1 Constraint Symmetry and Solution Symmetry Presented by Beau M. Christ Symmetry in CSP’s Spring 2010 Presented by Beau M. Christ Symmetry in CSP’s Spring 2010

2 OutlineOutline IntroductionDefinitions Constraint and Solution Symmetry Relationships Symmetry in CSPs with Few Solutions Symmetry in SAT Symmetry in Practice Conclusion

3 IntroductionIntroduction Symmetry can be problematic Thus, symmetry-breaking research is active And yet, how do we define symmetry?

4 OutlineOutline IntroductionDefinitions Constraint and Solution Symmetry Relationships Symmetry in CSPs with Few Solutions Symmetry in SAT Symmetry in Practice Conclusion

5 Definition #1 A CSP instance is a triple V,D,C where: V is a set of variables; V is a set of variables; D is a universal domain, specifying the possible values for those variables; D is a universal domain, specifying the possible values for those variables; C is a set of constraints. Each constraint c ∈ C is a pair c = σ,ρ where σ is a list of variables from V, called the constraint scope, and ρ is a ∣ σ ∣ -ary relation over D, called the constraint relation. C is a set of constraints. Each constraint c ∈ C is a pair c = σ,ρ where σ is a list of variables from V, called the constraint scope, and ρ is a ∣ σ ∣ -ary relation over D, called the constraint relation.

6 DefinitionsDefinitions An assignment of values to variables is a set {v 1,a 1,v 2,a 2,..., v k,a k } where {v 1,v 2,..., v k } ⊆ V and a i ∈ D, for all 1 ≤ i ≤ k. A solution to a CSP instance V,D,C is a mapping from V into D whose restriction to each constraint scope is in the corresponding constraint relation, i.e., is allowed by the constraint

7 DefinitionsDefinitions A general agreement for symmetry in CSPs Acts on variable-value pairs Symmetries map solutions to solutions and non-solutions to non-solutions But should any bijective mapping that preserves solutions be considered a symmetry, or just a consequence of leaving the constraints unchanged?

8 Definition #2 For any CSP instance P = V,D,C, a solution symmetry of P is a permutation of the set V × D that preserves the set of solutions to P.

9 DefinitionsDefinitions But we do not want to have to examine all solutions to a problem in order to identify symmetry. This leads us to the second viewpoint of symmetry, which uses the microstructure of the CSP.

10 Definition #3 For any CSP instance P = V,D,C, the microstructure complement of P is a hypergraph with set of vertices V× D. A set of vertices is a hyperedge of the microstructure complement if it represents an assignment disallowed by a constraint, or consists of a pair of incompatible assignments for the same variable.

11 DefinitionsDefinitions Note: The vertices of the microstructure complement are variable-value pairs of the CSP, and a set of vertices {v 1,a 1,v 2,a 2,..., v k,a k } is a hyper edge if and only if: {v 1,v 2,..., v k } is the set of variables in the scope of some constraint, but the constraint disallows the assignment {v 1,a 1,v 2,a 2,..., v k,a k }; or {v 1,v 2,..., v k } is the set of variables in the scope of some constraint, but the constraint disallows the assignment {v 1,a 1,v 2,a 2,..., v k,a k }; or k=2, v 1 = v 2, and a 1 ≠ a 2 k=2, v 1 = v 2, and a 1 ≠ a 2

12 Definition #4 For any CSP instance P = V,D,C, a constraint symmetry is an automorphism of the microstructure complement of P (or, equivalently, of the microstructure).

13 Example #1: n-queens Standard formulation V = {r 1,r 2,..., r n } representing the rows D = {1,2,..., n} representing the columns C (intension) for all i, j, 1 ≤ i < j ≤ n, r i ≠ r j ; for all i, j, 1 ≤ i < j ≤ n, ∣ r i - r j ∣ ≠ ∣ i - j ∣

14 Example #1: n-queens Chessboard has 8 symmetries (x, y, d1, d2, r90, r180, r270, id) Symmetries map vvp r i,k to (respectively): r n+1-i,k, r i,n+1-k, r i,i, r n+1-k,n+1-i, r k,n+1-i, r n+1-i,n+1-k, r n+1-k,i, r i,k

15 Example 1: n-queens This hides symmetry between rows and columns This symmetry is restored using the microstructure complement, as it treats both reasons for a disallowed pair of assignments equally

16 OutlineOutline IntroductionDefinitions Constraint and Solution Symmetry Relationships Symmetry in CSPs with Few Solutions Symmetry in SAT Symmetry in Practice Conclusion

17 Relationships: Theorem 1 The group of constraint symmetries of a CSP instance P is a subgroup of the group of solution symmetries of P. Note: The converse is not true.

