ACL2 Meets the GPU: Formalizing a CUDA-based Parallelizable All-Pairs Shortest Path Algorithm in ACL2

TL;DR

This paper formalizes a GPU-based all-pairs shortest path algorithm in ACL2, demonstrating correctness, efficiency, and the ability to recover shortest paths, bridging GPU programming and formal verification.

Contribution

It introduces a formal ACL2 specification of a CUDA-based APSP algorithm, including path recovery, and demonstrates its correctness and performance compared to C and CUDA implementations.

Findings

ACL2 specification processes millions of vertices efficiently.

ACL2 version runs at one-sixth the speed of C implementation.

Porting ACL2 code back to C and CUDA shows minimal or slight performance changes.

Abstract

As Graphics Processing Units (GPUs) have gained in capability and GPU development environments have matured, developers are increasingly turning to the GPU to off-load the main host CPU of numerically-intensive, parallelizable computations. Modern GPUs feature hundreds of cores, and offer programming niceties such as double-precision floating point, and even limited recursion. This shift from CPU to GPU, however, raises the question: how do we know that these new GPU-based algorithms are correct? In order to explore this new verification frontier, we formalized a parallelizable all-pairs shortest path (APSP) algorithm for weighted graphs, originally coded in NVIDIA's CUDA language, in ACL2. The ACL2 specification is written using a single-threaded object (stobj) and tail recursion, as the stobj/tail recursion combination yields the most straightforward translation from imperative programming languages, as well as efficient, scalable executable specifications within ACL2 itself. The ACL2 version of the APSP algorithm can process millions of vertices and edges with little to no garbage generation, and executes at one-sixth the speed of a host-based version of APSP coded in C- a very respectable result for a theorem prover. In addition to formalizing the APSP algorithm (which uses Dijkstra's shortest path algorithm at its core), we have also provided capability that the original APSP code lacked, namely shortest path recovery. Path recovery is accomplished using a secondary ACL2 stobj implementing a LIFO stack, which is proven correct. To conclude the experiment, we ported the ACL2 version of the APSP kernels back to C, resulting in a less than 5% slowdown, and also performed a partial back-port to CUDA, which, surprisingly, yielded a slight performance increase.