availability. The most intricate proof concerns the fork rule, where the authors must show that, regardless of the permutation θ governing thread execution order, the optimized program preserves the semantics of the original. They achieve this by inductively applying the lemmas for modified analysis and the properties of C, demonstrating that the optimizer’s transformations are semantics‑preserving even in the presence of arbitrary interleavings.

In the related‑work section, the paper surveys classic PRE literature (Morel & Renvoise, maximum‑flow formulations, strength reduction) and prior type‑system approaches to program analysis and optimization. It also reviews analyses of multi‑threaded programs such as data‑race detection, dead‑lock analysis, and pointer analysis, noting that none of these address PRE. The authors claim that this is the first work to combine a type‑system framework with PRE for multi‑threaded code.

The conclusion restates the contribution: a sound, type‑based PRE technique for a language with parallel threads. Limitations are acknowledged: the analysis assumes shared global variables only, does not model synchronization primitives (mutexes, semaphores), and has not been implemented in a full compiler or evaluated on real benchmarks. Future work includes extending the framework to richer languages, incorporating synchronization, and performing empirical performance studies.

Overall, the paper offers a solid theoretical foundation for extending classic compiler optimizations to concurrent settings, demonstrating that type systems can serve as a clear and provably correct vehicle for such extensions.