Bounded Universal Expansion for Preprocessing QBF

Transcription

1 SAT 2007 Bounded Universal Expansion for Preprocessing QBF Uwe Bubeck and Hans Kleine Büning International Graduate School Dynamic Intelligent Systems, University of Paderborn

2 Outline Introduction Bounded Expansion Variable Selection Refinements Results and Conclusion Uwe Bubeck (U Paderborn) Bounded Universal Expansion 2

3 Bounded Universal Expansion Introduction

4 Universals and Complexity Universals make QBF more difficult: must consider both values for each universal existentials may have different values depending on preceding universals x 1 x 2 y (y x 1 x 2 ) ( y x 1 ) ( y x 2 ) y depends on x 1 and x 2 quantifier ordering must be respected x y (x y) ( x y) y x (x y) ( x y) So let us focus on the universals! Uwe Bubeck (U Paderborn) Bounded Universal Expansion 4

5 Semantics of Quantifiers Quantifiers are actually abbreviations: x φ(x) is true if and only if φ[x/0] is true and φ[x/1] is true x φ(x) is true if and only if φ[x/0] is true or φ[x/1] is true we can eliminate them by expansion. Uwe Bubeck (U Paderborn) Bounded Universal Expansion 5

6 Bounded Universal Expansion Bounded Expansion

7 Expansion Procedure 1/3 x φ(x) is true if and only if φ[x/0] is true and φ[x/1] is true Duplicate the matrix of the formula. For dominated existentials, we need two separate instances. Example: y 1 x 1 x 2 y 2 φ(x 1, x 2, y 1, y 2 ) y 1 x 2 y (0) 2 y (1) 2 φ(0, x 2, y 1, y (0) 2 ) φ(1, x 2, y 1, y (1) 2 ) Uwe Bubeck (U Paderborn) Bounded Universal Expansion 7

8 Expansion Procedure 2/3 Longer Example: Exponential Growth! Uwe Bubeck (U Paderborn) Bounded Universal Expansion 8

9 Expansion Procedure 3/3 How to avoid exponential blowup? Do not expand all universals! Try to eliminate as many as possible under bounded expansion costs. Leave expensive universals to other solvers / techniques. Uwe Bubeck (U Paderborn) Bounded Universal Expansion 9

10 Expansion for Preprocessing Hypothesis: we can make it easier for QBF solvers by first removing carefully selected cheap or rewarding universals. QBF Formula Preprocessing by expansion Original QBF Solver Crucial for success: accurate a priori cost estimates suitable selection strategy Uwe Bubeck (U Paderborn) Bounded Universal Expansion 10

11 Bounded Expansion Enforce tight bounds: global size limit C global : Φ cur C global Φ Suitable values: 2 4 prevents flooding solver with huge formulas individual cost limit C single : c x C single Φ cur Suitable values: 0.5 leave expensive universals to the solver Uwe Bubeck (U Paderborn) Bounded Universal Expansion 11

12 Bounded Universal Expansion Variable Selection

13 Variable Locality x 1 y 1 x 2,x 3 y 2,y 3 (φ(x 1, y 1, x 2, y 2 ) ψ(x 1, y 1, x 3, y 3 )) x 1 and y 1 occur in the whole formula, x 2 / y 2 and x 3 / y 3 are only used locally in φ resp. ψ. when expanding x 3, we only need to duplicate y 3 and clauses in ψ. take into account variable connectivity: start with locally connected variables and compute the transitive closure (see Quantor [Biere 04]) Uwe Bubeck (U Paderborn) Bounded Universal Expansion 13

14 Expansion Order 1/2 Quantor always expands from the innermost block. X φ Our Hypothesis: For a preprocessor, it might be rewarding to expand from further outside. X φ Reasons: subsequent solver might do inner universals more efficiently preprocessor has more time for cost computations universals further outside might actually be cheaper Uwe Bubeck (U Paderborn) Bounded Universal Expansion 14

15 Expansion Order 2/2 Φ = x 1 y 1 x 2,x 3 y 2,y 3 x 4,x 5 y 4,y 5 (φ ψ) In which order to expand the universals? Further left, we have more dependent existentials right to left x 5, x 4, x 3,... appears best but what if Φ can be rewritten as Φ = x 1 y 1 ( x 2 y 2 x 4 y 4 φ) ( x 3 y 3 x 5 y 5 ψ)? If φ < ψ, it is cheaper to expand x 4,x 2 before x 5,x 3... or what if x 2 or x 3 are particularly rewarding? ( goal orientation) Uwe Bubeck (U Paderborn) Bounded Universal Expansion 15

