Dynamic Symmetry breaking with AI and Algebra

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Dynamic Symmetry breaking with AI and Algebra
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Dynamic Symmetry breaking with AI and Algebra

• Iain McDonald
• iain_at_dcs.st-and.ac.uk

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Symmetry Breaking

• Cut branches of the search tree
• SBDS

• Partial assignments are symmetrically equivalent

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Symmetry Example

• This assignment results in failure

• From this we can infer that X is also no good

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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)

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Problems with this method

• Overhead increases with the number of symmetries
• At the root of the search tree all symmetries are
  equivalent

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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)

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Orbit finding Algorithm

• Finds all the places that a partial assignment
  can go to

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Advantages of Algebra

• Does not have overhead
• Store only the generators
• Find unique symmetries at each node

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Problems with Algebra

• Re-generating symmetries at each node
• Symmetries may be invalid

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Ideas for the future

• Current progress
• Greater integration of the two techniques

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Thank you