Combining Small-Step and Big-Step Semantics to Verify Loop Optimizations

TL;DR

This paper introduces a hybrid semantics approach combining small-step and big-step semantics to verify loop optimizations, enabling more effective structural transformations while preserving program behavior.

Contribution

It proposes a novel framework that integrates both semantics types with an abstract interface, extending big-step semantics to handle divergence and enabling verified loop optimizations in CompCert.

Findings

Successfully verified loop optimizations including unrolling in CompCert

Extended big-step semantics to handle divergence with infinite traces

Demonstrated practical integration of semantics in compiler transformations

Abstract

Verified compilers aim to guarantee that compilation preserves the observable behavior of source programs. While small-step semantics are widely used in such compilers, they are not always the most convenient framework for structural transformations such as loop optimizations. This paper proposes an approach that leverages both small-step and big-step semantics: small-step semantics are used for local transformations, while big-step semantics are employed for structural transformations. An abstract behavioral semantics is introduced as a common interface between the two styles. Coinductive big-step semantics is extended to correctly handle divergence with both finite and infinite traces, bringing it on par with the expressiveness of small-step semantics. This enables the insertion of big-step transformations into the middle of an existing small-step pipeline, thereby fully preserving all top-level semantic preservation theorems. This approach is practically demonstrated in CompCert by implementing and verifying a few new loop optimizations in big-step Cminor, including loop unswitching and, notably, full loop unrolling.