1. PhD March 20071 The Evergreen Project: How To Learn From Mistakes Caused by Blurry Vision in MAX-CSP Solving Karl J. Lieberherr Northeastern University Boston joint work with Ahmed Abdelmeged, Christine Hang and Daniel Rinehart

2. PhD March 20072 Where we are Introduction Look-forward Look-backward Packed truth tables SPOT: how to use the look-ahead polynomials (look-forward) together with superresolution (look-backward).

3. PhD March 20073 Problem Snapshot SAT: classic problem in complexity theory SAT & MAX-SAT Solvers: working on CNFs (a multi-set of disjunctions). CSP: constraint satisfaction problem –Each constraint uses a Boolean relation. –e.g. a Boolean relation 1in3(x y z) is satisfied iff exactly one of its parameters is true. CSP & MAX-CSP Solvers: working on CSP instances (a multi-set of constraints).

4. PhD March 20074 Introduction Boolean MAX-CSP(G) for rank d, G = set of relations of rank d –Input Input = Bag of Constraint Constraint = Relation + Set of Variable Relation = int. // Relation number < 2 ^ (2 ^ d) in G Variable = int –Output (0,1) assignment to variables which maximizes the number of satisfied constraints. Example Input: G = {22} of rank 3 –22:1 2 3 0 –22:1 2 4 0 –22:1 3 4 0 1in3 has number 22 M = {1 !2 !3 !4} satisfies all

5. PhD March 20075 Variation MAX-CSP(G,f): Given a MAX-CSP(G) instance expressed in n variables which may assume only the values 0 or 1, find an assignment to the n variables which satisfies at least the fraction f of the constraints. Example: G = {22} of rank 3 MAX-CSP({22},f): 22:1 2 3 0 22:1 2 4 0 22:1 3 4 0 22: 2 3 4 0

6. PhD March 20076 Our Approach Superresolution & P-Optimality Based MAX-CSP Solver Highlights –Look Forward (in Abstract Representation) –Look Backward (in Transition System) –Packed Truth Tables (in Intermediate Representation)

7. PhD March 20077 Where we are Introduction Look-forward Look-backward Packed truth tables SPOT: how to use the look-ahead polynomials together with superresolution.

8. PhD March 20078 Look Forward Why? –To make informed decisions How? –Abstract representation based on look-ahead polynomials

9. PhD March 20079 Look-ahead Polynomial (Intuition) The look-ahead polynomial computes the expected fraction of satisfied constraints among all random assignments that are produced with bias p.

10. PhD March 200710 Consider an instance: 40 variables, 1000 constraints (1in3) 1, …,40 22: 6 7 9 0 22: 12 27 38 0 Abstract representation: reduce the instance to look-ahead poly. 3p(1-p) 2

11. PhD March 200711 3p(1-p) 2 for MAX-CSP({22})

12. PhD March 200712 Look-ahead Polynomial (Definition) F is a MAX-CSP(G) instance. N is an arbitrary assignment. The look-ahead polynomial la F,N (p) computes the expected fraction of satisfied constraints of F when each variable in N is flipped with probability p.

13. PhD March 200713 The general case MAX-CSP(G) G = {R 1, … }, t R (F) = fraction of constraints in F that use R. x = p

14. PhD March 200714

15. PhD March 200715 Look-ahead Polynomial in Action Focus on purely mathematical question first Algorithmic solution will follow Mathematical question: Given a MAX- CSP(G) instance. For which fractions f is there always an assignment satisfying fraction f of the constraints? In which constraint systems is it impossible to satisfy many constraints?

16. PhD March 200716 Remember? MAX-CSP(G,f): Given a MAX-CSP(G) instance expressed in n variables which may assume only the values 0 or 1, find an assignment to the n variables which satisfies at least the fraction f of the constraints. Example: G = {22} of rank 3 MAX-CSP({22},f): 22:1 2 3 0 22:1 2 4 0 22:1 3 4 0 22: 2 3 4 0

17. PhD March 200717 Simple example MAX-CSP({22},f): For f <= u: problem has always a solution For f = u +  : problem has not always a solution,    u  critical transition point always (fluid) not always (solid)

18. PhD March 200718 The Magic Number u = 4/9

19. PhD March 200719 3p(1-p) 2 for MAX-CSP({22})

20. PhD March 200720 Produce the Magic Number Use an optimally biased coin –1/3 in this case In general: min max problem

21. PhD March 200721 General Dichotomy Theorem MAX-CSP(G,f): For each finite set G of relations there exists an algebraic number t G For f <= t G : MAX-CSP(G,f) has polynomial solution For f = t G +  : MAX-CSP(G,f) is NP-complete,   t G  critical transition point easy (fluid) hard (solid) due to Lieberherr/Specker polynomial solution: Use optimally biased coin. Derandomize. P-Optimal. 

