Unformatted text preview: harder than average uniform 3cnf problems at their crossover point. Directional resolution, on the other hand, behaved in a tamed way on chains and was sometimes more than 1000 times faster than DPbacktracking. In Table 3 we compare DPbacktracking with DR on the same chain problems as in Table 2. A detailed illustration in Table 4 lists the results on selected hard instances (number of 23 Table 4: DR and DPbacktracking on hard chain instances (number of deadends > 5000) 3cnf chains with 125 variables: 25 subtheories, 5 variables in each Num SAT: of cls 349 349 349 399 399 399 399 399 399 449 449 449 0 0 0 0 0 0 0 0 0 1 0 0 0 or 1 DPbacktracking
Time: 1st solution Dead ends DR
Time: 1st solution 1.5 2.4 1.9 3.6 3.1 3.1 3.0 2.2 2.9 5.2 3.0 3.5 41163.8 3779913 102615.3 9285160 55058.5 5105541 74.8 87.7 149.3 11877.6 841.8 655.5 2549.2 289.7 6053 7433 12301 975170 70057 47113 181504 21246 37903.3 3079997 deadends exceeds 5000) from Table 3. We see that extremely hard instances ( more than 100000 deadends), although rare, contribute most to the mean values. All the experiments reported so far used mindiversity ordering. When experimenting with dierent orderings (input ordering and minwidth ordering) we observed similar results (Figure 9). We also ran a small set of experiments with the actual code of tableau [4], that implements the DavisPutnam procedure with various heuristics. Its behaviour on chain problems was similar to the one of our implementation. Some problem instances that were hard for our version of DPbacktracking were easy for tableau, however there was a subset of instances that were extremely dicult for both algorithms. 24 3CNF CHAINS 15 subtheories, 4 variables in each ’Initial ’ ordering 500 experiments 3CNF CHAINS 15 subtheories, 4 variables in each Minwidth ordering 100 experiments 100 100 CPUtime (logscale) CPUtime (log scale)
DPsplitting DPelimination 10 10 1 1 .1 DPbacktracking DR .01 .1 0 5 10 15 Clauses per subtheory 20 0 5 10 15 Clauses per subtheory 20 (a) input ordering (b) minwidth ordering Figure 9: DR and DPBacktracking on chains with dierent orderings Almost all the hard chain problems (for DPbacktracking) were unsatisable. One plausible explanation is the existence of an unsatisable subtheory that is last in the ordering. If all other subtheories are satisable, then DPbacktracking will try to reinstantiate variables from the satisable subtheories each time it encounters a deadend. Figure 10 shows an example of a chain of satisable theories where an unsatisable one appears almost at the very end of ordering. Mindiversity and minwidth orderings do not guarantee that we avoid a situation like that. Not knowing the structure hurts DPbacktracking. Choosing the right ordering would help but this may be hard to recognize without some preprocessing. Other variants of backtracking that are capable of exploiting the structure like backjumping[13, 9] would avoid such useless reinstantiation of variable...
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 Spring '09
 W. Alabama
 Analysis of algorithms, Conjunctive normal form, Qn, directional resolution

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