Computing the Density of States of Boolean Formulas

1. Computing the Density of States of Boolean Formulas Stefano Ermon, Carla Gomes, and Bart Selman Cornell University, September 2010

2. Motivation:Significant progress in SAT From 100 variables, 200 constraints (early 90’s) to over 1,000,000vars. and 5,000,000clauses in 20 years. Applications: Hardware and Software Verification, Planning, Scheduling, Optimal Control, Protocol Design, Routing, Multi-agent systems, E-Commerce (E-auctions and electronic trading agents), etc. SAT: Given a Boolean formula Φin CNF, Φ=C1ΛC2 Λ…ΛCm does Φ have a satisfying assignment?

3. Extending SAT technology How can we combine both challenges? • Model counting problem (number of distinct satisfying assignments): • probabilistic inference problems • multi-agent / adversarial reasoning (bounded) [Roth ‘96, Littman et. al. ‘01, Sang et. al. ‘04, Darwiche ‘05, Domingos ‘06] • MAX-SAT and Weighted MAX-SAT: find a truth assignment that maximizes the number of satisfied clauses or the sum of their weights • beyond decision (NP) [Hansen at al. ’90] • hard and soft constraints [Heras et al. ’08, Cohen et al. ’06]

4. Density of states • Given a Boolean formula Φin CNF, Φ=C1ΛC2Λ…ΛCm with m clauses the density of states is a function that gives the number of truth assignments that violate exactly i clauses, for i =0,..,m • n(0) = number of assignments that violate 0 clauses (models) • n(1) = number of assignments that violate exactly 1 clause

5. Density of states: a challenging problem • Generalizes SAT • Decision problem: Φ is satisfiable if and only if n(0)>0 • Generalizes MAX-SAT • MAX-SAT is the minimum i such that n(i)>0 • Generalizes #SAT • Number of models = n(0)

6. Statistical physics • More generally, the density of states (DOS) gives the number of microstates with energy E • Microstates = truth assignments • Energy = number of violated clauses • Ground states = maximally satisfying assignments • Compact, very informative characterization of a physical system • Macroscopic thermodynamic quantities (free energy, internal energy,..) • Partition function, phase transitions,..

7. Motivation • DOS provides a finer characterization of the structure of a combinatorial search space • Statistical physics and CSPs: • Insights on problem structure, hardness, new algorithms, Survey Propagation[Montanari et. al. ‘07, Monasson et. al. ‘96, Mézard’02, Parisi‘02] • By defining different energy functions, it can be naturally used for probabilistic style inference (e.g. Markov Logic,[Domingos ‘06] )

8. Talk Outline • Prior work • A novel sampling strategy: MCMC-FlatSat • Empirical Validation • Small formulas with ground truth • Synthetic formulas • Random 3-SAT • Large structured instances • Model counting • Conclusions

9. Density of states: prior work • Exact Method: Enumeration (exponential) • Approximate • Uniform Sampling [Belaidouni et. al. ‘02] • Sample density from K random truthassignments • Impractical, unlikely to hit rare assignments (e.g. solutions) • Metropolis Sampling [Rose et. al. ‘96] • In theory, the density can be extracted from the Boltzmann distribution • Impractical, difficult choice of the temperature and slow mixing times[Wei et. al. ‘04 ] How can we improve the sampling strategy?

10. The flat histogram idea Idea: Set up a Markov Chain that visits all energy levels equally often [F. Wang and Landau ’02, J. Wang et. al. ’99, De Oliveira et. al. ’96] e.g. an equal amount of time at the set of truth assignments with 0 unsat clauses, 1 unsat clause, ... How? Flip a variable, accept new state σ’ with probability Always accepts “rarer” states (when n(E’)<n(E))

11. Example RED↔ 0 unsat clauses, n(0)=1 GREEN↔ 1 unsat clauses, n(1)=3 BLUE↔ 2 unsat clauses, n(2)=10 1 1 1 1 1 Goal: visit all energy levels (colors) equally often

12. Example RED↔ 0 unsat clauses, n(0)=1 GREEN↔ 1 unsat clauses, n(1)=3 BLUE↔ 2 unsat clauses, n(2)=10 3/10 3/10 3/10 3/10 Goal: visit all energy levels (colors) equally often

