Dynamic Symmetry breaking with AI and Algebra

1. Dynamic Symmetry breaking with AI and Algebra

2. Dynamic Symmetry breaking with AI and Algebra Iain McDonald iain@dcs.st-and.ac.uk

3. rotate 90º Symmetry Breaking • Cut branches of the search tree • SBDS • Partial assignments are symmetrically equivalent

4. Symmetry Example • This assignment results in failure • From this we can infer that X is also no good

5. SBDS using AI • State of partial assignment, A • Next assignment of a value to a variable, var = val • Symmetric equivalent of A is g(A) • We can now say: • A and (var val) and g(A)  g(var  val)

6. flip rotate 90º Problems with this method • Overhead increases with the number of symmetries • At the root of the search tree all symmetries are equivalent

7. rotate 90º Using algebraic methods • A symmetry is represented by a group element • This permutation is: (1 4 16 13)(2 8 15 9)(3 12 14 5)(6 7 11 10)

8. Orbit Orbit finding Algorithm • Finds all the places that a partial assignment can go to

9. Advantages of Algebra • Does not have overhead • Store only the generators • Find unique symmetries at each node

10. Problems with Algebra • Re-generating symmetries at each node • Symmetries may be invalid

11. Ideas for the future • Current progress • Greater integration of the two techniques

12. Thank you…