Abstract
Modern compilers employ sophisticated optimization passes, yet fundamental inefficiencies
remain due to overly conservative treatment of instruction ordering. This paper introduces
a novel approach that applies noncommutative ring theory to compiler optimization
and quality assurance. By mapping instructions to ring elements and computing commutators
[A,B] = AB − BA, we classify operations as commutative (order-invariant) or
non-commutative (order-dependent). Commutative operations can be freely reordered, parallelized,
and optimized without semantic change.
Key contributions: (1) Algebraic framework for commutativity analysis in compilers; (2)
Instruction reordering with mathematical guarantees; (3) Automatic parallelization of commutative
regions; (4) Order-dependent bug detection via non-commutativity identification;
(5) Integration strategy for LLVM infrastructure.
This work represents a paradigm shift from heuristic-based optimization to algebrabased
optimization with p