Survey Propagation: an Algorithm for Satisfiability

1. Survey Propagation: an Algorithm for Satisfiability A. Braunstein, M. Mezard, R. Zecchina Hannaneh Hajishirzi : hajishir@uiuc.edu

2. Content • SAT Problem • Warning Passing • Survey Propagation • Belief Propagation

3. SAT Problem • CNF • Clause a = (zi1 zi2 … zik) And: Jair = 1 if zir= ~xir Jair = -1 if zir = xir

4. Factor Graph • Bipartite Graph • Variables  Variable nodes • Clauses  Function nodes Edge(a - i): variable “i” appears in clause “a”

5. Variable Node Function Node

6. Some Notions • V(i): neighbors of variable i • V+(i): clauses that i appears un-negated • V- (i): clauses that i appears negated • V(i)\b: set V(i) without a node b

7. Warning Passing j hja uai a = (i  j  k  l) k a i hka hla l uai can be a warning

8. Formulae hja : Cavity Field uai : Cavity Bias [1]

9. WP algorithm • T = 0: Randomly initialize cavity bias(uai ) • For t = 1 to t = tmax 2.1 Update all uai(t) sequentially. (update method) 2.2 If uai(t) = uai(t-1) for all edges, u*ai = uai(t). Goto 3. • If t = tmax Return UN-CONVERGED. else output cavity biases u*ai.

10. Update Method • Update(uai ) • For every j  V(a) \ i, compute hja • Using hja, compute uai

11. Convergence • Local Field: Hi • Contradiction Number: ci

12. b a 1 1 0 0 1 2 0 0 c 0 0 3 d

13. b a 1 1 1 2 1 0 0 c 0 0 0 3 d

14. b a 1 1 1 2 -1 0 0 c 0 0 0 3 d

15. b a 1 1 1 2 1 -1 0 0 c 0 1 1 0 3 d

16. WP on Tree • If Factor graph: Tree • WP converges •  i: ci > 0 UNSAT • Otherwise: SAT

17. 1. Remove edge a-i2. Level of ‘a’ = 0u = 13. Level of ‘a’ = 1 u = 04. Level of ‘a’ = r can be determined from u’s at level r-2

18. Solving SAT with WP 1. Run WP. 2. Check if there are conflicts. 3. Fix all constrained variables Clean the graph. 4. Choose randomly an un-biased variable,Fix it to one value (0, 1), Clean the graph. 5. Run again WP on the new instance.

19. Example x1 = 1, x2 = 0 x3 = 1, x4 = 1 x7 {0, 1}

20. Example x1 = 1, x2 = 0 x3 = 1, x4 = 1 x7 {0, 1} x5 = 1 x6 = 1 1 1 [3]

21. Clustering • Random SAT with M =  N •  < d = 3.921 Set of all satisfying assignmentsis connected (1 cluster) Local Search

22. Clustering • d <  < c = 4.267 (hard-SAT region) - Set of all satisfying assignmentsbecomes divided into subsets (clusters) - Proliferation of meta-stable clusters Difficult for Local Search Methods

23. Clustering •  = c Number of Clusters = 0 •  > c Almost unsatisfiable  Survey Propagation tries to solve SAT problem in hard-SAT region.

24. Definition of Surveys • Survey uai uai in cluster l Ncl :Number of clusters (x, y) = 1 if x = y,otherwise = 0

25. Surveys u  {0, 1} Binomial Distribution For any set of cavity biases: Q(u, v, w,…): Joint Probability

26. [1]

27. Jaj = 1: [2] Jaj = -1:V+(i) & V-(i) exchange

29. Factorized Form Normalization Factor Constraint All values for all ubj

30. Compute C • Class u: ubj = 1, b  Vua(j) 2. Class s: ubj = 1, b  Vsa(j)

31. Compute C • 3. Class 0: ubj = 0, b  V(j)

32. Closed Formula

33. SP Algorithm 1. T = 0: Randomly initialize cavity bias(ai ) 2. For t = 1 to t = tmax 2.1 Update all ai(t) sequentially. (update method) 2.2 If |ai(t) - ai(t-1)| < for all edges, *ai = ai(t). Goto 3. 3. If t = tmax Return UN-CONVERGED. else output cavity biases *ai.

34. Discussion on SP • If factor-graph = tree, the same as WP • No proof of convergence • Experimental: converges where WP doesn’t • Useful for large N

35. Find SAT Assignment • Simple Algorithm (at most 2N SP calls) • Fix one variable • Run SP for size N-1 • If sub-problem is SAT keep the assignment • Otherwise change the value • Drawbacks • Imprecision of determining if problem is Satisfiable or not • Not using information from “surveys”

36. Properties of Variables Fraction of clusters that xj is positive:

37. Properties of Variables

38. Categories of Variables • Under-constrained: Fix it: Affects internal structure of clusters • Biased: Fix it: Few clusters are eliminated • Balanced: Fix it: Enormous effect or and

39. Search Algorithm 1. Run SP 2. Evaluate all Wi+,Wi-,Wi0. 3. Simplify & Search 3.1 If (exists  <> 0) , fix with largest |Wi+ - Wi-| 3.2 If (all  = 0) , Run WalkSAT 3.3 SP doesn’t converge, “Probably UNSAT” 4. If Done, output “SAT” If no contradiction  goto 1

40. BP in SAT Problem • ia(xi): variable takes xi, in the absence of clause “a” • ai(xi): Clause “a” becomes satisfied, given value for xi For all values of variables xj 1 if X={xi} satisfies “a”Otherwise 0

41. BP in SAT Problem • ia(xi): Probability that variable takes xi and violates clause “a” • ai: Probability that all variables in “a” except “i”, violate “a” i appears un-negated

42. BP in SAT Problem Probability that xj = 1

43. Example  1 = 1, 2 = 0  3 = 1, 4 = 1  7 =3/4 ai ia  5 = 1/2  6 = 3/4 [3]

44. Null Message • Null message onto a variable means: • Receives no warning • Under-constrained • 3 states for a variable • 0, 1, * (joker state)

45. New BP • No warning from Vua(i) & Vsa(i) • No warning from Vsa(i), but at least 1 from Vua(i)

46. New BP • No warning from Vua(i), but at least 1 from Vsa(i) • At least 1 warning from Vsa(i), and 1 from Vua(i) New Formula:

47. Loopy Factor Graph 2 possible solutions for WP,2 generalized assignments (1, 1, *), (0, 0, 1) In SP:a1 = b2 = x a2 = b1= yc1 = c2= 0c3= (1-x)2y2 / (1-xy)2

48. When is SP useful? • Maybe in small cases WP is better • Useful in difficult cases for WP • When messages are sent according to different clusters = • When graph is not well connected SP performs better

49. Conclusion • Message Passing Methods for SAT • Advantage of SP • More theoretical & empirical work needed