13. Example RED↔ 0 unsat clauses, n(0)=1 GREEN↔ 1 unsat clauses, n(1)=3 BLUE↔ 2 unsat clauses, n(2)=10 3/10 3/10 3/10 3/10 Goal: visit all energy levels (colors) equally often

14. Example RED↔ 0 unsat clauses, n(0)=1 GREEN↔ 1 unsat clauses, n(1)=3 BLUE↔ 2 unsat clauses, n(2)=10 1 1 1 Goal: visit all energy levels (colors) equally often

15. Example RED↔ 0 unsat clauses, n(0)=1 GREEN↔ 1 unsat clauses, n(1)=3 BLUE↔ 2 unsat clauses, n(2)=10 1 1 1 1 1 Goal: visit all energy levels (colors) equally often

16. Example RED↔ 0 unsat clauses, n(0)=1 GREEN↔ 1 unsat clauses, n(1)=3 BLUE↔ 2 unsat clauses, n(2)=10 1 1 1 1 1 Goal: visit all energy levels (colors) equally often

17. Example RED↔ 0 unsat clauses, n(0)=1 GREEN↔ 1 unsat clauses, n(1)=3 BLUE↔ 2 unsat clauses, n(2)=10 1 1 1 1 1 Goal: visit all energy levels (colors) equally often

18. Example RED↔ 0 unsat clauses, n(0)=1 GREEN↔ 1 unsat clauses, n(1)=3 BLUE↔ 2 unsat clauses, n(2)=10 1/10 1/3 1/10 1/10 1 Goal: visit all energy levels (colors) equally often

19. Example RED↔ 0 unsat clauses, n(0)=1 GREEN↔ 1 unsat clauses, n(1)=3 BLUE↔ 2 unsat clauses, n(2)=10 1/10 1/3 1/10 1/10 1 Goal: visit all energy levels (colors) equally often

20. Example …and finally…

21. Example RED↔ 0 unsat clauses, n(0)=1 GREEN↔ 1 unsat clauses, n(1)=3 BLUE↔ 2 unsat clauses, n(2)=10

22. Flat histogram: Intuition (Red, Blue, Green) Note: in contrast, Simulated Annealing concentrates sampling around low energy states (more greedy!) Detailed balance holds with thosetransition probabilities Properly biased towards “rare” states But there are few “rare” states The total time spent in each type of state is the same (flat visit histogram).

23. Flat histogram But… how can we even run the Markov Chain? Acceptance probability: The density n() is unknown and is precisely what we want to compute!

24. Adaptive sampling • Start with an initial guess g (our estimate of the true density n) • Random walk: • Use g for guidance (acceptance probability) • Chain will initially not sample uniformly across energy levels • Each step, adjust gusing a modification factor F • Keep track of the visit histogram H • When we see a flat H, we must have the right density!

25. The modification factor, F • F controls the tradeoff between convergence rateand accuracy • Use large modification factors F at the beginning to get • rough estimates • fast convergence • Keep reducing F to get finer estimates • Analogous to an annealing process

26. MCMC-FlatSat Initialization Inner Loop Inner Loop : adaptive sampling until the visit histogram is flat (g becomes our new guess for n)

27. MCMC-FlatSat OuterLoop Reduce modification factor and repeat inner loop until g≈n

28. Outline of empirical validation • Empirical validation on combinatorial problems • Convergence • Efficiency (number of samples vs search space size) • Accuracy • We study: • Small formulas with ground truth • Large synthetic formulas • Random 3-SAT • Large structured instances • Model counting

29. Formulas with known ground truth • Instances from MAXSAT2007 competition (Ramsey, Spin Glass, Max Clique) • Direct enumeration is possible (n<=28), so we can compare our estimate with ground truth • Metrics: • KL divergence: • Relative error per point

30. n – ground truth g – estimate X 3.1 % maximum relative error Rel. error (%) Energy • Spin glass instance, 27 variables, 162 clauses • Needs ~ 8 *106 flips<< search space size (227≈1.3 *108) Log-density Energy (# unsat clauses)

31. n – ground truth g – estimate n g X X Max Clique Ramsey Log-density Log-density Energy (# unsat clauses) Energy (# unsat clauses)

