Modeling Layout Abstractions Using Integer Set Relations

2025-11-13

AI Summary

In deep learning compilers, heterogeneous mathematical foundations and disjoint modeling approaches for CuTe and Triton linear layouts impede unified analysis and cross-system optimization. Method: This paper introduces the first unified layout algebraic model based on integer set relations (ISRs), formally representing the mapping from logical tensor structures to physical memory as integer set transformations expressible in the ISL (Integer Set Library) framework. The model precisely captures strided access patterns, bit-level swizzling, and F₂-vector-space transformations. Contribution/Results: It supports cross-paradigm operations—including layout composition, inverse transformation, and complement—enabling rigorous reasoning across diverse layout families. We validate its correctness, expressive completeness, and cross-system inferential capability across a full spectrum of layouts, from identity to high-dimensional complex configurations. This work establishes the first verifiable, composable, and theoretically grounded unifying framework for layout optimization in deep learning compilation.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Computer Vision: Learning & Optimization for CV

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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.

Problem

• Unifying mathematical representations of CuTe and Triton layout systems
• Enabling formal analysis and verification of tensor layout transformations
• Modeling complex stride calculations and swizzle operations mathematically

Innovation

• Unified layout modeling via integer set relations
• ISL-based algorithms for composition and inversion
• Handles complex stride configurations and swizzle patterns