Resolution versus Search: Two Strategies for SAT

1. Resolution versus Search:Two Strategies for SAT Brad Dunbar Shamik Roy Chowdhury

2. Propositional Satisfiability Problems • Propositional satisfiability Algorithms with good average performance has been focus of extensive research. • Davis Putnam Algorithm for deciding propositional satisfiability Directional Resolution. • Worst Case Time /Space complexity of DR : • O( n.exp(w*) ) where • n : number of variables • W* : induced width

3. Backtracking Vs Resolution

4. What makes DR a good algorithm: • Decides satisfiability and finds solution ( model ). • Given input theory and a variable ordering Knowledge Compilation Algorithm : • Generation equivalent theory ( directional extension )  • Each model can be found in linear time. • All models can be found in the time linear in the number of models. • Performs better on structured algorithms. • k-tree embeddings having induced width. • w* < n ( generally ) • DR ( worst case bound) < DP ( worst case bound )

5. An Example Resolution : Resolution over A • Node : Each propositional variable. • Edge : Between variables of the same clause. • Resolution over clauses ( a V Q ) and ( b V ~Q ) => a V b ( Resolvent ). • Resolution over A ( adj. Fig. ) => (B V C V E ) … introduces edge C – E.

6. Directional Resolution – An ordering based algorithm

7. Execution of Directional Resolution (DR):Knowledge Compilation & model generation

8. Complexity of Directional Resolution(DR) Algorithm: Change of E(Q) with ordering

9. Complexity : Induced Width

10. Dependence of complexity on Induced Width • Theorem 4: • Given Theory(Q) and an ordering of its variables (o). • Directional Resolution(DR) time complexity along ‘o’ • Size of at most • where is the induced width of interaction graph.

11. Change of Induced Width with Variable Ordering

12. Change of Induced Width with Variable Ordering

13. Change of Induced Width with Variable Ordering

14. Ordering Heuristics : Which Ordering gives Minimum Induced Width ? • Finding an ordering which yields smallest induced width is NP-HARD. • Ordering Heuristics : • Polynomial Time Greedy Algorithm.  • Computation/Generation of min-width ordering. 

15. Diversity • Upper bound on the number of resolution operation. • Based on fact : Proposition resolved only when it appears both positively and negatively in different clauses. • Div(o) – largest diversity of its variables relative to ‘o’. • Div(of a theory) – minimum diversity among all orderings • bounds number of clauses generated in each bucket. • Eg: If ordering (o) has 0 diversity, then algorithm DR adds no clauses to the theory regardless of its induced width .

16. Diversity computation

17. Ordering Heuristics : Algorithm to generate ordering giving minimum Diversity • Finding an ordering which yields minimum- induced diversity is NP-HARD. • Ordering Heuristics : • Polynomial Time Greedy Algorithm.  • Computation/Generation of min-diversity ordering. 

18. Directional Resolution and Tree Clustering

19. Directional Resolution and Tree Clustering

20. Directional Resolution and Tree Clustering

21. Directional Resolution and Tree Clustering

22. Directional Resolution and Tree Clustering

23. Directional Resolution and Tree Clustering

24. Backtracking (DP) Algorithm

25. Comparison of Backtracking and Resolution

26. Random Problem Generators

27. DR vs DP, 3-cnf Chains

28. DR vs DP, > 5000 Dead-Ends

29. DP vs DR, Uniform Random 3-cnfs

30. DR and DP on 3-cnf Chains, Different Ordering

31. Numer of Deadends

32. DP vs Tableau (Uniform Random)

33. DP vs Tableau (Chains)

34. Bounded Directional Resolution - BDR(i)

35. Dynamic Conditioning + Directional Resolution - DCDR(b)

38. Conclusions • DP Performs much better on random uniform k-cnfs • DR Performs much better on k-cnf chains and (k,m) trees • A hybrid model can perform better than DR and DP for certain cases

39. References • Rish and Dechter (Irina Rish and RinaDechter. "Resolution versus Search: Two Strategies for SAR." Journal of Automated Reasoning, 24, 215—259, 2000.) • (Davis, M. and Putnam, H. (1960). "A computing procedure for quantification theory." Journal of the ACM, 7(3): 201--215.) • (Davis, M., Logemann, G., and Loveland, D. (1962). "A machine program for theorem proving." Communications of the ACM, 5(7): 394--397)