16 Cost Estimates 1/2 Scheduling of eliminations requires tight a priori estimates of expansion costs: duplicate clauses with dependent existentials D x when x = 0: remove clauses with x and occurrences of +x when x = 1: remove clauses with +x and occurrences of x Estimate: c x s(d x ) - s( x) - o(x) - s(x) - o( x) In Quantor, D x = Y r (innermost existentials). We expand far less universals and can therefore afford to compute D x. Uwe Bubeck (U Paderborn) Bounded Universal Expansion 16

17 Cost Estimates 2/2 Only subsequent existentials propagate connectivity, universals never do. Example: y 1 x 1 y 2 x 2 y 3,y 4 (x 1 y 2 ) (y 2 y 3 ) ( y 2 x 2 ) (y 4 x 2 ) ( y 3 y 1 ) ( y 1 y 4 ) (x 1 y 2 ) (y 2 y 3 ) ( y 2 x 2 ) (y 4 x 2 ) ( y 3 y 1 ) ( y 1 y 4 ) X (x 1 y 2 ) (y 2 y 3 ) ( y 2 x 2 ) (y 4 x 2 ) ( y 3 y 1 ) ( y 1 y 4 ) (x 1 y 2 ) (y 2 y 3 ) ( y 2 x 2 ) (y 4 x 2 ) ( y 3 y 1 ) ( y 1 y 4 ) Only y 2 and y 3 depend on x 1, only underlined clauses are affected by expansion X Uwe Bubeck (U Paderborn) Bounded Universal Expansion 17

18 Bounded Universal Expansion Refinements

19 Goal Orientation 1/2 Expanding just because it is cheap lacks foresight. we need more goal orientation Suitable goal: obtaining unit literals Φ = x y 1,y 2 (x y 1 ) ( y 1 y 2 ) y 2 must be true propagate Φ = x y 1,y 2 (x y 1 ) y 1 Uwe Bubeck (U Paderborn) Bounded Universal Expansion 19

20 Goal Orientation 2/2 Discover new unit literals through expansion: x 1,x 2 y 1,y 2 (x 1 y 1 ) ( y 1 y 2 ) (x 2 y 1 y 2 ) When x 1 = 0, we obtain a new unit literal y 1 formula collapses But x 2 has a lower cost estimate! take into account units obtained for x i = 0 or x i = 1 No need to balance costs and goals: unit propagations have direct impact on costs Uwe Bubeck (U Paderborn) Bounded Universal Expansion 20

21 Q-Resolution 1/5 We can also eliminate existential variables y i through resolution: resolve all clauses containing positive y i with all clauses containing negative y i y i y i y i y i Uwe Bubeck (U Paderborn) Bounded Universal Expansion 21

22 Q-Resolution 2/5 Resolution is often problematic: generates large clauses produces excessive numbers of new clauses Solution: estimate resolution costs c y and only resolve if resolution is cheap: c y < C res Φ cur, C res 0 resolution reduces expansion costs Uwe Bubeck (U Paderborn) Bounded Universal Expansion 22

23 Q-Resolution 3/5 Before expanding a universal x, resolve only on dependent existentials y D x. eliminates y and its soon-to-be-created copy y. may save duplicating some clauses in the expansion. Example: let {(y y 2 ), (y y 3 ), ( y y 4 )} φ and y, y 2 D x, but y 3, y 4 D x. resolvents: {(y 2 y 4 ), ( y 3 y 4 )} Cost estimate (δ: assumed rate of duplication): c y x (1 + δ) c y (1 - δ) (s(y) + s( y)) Uwe Bubeck (U Paderborn) Bounded Universal Expansion 23

24 Q-Resolution 4/5 Resolution can also cut dependencies: x D x y y connects x to the existentials in D x y D x y no dependent existentials Uwe Bubeck (U Paderborn) Bounded Universal Expansion 24

25 Q-Resolution 5/5 x no dependent no dependent existentials existentials D x D x existentials in D x no longer depend on x Expansion of x does not need to duplicate y when one phase of y only occurs in clauses with variables v D x and v x. Uwe Bubeck (U Paderborn) Bounded Universal Expansion 25