22. PhD March 200722 Observations The look-ahead polynomial look-forward approach has not been used in state-of- the-art MAX-SAT and Boolean MAX-CSP solvers. Often a fair coin is used. The optimally biased coin is often significantly better.

23. PhD March 200723

24. PhD March 200724 N 0 ={!v 1,!v 2,!v 3,!v 4 }

25. PhD March 200725 N 0 ‘ ={v 1,!v 2,!v 3,!v 4 }

26. PhD March 200726 SAT Rank 2 example 9 constraints 14 : 1 2 0 14 : 3 4 0 14 : 5 6 0 7 : 1 3 0 7 : 1 5 0 7 : 3 5 0 7 : 2 4 0 7 : 2 6 0 7 : 4 6 0 14: 1 2 = or(1 2) 7: 1 3 = or(!1 !3) What is the look-ahead polynomial?

27. PhD March 200727 appmean = lookahead is an approximation of the true mean Blurry vision What do we learn from the abstract representation? set 1/3 of the variables to true (maximize). the best assignment will satisfy at least 7/9 constraints. very useful but the vision is blurred in the “middle”. excellent peripheral vision

28. PhD March 200728 Where we are Introduction Look-forward Look-back Packed truth tables SPOT: how to use the look-ahead polynomials

29. PhD March 200729 Look Backward Why? –to avoid past mistakes How? –Transition system based on superresolution

30. PhD March 200730 Observation Optimally biased coin technique based on look-ahead polynomials is “best-possible”. If we could improve it by a trillionth in polynomial time, then P=NP. We improve it now by learning new constraints that will influence the polynomial.

31. PhD March 200731 Clause Learning Let’s go beyond what an optimally biased coin guarantees! Goal: satisfy the maximum number of constraints. Approach: Superresolution. –When to apply: number of constraints guaranteed to be unsatisfied doesn’t decrease A mistake is made. –Who to blame: the decision literals They are the culprits. –How to penalize: add the disjunctions of their negations as a superresolvent The gang of culprits is watched.

32. PhD March 200732 Transition Rules Semi-Superresolution (SSR): NewSR = V (¬k), where k M d M || F || SR || N → M || F || SR, NewSR || N if unsat(M,SR) > 0 or unsat(M,F) ≥ unsat(N,F).

33. PhD March 200733 Algorithm plan start with an arbitrary assignment N. while (proof incomplete) { –try to improve N by creating new assignment from scratch using optimally biased coin to flip the assignments; success: Update N; failure: learn a new constraint that will prevent same mistake and will “improve” the polynomial. }

34. PhD March 200734

35. PhD March 200735

36. PhD March 200736 Properties of TS TS finds the maximum in a finite number of steps. It creates a proof that we indeed found the maximum.

37. PhD March 200737 Optimized Semi-Superresolution Not all decision literals may be responsible for the “mistake”. Want to find a minimal superresolvent so that deleting one literal would destroy the superresolvent property. Can be implemented by a traversal back the implication graph that is built as part of unit propagation.

38. PhD March 200738 Where we are Introduction Look-forward Look-back Packed Truth Tables SPOT: how to use the look-ahead polynomials

39. PhD March 200739 Requirements for Packed Truth Tables The look-ahead polynomial can be computed efficiently. Requires efficient truth table analysis. Reduction of an instance must be efficient. Efficiently compute the forced variables. Each relation has a unique representation.

40. PhD March 200740 Packed Truth Tables 22 254

41. PhD March 200741 RelationI: implemented by bitwise operations int isForced(int variablePosition) boolean isIrrelevant(int variablePosition) int nMap(int variablePosition) int numberOfRelevantVariables() int q(int s) int reduce(int variablePosition, int value) int rename(int permutationSemantics, int... permutation)

42. PhD March 200742 Where we are Introduction Look-forward Look-back Packed truth tables SPOT: how to use the look-ahead polynomials with superresolution

43. PhD March 200743 Using the look-ahead polynomials Value Ordering –Decide: how to set the variable Variable Ordering –Which variable to set next

44. PhD March 200744 There is hope that the look-ahead polynomials are useful

45. PhD March 200745 What is new? New: Packed Truth Tables New: Superresolution for MAX-CSP New: Integration of look-ahead polynomials with superresolution Old: Superresolution for SAT (1977) Old: Look-ahead polynomials (1983)

46. PhD March 200746 Future work Exploring best combination of look-forward and look-back techniques. Find all maximum-assignments or estimate their number. Robustness of maximum assignments. Are our MAX-CSP solvers useful for reasoning about biological pathways?

