Symmetry within and between solutions

TL;DR

This paper explores how symmetry, both within and between solutions, can be leveraged to improve problem-solving efficiency and discover new theoretical results in AI, graph theory, and number theory.

Contribution

It reviews recent advances in applying symmetry to constraint satisfaction problems, highlighting methods for exploiting symmetries in solutions and constraints.

Findings

Symmetry can accelerate problem solving in AI.

Symmetry-based reasoning has led to new theoretical insights.

Symmetry application improves understanding in graph and number theory.

Abstract

Symmetry can be used to help solve many problems. For instance, Einstein's famous 1905 paper ("On the Electrodynamics of Moving Bodies") uses symmetry to help derive the laws of special relativity. In artificial intelligence, symmetry has played an important role in both problem representation and reasoning. I describe recent work on using symmetry to help solve constraint satisfaction problems. Symmetries occur within individual solutions of problems as well as between different solutions of the same problem. Symmetry can also be applied to the constraints in a problem to give new symmetric constraints. Reasoning about symmetry can speed up problem solving, and has led to the discovery of new results in both graph and number theory.