Modeling Layout Abstractions Using Integer Set Relations

TL;DR

This paper introduces a unified mathematical framework using integer set relations to analyze and verify different tensor layout systems in deep learning compilers, enabling formal reasoning and cross-system optimization.

Contribution

It develops a novel ISL-based approach to model CuTe and Triton layouts within a single formal system, facilitating correctness verification and future optimization strategies.

Findings

Successfully models complex layout transformations including swizzles

Enables formal analysis and verification of layout correctness

Handles diverse layout complexities from simple to sophisticated

Abstract

Modern deep learning compilers rely on layout abstractions to manage the complex mapping between logical tensor structures and physical memory arrangements. CuTe layouts and Triton linear layouts are widely adopted industry standards. However, these layout systems operate independently with distinct mathematical underpinnings, preventing unified formal analysis and cross-system reasoning. We bridge this gap by introducing a novel approach that leverages the Integer Set Library (ISL) to create a unified mathematical representation for both layout systems through integer set relations, thereby enabling rigorous formal analysis, correctness verification, and the foundation for future cross-system optimization strategies. Our approach models CuTe layouts through integer set relations that encode the transformation from multi-dimensional coordinates to linear indices using stride-based calculations, including sophisticated swizzle operations that perform bit-level manipulations for enhanced memory access patterns. For Triton linear layouts, we construct integer set relations that model the binary vector space transformations where arithmetic operations follow finite field F_2 rules. We implement a complete suite of layout manipulation algorithms for composition, inversion, complement using built-in operations in ISL to ensure mathematical correctness and preserve layout semantics. Experimental evaluation shows that the system handles the full spectrum of layout complexity, from elementary identity transformations to sophisticated multi-dimensional tensor arrangements with complex stride configurations and swizzle patterns, validating the mathematical modeling approach across different layout paradigms.