47. PhD March 200747 Conclusions Presented SPOT, a family of MAX-CSP solvers based on look-ahead polynomials and non-chronological backtracking. SPOT has a desirable property: P-optimal. Preliminary experimental results are encouraging.

48. PhD March 200748 end for now

49. PhD March 200749 Rank 2 example 14 : 1 2 0 14 : 3 4 0 14 : 5 6 0 7 : 1 3 0 7 : 1 5 0 7 : 3 5 0 7 : 2 4 0 7 : 2 6 0 7 : 4 6 0

50. PhD March 200750 appmean is an approximation of the true mean

51. PhD March 200751

52. PhD March 200752 Transition Manager

53. PhD March 200753

54. PhD March 200754 MAX-CSP: Superresolution and P-Optimality Karl J. Lieberherr Northeastern University Boston joint work with Ahmed Abdelmeged, Christine Hang and Daniel Rinehart

55. PhD March 200755

56. PhD March 200756 Binomial Distribution

57. PhD March 200757

58. PhD March 200758 Example x1 + x2 + x3 = 1 x1 + x2 + + x4 = 1 can satisfy 6/7 x1 + x3 + x4 = 1 x1 + x2 + + x5 = 1 x1 + x3 + x5 = 1 x2 + x3 + x5 =1

59. PhD March 200759 maximize 3x(1-x) 2

60. PhD March 200760 Organization of Solver look backlook forward

61. PhD March 200761 Transition Rules Unit-Propagation (UP): M || F || SR || N → Mk || F || SR || N if k is undefined in M, and unsat (M¬k,SR) > 0 or unsat(M¬k,F) ≥ unsat(N,F).

62. PhD March 200762 Transition Rules Decide (D): M || F || SR || N → Mk d || F || SR || N if k is undefined in M, and v(k) occurs in some constraint of F.

63. PhD March 200763 Transition Rules Update: M || F || SR || N → M || F || SR || M if M is complete, and unsat(M,F) < unsat(N,F).

64. PhD March 200764 Transition Rules Restart: M || F || SR || N → { } || F || SR || N

65. PhD March 200765 Transition Rules Finale: M || F || SR || N → M || F || SR || N if Φ SR or unsat(N,F) = 0.

66. PhD March 200766 Transition Rules Semi-Superresolution (SSR): NewSR = V (¬k), where k M d M || F || SR || N → M || F || SR, NewSR || N if unsat(M,SR) > 0 or unsat(M,F) ≥ unsat(N,F).

67. PhD March 200767 Transition Rules

68. PhD March 200768 Transition Rules (cont.)

69. PhD March 200769 Transition Rules Semi-Superresolution (SSR): NewSR = V (¬k), where k M d M || F || SR || N → M || F || SR, NewSR || N if unsat(M,SR) > 0 or unsat(M,F) ≥ unsat(N,F).

70. PhD March 200770 Transition Rules Semi-Superresolution (SSR): NewSR = V (¬k), where k M d M || F || SR || N → M || F || SR, NewSR || N if unsat(M,SR) > 0 or unsat(M,F) ≥ unsat(N,F).

71. PhD March 200771 Transition Rules Semi-Superresolution (SSR): NewSR = V (¬k), where kєM’ subset M d M || F || SR || N → M || F || SR, NewSR || N if mistake(M) and UP*(reduce(F,A(NewSR)))

72. PhD March 200772 Our Approach Superresolution & P-Optimality Based MAX-CSP Solver Highlights –Optimally Biased Coin (in Abstract Representation) –Clause Learning (in Transition System) –Bitwise Relation Reduction (in Intermediate Representation)

73. PhD March 200773 Clause Learning Let’s go beyond what an optimally biased coin guarantees! Goal: satisfy the maximum number of constraints. Approach: Superresolution. –When to apply: number of constraints guaranteed to be unsatisfied doesn’t decrease A mistake is made. –Who to blame: the decision literals They are the culprits. –How to penalize: add the disjunctions of their negations as a superresolvent The gang of culprits is watched.

74. PhD March 200774 Sudoku