Random SAT Instances a la Carte

Transcription

1 Random SAT Instances a la Carte Carlos Ansótegui María Luisa Bonet DIEI, UdL, Lleida, Spain LSI, UPC, Barcelona, Spain Jordi Levy IIIA, CSIC, Barcelona, Spain CCIA 08, Empuries Random SAT Instances a la Carte p.1

2 SAT SAT is a central problem in computer science The problem is NP-complete in the worst case State-of-the-art solvers are of practical use with real-world instances Objective: Design solvers that perform well on industrial instances Random SAT Instances a la Carte p.2

3 SAT Competitions SAT competitions evaluate ideas, techniques and solvers Competitions use benchmarks: Randomly generated Industrial Crafted Random SAT Instances a la Carte p.3

4 SAT Competitions SAT competitions evaluate ideas, techniques and solvers Competitions use benchmarks: Randomly generated Unlimited in number Families of instances: one for every number of vars Generated on demand: fair in competitions Parameterized degree of dificulty Industrial Crafted Random SAT Instances a la Carte p.3

5 SAT Competitions SAT competitions evaluate ideas, techniques and solvers Competitions use benchmarks: Randomly generated Unlimited in number Families of instances: one for every number of vars Generated on demand: fair in competitions Parameterized degree of dificulty Industrial Limited in number Specially valuable Crafted Random SAT Instances a la Carte p.3

6 General Objective Create generators of random instances with properties similar to industrial ones to test solvers Stated as 10th challenge by Kautz&Sellman in Ten Challenges in Propositional Reasoning and Search : Develop a generator for problem instances that have computational properties that are more similar to real-world instances[...] While hundreds of specific [industrial] problems are available, it would be useful to be able to randomly generate similar problems by the thousands for testing purposes Also Rina Dechter in her book proposes the same objective Random SAT Instances a la Carte p.4

7 Progress on this Objective Number of occurrence of variables follow similar distributions as in industrial Similar ratio clause/variables Instances generated around the transition point (not trivially sat or unsat) Use of an external measure of complexity (Strahler) to generate instances parameterized by dificulty Strahler similar to industrial Similar behaviour w.r.t. solvers Sizes of clauses follows similar distribution as in industrial Study structure in industrial instances and try to obtain similar random instances Random SAT Instances a la Carte p.5

8 Strahler as a Measure of the Hardness Strahler of a Tree (1) str( ) = 0 ( ) str t 1 t = 2 str( t 1 ) + 1 if str( t 1 ) = str( t 2 ) max{str( t 1 ),str( t 2 )} otherwise (2) Depth of the biggest complete tree that can be embedded (3) Minimum number of pointers (memory) that we need in order to traverse it Random SAT Instances a la Carte p.6

9 Strahler as a Measure of the Hardness Strahler of a Tree (1) str( ) = 0 ( ) str t 1 t = 2 str( t 1 ) + 1 if str( t 1 ) = str( t 2 ) max{str( t 1 ),str( t 2 )} otherwise (2) Depth of the biggest complete tree that can be embedded (3) Minimum number of pointers (memory) that we need in order to traverse it Random SAT Instances a la Carte p.6

10 Strahler as a Measure of the Hardness Strahler of a Tree (1) str( ) = 0 ( ) str t 1 t = 2 str( t 1 ) + 1 if str( t 1 ) = str( t 2 ) max{str( t 1 ),str( t 2 )} otherwise (2) Depth of the biggest complete tree that can be embedded (3) Minimum number of pointers (memory) that we need in order to traverse it Strahler of an (unsatisfiable) Formula = minimum Strahler of a refutation Random SAT Instances a la Carte p.6

11 Strahler as a Measure of the Hardness Strahler of a Tree (1) str( ) = 0 ( ) str t 1 t = 2 str( t 1 ) + 1 if str( t 1 ) = str( t 2 ) max{str( t 1 ),str( t 2 )} otherwise (2) Depth of the biggest complete tree that can be embedded (3) Minimum number of pointers (memory) that we need in order to traverse it Strahler of an (unsatisfiable) Formula = minimum Strahler of a refutation = Space needed to refute a formula [Ben-Sasson, Galesi 2003; Esteban, Toran 2001] = Hardness of a formula [Kullmann 2003] Random SAT Instances a la Carte p.6

