Modeling Layout Abstractions Using Integer Set Relations

TL;DR

This work addresses the lack of a unified mathematical framework for diverse tensor layout abstractions by modeling CuTe and Triton linear layouts as integer set relations in the ISL library. It develops a cohesive approach that expresses coordinate, index, and layout mappings as ISL relations, enabling formal analysis, verification, and cross-system reasoning. Key contributions include a complete suite of ISL-based layout operations (composition, inversion, complement), hierarchical layout handling, and practical tooling (isl-layout) that translates CuTe and linear layouts into ISL relations. The framework lays foundational groundwork for cross-system optimization and formal verification in deep learning compilers, with potential integration into broader compiler ecosystems such as MLIR.

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.

Figures (3)

• Figure 1: Mappings defined by the CuTe layout $H = (4, 2, 2) : (2, 1, 8)$
• Figure 2: Layout mapping for CuTe swizzle $H_{1, 2, 1}$
• Figure 3: Linear layout mapping for $H=(crd=(4,4), idx=(4,4), vals=[(1,1), (2,2), (0,1), (0,2)])$