Improving stochastic local search for SAT with a new probability distribution

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1 Adrian Balint July 13, 2010 Andreas Fröhlich Improving stochastic local search for SAT with a new probability distribution

2 Page 2 Improving stochastic local search for SAT with a new probability distribution Balint, Fröhlich July 13, 2010 Introduction SAT Formula F as a set of clauses. Each clause as a set of literals. Is there an assignment α, so that F (α) = 1? SLS-solvers Operate on complete assignments. Try to change an initial assignment α 0 in a way, so that it becomes satisfying after several steps. Use local information and heuristics to decide how to modify the assignment in each search step.

3 Page 3 Improving stochastic local search for SAT with a new probability distribution Balint, Fröhlich July 13, 2010 Pseudocode of a typical SLS-solver INPUT: formula F, cutoff. OUTPUT: model for F or UNKNOWN. SLS(F, cutoff) for i =1 to maxtries α = α 0 = random assignment; for j =1 to maxflips if (α is a model for F) return α; var = pickvar (); α[var] = 1 α[var]; return UNKNOWN;

4 Page 4 Improving stochastic local search for SAT with a new probability distribution Balint, Fröhlich July 13, 2010 SLS architectures Differ in how to select a variable for flipping. GSAT Considers all variables. WalkSAT DLS Only considers variables from a randomly chosen unsatisfied clause. Similar to GSAT. Additionally uses clause-weighting.

5 Page 5 Improving stochastic local search for SAT with a new probability distribution Balint, Fröhlich July 13, 2010 Modern SLS-solvers Can not be put into just one category. Are hybrids of the different architectures e.g.: adaptg2wsat = GSAT + WalkSAT + adaptive scheme [Li07] gnovelty+ = adaptg2wsat + DLS-component [Pham07] TNM = GSAT + WalkSAT-version + two noise schemes to switch between [Wei09] they all use a Novelty-version to escape from local minima.

6 Page 6 Improving stochastic local search for SAT with a new probability distribution Balint, Fröhlich July 13, 2010 Escaping from local minima Novelty Choose a random unsatisfied clause c F Pick a variable x from c according to the following heuristic: best score variable has min age? no yes choose best variable noise 1-noise choose second best variable choose best variable

7 Page 7 Improving stochastic local search for SAT with a new probability distribution Balint, Fröhlich July 13, 2010 Escaping from local minima adaptnovelty+ Additionally implements a random walk with probability wp. Uses an adaptive scheme for setting the noise probability [Hoos02].

8 Page 8 Improving stochastic local search for SAT with a new probability distribution Balint, Fröhlich July 13, 2010 Novelty as a probability distribution This can be seen as a probability distribution p on the variables of c. p(x) can take only 4 different values: Always chosen; p(x) = 1 Chosen, when no noise step occurs; p(x) = (1 noise) Chosen, when a noise step occurs; p(x) = noise Never chosen; p(x) = 0 Information used to decide: score, age Does not take into account the exact values of score and age. Does not put different variables into relation.

9 Page 9 Improving stochastic local search for SAT with a new probability distribution Balint, Fröhlich July 13, 2010 Example c = {x 1,x 2,x 3 } Case 1 score(x 1 ) = 0,score(x 2 ) = 3,score(x 3 ) = 3 age(x 1 ) = ,age(x 2 ) = 10 4,age(x 3 ) = Case 2 score(x 1 ) = 1,score(x 2 ) = 2,score(x 3 ) = 3 age(x 1 ) = 10 3,age(x 2 ) = 10 4,age(x 3 ) = 10 7 Both cases are identical for Novelty+. The only chance for x 3 to be selected is by a random walk.

10 Page 10 Improving stochastic local search for SAT with a new probability distribution Balint, Fröhlich July 13, 2010 New probability distribution Define a probability distribution as a concatenation of continuous functions on the variables of c. Choose a random variable according to this distribution. The form of this distribution p(x) := p s(x) p a (x) x c p s ( x) p a ( x). p s (x) is a function considering the score of x. p a (x) is a function considering the age of x. p a (x),p s (x) > 0

11 Page 11 Improving stochastic local search for SAT with a new probability distribution Balint, Fröhlich July 13, 2010 New probability distribution Define a probability distribution as a concatenation of continuous functions on the variables of c. Choose a random variable according to this distribution. The form of this distribution p(x) := p s(x) p a (x) x c p s ( x) p a ( x). p s (x) := c 1 score(x) p a (x) := ( age(x) c 3 ) c2 + 1 c 1 controls the influence of score on the decision. c 2 and c 3 specify, how quick and how strong age starts to affect the decision.

12 Page 12 Improving stochastic local search for SAT with a new probability distribution Balint, Fröhlich July 13, 2010 Properties of p Like Novelty Prefers variables with high score. Prefers variables with high age. Additionally Takes into account the exact values of score and age. Puts different variables into relation. Does not need an explicit random walk. Implicit random walk because of p s (x),p a (x) > 0. Does not need an explicit noise scheme. Growing age automatically leads to new decisions when getting stuck in a certain part of the search space.

13 Page 13 Improving stochastic local search for SAT with a new probability distribution Balint, Fröhlich July 13, 2010 Implementation details Sparrow gnovelty+ as the basic module. gnovelty+ consists of: Gradient-walk as in G2WSAT. adaptnovelty+ to escape local minima. Additive weighting scheme. Replaced the adaptnovelty+ component by a WalkSAT-algorithm using the presented probability distribution. Specifying the parameters c 1 = 4,c 2 = 4,c 3 = Constants fixed for all instances. Empirical values. Not optimized yet.

14 Page 14 Improving stochastic local search for SAT with a new probability distribution Balint, Fröhlich July 13, 2010 Evaluation Empirical Tests 2 sets of large random 3-SAT instances with ratio = 4.2 Not optimized for k-sat, k 3. Benchmark 1: 64 different instances from SAT 2009 Competition (2k - 18k variables) Each instance is solved 100 times (cutoff 1200 sec.) Benchmark 2: 40 additional instances from SAT 2009 Competition (20k - 26k variables) Each instance is solved 50 times (cutoff 2400 sec.)

15 Page 15 Improving stochastic local search for SAT with a new probability distribution Balint, Fröhlich July 13, 2010 Results: Sparrow vs gnovelty+ Numbers of flips Time in seconds Success runs gnovelty+2t 0.0e e e e e e Sparrow

16 Page 16 Improving stochastic local search for SAT with a new probability distribution Balint, Fröhlich July 13, 2010 Results: Sparrow vs TNM Numbers of flips Time in seconds Success runs TNM 0.0e e+08 0e+00 3e e e e+00 2e+08 4e Sparrow

17 Page 17 Improving stochastic local search for SAT with a new probability distribution Balint, Fröhlich July 13, 2010 Results: Sparrow vs hybridgm3 hybridgm3 0.0e e e e+00 Numbers of flips Time in seconds Success runs 1.0e e e e Sparrow

18 Page 18 Improving stochastic local search for SAT with a new probability distribution Balint, Fröhlich July 13, 2010 Conclusion Summary Escaping local minima is a key feature for SLS-solvers Variable selection in local minima should not be done in a deterministic way Detailed differentiation useful for variable selection. Outlook Implement more attributes. Arbitrary complex function. Variable expressions [Tompkins10].

19 Page 19 Improving stochastic local search for SAT with a new probability distribution Balint, Fröhlich July 13, 2010 Thank you for your attention!

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