12 Strahler of (Classical) Random 3-CNF % strahler / vars 2 % strahler / vars vars 400 vars 500 vars ratio clauses / var vars Random SAT Instances a la Carte p.7

13 Strahler of Industrial Instances instance unsat/sat #vars(n) space (s) 100 s/n sat solver competition 2005 vmpc27 sat vmpc30 sat depots3_ks99i sat driverlog2_v0li sat ferry6_ks99i sat ferry6_ks99a sat ferry7_ks99a sat satellite2_v0li sat ssa cnf unsat ssa cnf unsat ssa cnf unsat bf cnf unsat bf cnf unsat bf cnf unsat bf cnf unsat Random SAT Instances a la Carte p.8

14 Generator: Probability Distribution f(x) = ln(b)/(b-1)*b^x Given a continuous prob. distribution function φ(x;b) = ln(b) b 1 bx Define P(v;b,n) φ(v/n;b) taking n equidistant points: P(v;b,n) = n 1 i=0 ln(b) b 1 bv/n ln(b) b 1 1 b1/n = bi/n 1 b b v/n Random SAT Instances a la Carte p.9

15 Input: Output: F = for i = 1 to m do repeat Ci = Geometric Generator n,m,k,b a k-sat instance with n variables and m clauses for j = 1 to k do c = rand([0 1)) v = 0 while c > Pr(v;b,n)do v = v + 1 c = c P(v;b,n) rand([0 1)) C i = C i ( 1) rand({0,1}) v until C i is not a tautology or simplifiable F = F {C i } 0 P(0; b, n) i P(1; b, n) i P(2; b, n) 1 ] P(n 1; b, n) Random SAT Instances a la Carte p.10

16 Input: Output: Geo-Regular Algorithm n,m,k,b a k-sat instance with n variables and m clauses bag = for v = 1 to n do bag = bag {P(v;B,n) k m 2 copies of v} bag = bag {P(v;B,n) k m 2 copies of v} endfor repeat F = for i = 1 to m do C i = random multiset of k literals from bag bag = bag \ C i F = F {C i } until F does not contain tautologies or simplifiable clauses Random SAT Instances a la Carte p.11

17 Percentage of unsat vs. clause/variable % unsatisfiable 50 geometric % unsatisfiable 50 geo-regular 25 b=1 b=2 b=4 b=8 b=16 25 b=1 b=2 b=4 b=8 b= clause/420 ratio clause/300 ratio Random SAT Instances a la Carte p.12

18 Phase transition point as a function of b 4.5 geometric geo-regular clause/variable ratio at crossover point b Random SAT Instances a la Carte p.13

19 Strahler vs. number of variables b=1 b=2 b=4 b=8 b=16 geometric 15 b=1 b=2 b=4 b=8 b=16 geo-regular strahler strahler variables variables Random SAT Instances a la Carte p.14

20 Strahler/vars vs. vars geometric b=1 b=2 b=4 b=8 b=16 geo-regular b=1 b=2 b=4 b=8 b= strahler/variable 0.03 strahler/variable variables variables Random SAT Instances a la Carte p.15

21 Strahler/vars vs. as a function of b geo-reg, n=300 geometric, variable= strahler/n b Random SAT Instances a la Carte p.16

22 Powerlaw distribution Recent Progress Have phase transition point For α = 0.75 we get m/n = 2.87 We get Strahler/vars 0.59% for vars 2000 therefore problems are easy We have studied the quotient time needed by minisat time needed by kcnfs For geometric distr. the quotient is bigger than 1 (even for big b) For powerlaw distr. the quotient is smaller than 1 for b=0.75 Random SAT Instances a la Carte p.17