18 Relationships: Definition #5 For any CSP instance P, a k-ary nogood is an assignment to k variables of P that cannot be extended to a solution of P. The k-nogood hypergraph of P is a hypergraph whose set of vertices is V× D and whose set of edges is the set of all m-ary nogoods for all m ≤ k.

19 Relationships: Theorem 2 For any k-ary CSP instance P, the group of all solution symmetries of P is equal to the automorphism group of the k-nogood hypergraph of P. Thus, the k-nogood hypergraph of a CSP instance has the same vertices as the microstructure component.

20 Example 2: 4-queens Fig. 1: The 4-queens solutions and the complement of the binary no-good hypergraph

21 Example 2: 4-queens The automorphisms are: vertices within either clique can be permuted vertices in one clique can be swapped with the other eight isolated vertices can be permuted compose these permutations

22 Example 2: 4-queens Since the automorphisms of a graph are the same as the automorphisms of its complement, then theorem 2 tells us these are the solution symmetries for the 4-queens problem.

23 OutlineOutline IntroductionDefinitions Constraint and Solution Symmetry Relationships Symmetry in CSPs with Few Solutions Symmetry in SAT Symmetry in Practice Conclusion

24 CSPs with Few Solutions Any permutation of vvp’s is a solution symmetry when a CSP instance has no solutions. Furthermore, CSP instances with very few solutions must have many solution symmetries.

25 Example 3 Suppose a CSP instance with n variables and d values for each variable has only one solution Any permutation of the n vvp’s in the solution is a solution symmetry The permutations of the n(d-1) vvp’s not in the solution are also solution symmetries Thus, the solution symmetry group has n! × (n(d- 1))! elements

26 Example 4 Suppose we have a CSP instance with 2 solutions, where each solution has k assignments in common A solution symmetry must permute the k common assignments amongst themselves The remaining (n-k) assignments in each solution can also be permuted

27 Example 4 The two solutions can be swapped The other (nd-2n+k) assignments can also be permuted. Thus, we have k! × (n-k)! 2 × 2 × (nd-2n+k)! solution symmetries Note: This is 4-queens with k=0

28 CSPs with Few Solutions The basic idea is that it is often more difficult to construct permutations with more solutions For n-queens where n >6, it appears that the solution symmetries are exactly the 8 symmetries of the constraint symmetry group

29 CSPs with Few Solutions One way of dealing with constraint symmetry is to add symmetry-breaking constraints Symmetry breaking can eliminate constraint symmetry but lead to a much larger solution symmetry group

30 OutlineOutline IntroductionDefinitions Constraint and Solution Symmetry Relationships Symmetry in CSPs with Few Solutions Symmetry in SAT Symmetry in Practice Conclusion

31 Symmetry in SAT A SAT instance can be viewed as a CSP instance We can define a symmetry of a SAT instance as a permutation of the variables that leaves the set of clauses unchanged (Aloul et al. 2003) allow symmetries to act on literals, and thus is similar to our definition of constraint symmetry

32 Symmetry in SAT They construct a graph to represent the set of clauses, whose automorphisms are the symmetries of the SAT instance The graph is similar to the microstructure complement, but has vertices for clauses as well as literals, and colors for different types of vertex

33 OutlineOutline IntroductionDefinitions Constraint and Solution Symmetry Relationships Symmetry in CSPs with Few Solutions Symmetry in SAT Symmetry in Practice Conclusion

34 Symmetry in Practice Identifying symmetry helps ease search Add symmetry breaking constraints OR use a dynamic method that adds symmetry breaking constraints upon backtracking In practice, researchers identify constraint symmetries rather than solution symmetries

35 InterchangeabilityInterchangeability Freuder’s interchangeability is an explicit form of solution symmetry Full interchangeability can determine values that are, in a sense, equivalent This, in general, requires finding all solutions

36 InterchangeabilityInterchangeability Freuder defines local forms of interchangeability (such as neighborhood interchangeability, which is a form of constraint symmetry) Future work aims to identify values that are interchangeable in the set of solutions, and not just the constraints

37 OutlineOutline IntroductionDefinitions Constraint and Solution Symmetry Relationships Symmetry in CSPs with Few Solutions Symmetry in SAT Symmetry in Practice Conclusion

38 ConclusionConclusion Constraint symmetry: automorphism groups of the microstructure complement Solution symmetry: automorphism groups of the k-ary nogood hypergraph Solution symmetry may be less useful than constraint symmetry Although most research has dealt with constraint symmetry, new research is being done with interchangeability

39 OutlineOutline IntroductionDefinitions Constraint and Solution Symmetry Relationships Symmetry in CSPs with Few Solutions Symmetry in SAT Symmetry in Practice Conclusion

40 ReferenceReference “Constraint Symmetry and Solution Symmetry” ---Cohen et. al--- AAAI 2006

41 That’s all, folks! Questions?Comments?Remarks?