26 Quantifier Shifting 1/2 After expanding a universal x i, we can reorder the prefix by moving existentials y j D xi : x 1 y 1 x 2 y 2 y 3 x X 3 y 4 x 4 y 5 Duplicate dependent existentials in situ x 1 y 1 x 2 y 2 y 3 y 4 (0) y 4 (1) x 4 y 5 (0) y 5 (1) Assume y 4,y 4 D x2 x 1 y 1 y 4 (0) y 4 (1) x 2 y 2 y 3 x 4 y 5 (0) y 5 (1) Uwe Bubeck (U Paderborn) Bounded Universal Expansion 26

27 Quantifier Shifting 2/2 Quantifier Shifting Strategy: Given an existential y j, if the next universal x i further outside does not dominate y j (y j D xi ), move y j in front of x i. Information about variable dependencies already available for universal expansion solvers can benefit from these calculations Uwe Bubeck (U Paderborn) Bounded Universal Expansion 27

28 Bounded Universal Expansion Results and Conclusion

29 Implementation written in Java on top of our logic framework ProverBox [ can perform standard simplifications: unit propagation pure literal detection universal reduction dual binary clause detection subsumption (forward only initially) fully automatic with reasonable default parameters Uwe Bubeck (U Paderborn) Bounded Universal Expansion 29

30 Experiments Chosen Parameters: Global size limit C global = 2 Individual cost limit C single = 0.5 Resolution cost limit C res = 0.05 Experiment Setup: state-of-the-art solvers skizzo and SQBF 12 benchmark families from the QBFLIB, in total 688 instances time limit 300 sec. for each instance, give-up mode Uwe Bubeck (U Paderborn) Bounded Universal Expansion 30

31 Results - skizzo 1/2 Increase in number of solved problems: Adder ASP Blocks Connect3 Counter CounterFactual Evader-Pursuer k_branch_n k_path_n RobotsD2 Sortnet Szymanski Uwe Bubeck (U Paderborn) Bounded Universal Expansion 31

32 Results - skizzo 2/2 Speedup (including preprocessing time): 100,0 52,2 10,0 6,3 5,9 2,0 2,1 1,5 1,4 1,0 1,0 1,0 1,0 1,0 0,8 0,4 0,1 Adder ASP Blocks Connect3 Counter CounterFactual Evader-Pursuer k_branch_n k_path_n RobotsD2 Sortnet Szymanski Total Uwe Bubeck (U Paderborn) Bounded Universal Expansion 32

33 Results - SQBF 1/2 Increase in number of solved problems: Adder ASP Blocks Connect3 Counter CounterFactual Evader-Pursuer k_branch_n k_path_n RobotsD2 Sortnet Szymanski Uwe Bubeck (U Paderborn) Bounded Universal Expansion 33

34 Results - SQBF 2/2 Speedup (including preprocessing time): 100,0 10,0 1,0 21,7 14,6 4,8 3,6 3,8 3,8 1,7 2,0 1,0 1,1 1,4 1,0 0,7 0,1 Adder ASP Blocks Connect3 Counter CounterFactual Evader-Pursuer k_branch_n k_path_n RobotsD2 Sortnet Szymanski Total Uwe Bubeck (U Paderborn) Bounded Universal Expansion 34

35 Preprocessing Costs % of time spent on preprocessing (SQBF): Adder ASP Blocks Connect3 Counter CounterFactual Evader-Pursuer k_branch_n k_path_n RobotsD2 Sortnet Szymanski Total Uwe Bubeck (U Paderborn) Bounded Universal Expansion 35

36 Summary Summary of Results: skizzo skizzo+pre SQBF SQBF+pre solved time solved time solved time solved time , , , ,014 skizzo+pre: 10% more problems in 48% of time SQBF+pre: 43% more problems in 27% of time Only 2 benchmark families where preprocessing decreased number of solved instances boundedness lowers risk of negative effects Uwe Bubeck (U Paderborn) Bounded Universal Expansion 36

37 Conclusion QBF Solvers indeed benefit from expanding carefully selected universal variables in a preprocessing step. Future Work: other strategies for variable selection more sophisticated quantifier shifting Uwe Bubeck (U Paderborn) Bounded Universal Expansion 37