When a Loop Becomes an Integer Problem: Presburger Reasoning in MLIR Affine

Mon, 27 Jul 2026 00:00:00 +0300

I started with a small compiler question: can this loop run in parallel?

affine.for %i = 0 to %N {
 %x = affine.load %A[2 * %i] : memref<?xf32>
 affine.store %x, %A[2 * %i + 1] : memref<?xf32>
}

Parallelizing it may change the order in which its memory accesses happen, so the compiler needs to check whether one iteration depends on another. Both accesses use the same memref and one of them writes, which makes the question worth asking.

Assume %A has room for $2N$ elements. Every iteration reads an even address and writes the following odd address. By inspection, different iterations never touch the same element. What made me curious was how MLIR could reach that answer from the Affine IR. Along the way, loop syntax turns into matrix rows, floor division creates a hidden integer witness, and a parallelization question becomes a question about whether an integer set is empty. This post is about that path.

Compilers on onur.one

When a Loop Becomes an Integer Problem: Presburger Reasoning in MLIR Affine