32. Synthetic formulas • Ground truth for larger formulas? • Construct synthetic formulas • Formulas for which we derive a closed form solution for the density • Result on the composition (logical conjunction) of independent formulas (donot share variables) • The density of F is the convolution of the density of Φwith itself l times

33. n – ground truth g – estimate n g X X Convolution of uniform densities Log-density Log-density Convolution of pigeon hole formulas Energy (# unsat clauses) Energy (# unsat clauses) Needs ~ 2*107 flips << state space size (250≈1015)

34. Random k-SAT formulas • Well known phase transitions for the satisfiability property in terms of the ratio α (Φ is satisfiable ↔ nΦ(0)>0 ) • Analytic result on the averagedensity given a truth assignment, the probability of having a clause that is violated is 1/2k for random k-SAT

35. Average Densities • n=50 variables (average over 1000 instances) • Needs ~ 108 flips << state space size (250≈1015) Log-density Ratio clauses to var. α

36. Large structured instances • No ground truth known • Consistency checks • Number of models, when exact model counting is feasible (# models=n(0)) • Method of the moments: • Sample K assignments at random, compute their energies (unsat clauses) • Compute sample moments (e.g. average energy, 2nd order moment of energy, ..) • Compare with moments obtained using the estimated density g

37. Large structured instances

38. Logistic instance Log-density • Planning problem: • n=459 variables • m=4675 clauses • Huge search space (2459 truth assignments), but MCMC-FlatSat returns within hours • Remarkable precision: • Finds the only existing model g(0)=n(0)=1! • The mode of the estimated distribution is e300 times larger than the number of models g(0) counts the needles and the haystack! (the moments method indicates g is accurate) Energy (# unsat clauses)

39. Model counting • Comparison with state-of-the-art model counters • SampleCount[Gomes et. al. ‘06] • SampleMiniSAT[Gogate et. al. ‘07] • MCMCFlatSat is very accurate • Timings are competitive when ratio clauses to variables is not too large • DOS provides guidance • Information on what is not a model • Overhead because it provides more information

40. Model counting comparison

41. Conclusions • Computing the density of states is a hard problem (encompasses SAT, MAX-SAT, #SAT) • MCMCFlatSat: sampling strategy adapted from physics for combinatorial spacesthat adaptively explores the space while collecting statistics • Extremely accurate, very efficient (few samples) • Provides a compact, rich descriptionof the search space. New insights about structure and local search • Very general method: any property can be used for search space partitioning. • Many applications to counting and inference problems

42. Extra slides

43. Future work • SAT-specific improvements • Energy saturation • Energy barriers (and related normalization issues) • Walksat heuristics (with Metropolis-Hastings updates) • Direct application to inference in Markov Logic • Formal proof of convergence / counterexamples • Application to other counting problems in combinatorial spaces

44. Runtime for random 3-SAT Search space 2^50 Flips 10^8 ≈ 2^26

45. Related work on random 3-SAT • Lots of work on the i=0 (i.e. SAT/UNSAT) case [Gent et. al. ‘94] • Previous experimental work for i>0 [Zhang ‘01] • Different definition: “no more than iunsat clauses” versus “exactly iunsat clauses” • Location on the phase transitions • Apparently, same location • Analytic results [Achlioptas et al. ‘05] • We can see two phase transitions: assignments are unlikely to violate a large number of clauses

46. Other phase transition 50 variables, 1000 instances

47. Other structured instances

48. Histogram flatness (formal) • Flatness condition of the visit histogram H is a necessary condition for convergence • If g is equal to the true density n, the detailed balance is satisfied by upon convergence, the steady state probability is proportional to the reciprocal of the density of the corresponding energy level

49. Histogram flatness (formal) • Flatness condition of the visit histogram H is a necessary condition for convergence If g=n, the visit histogram H will be flat i.e. energy levels visited equally often • Problem: the density n() is unknown and is precisely what we want to compute!

50. Propositional Satisfiability (SAT) Satifiability (SAT): Given a formula in propositional calculus, does it have a model, i.e., is there an assignment to its variables making it true? (abc) AND (bc) AND (ac) SAT: prototypical hard combinatorial search and reasoning problem. Problem is NP-Complete. (Cook 1971)