# Symmetry in Software Synthesis

## TL;DR

This paper introduces a formal, mathematically grounded framework to identify and leverage inherent symmetry in hardware and software for more efficient multi-core mapping and scheduling, improving performance significantly.

## Contribution

It presents a novel formal framework using group theory and inverse semigroups to automatically detect symmetries, enhancing scalability and effectiveness in software synthesis for complex architectures.

## Key findings

- Achieves up to 10x acceleration in mapping algorithms
- Reduces execution time of algorithms significantly
- Improves quality of mapping results

## Abstract

With the surge of multi- and manycores, much research has focused on algorithms for mapping and scheduling on these complex platforms. Large classes of these algorithms face scalability problems. This is why diverse methods are commonly used for reducing the search space. While most such approaches leverage the inherent symmetry of architectures and applications, they do it in a problem-specific and intuitive way. However, intuitive approaches become impractical with growing hardware complexity, like Network-on-Chip interconnect or heterogeneous cores. In this paper, we present a formal framework that can determine the inherent symmetry of architectures and applications algorithmically and leverage these for problems in software synthesis. Our approach is based on the mathematical theory of groups and a generalization called inverse semigroups. We evaluate our approach in two state-of-the-art mapping frameworks. Even for the platforms with a handful of cores of today and moderate-size benchmarks, our approach consistently yields reductions of the overall execution time of algorithms, accelerating them by a factor up to 10 in our experiments, or improving the quality of the results.