resolution versus search: two strategies for sat brad dunbar shamik roy chowdhury
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Resolution versus Search:Two Strategies for SAT
Brad DunbarShamik Roy Chowdhury
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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
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Backtracking Vs Resolution
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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 )
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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.
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Directional Resolution – An ordering based algorithm
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Execution of Directional Resolution (DR):Knowledge Compilation & model generation
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Complexity of Directional Resolution(DR) Algorithm: Change of E(Q) with ordering
BUCKET CLAUSES
A (B v A ) , ( C V ~A) , ( D V A) , ( E V ~A )
D ( C V D ) , ( D V E )
C B V C
B B V E
E
Theory (B V A ), ( C V ~A), ( D V A), (E V ~A)
Ordering { E, B, C, D, A }
E 8
BUCKET CLAUSES
E E V ~A
D D V A
C C V ~A
B B V A
A
Theory (B V A ), ( C V ~A), ( D V A), (E V ~A)
Ordering { A, B, C, D, E}
E 4
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Complexity : Induced Width
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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.
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Change of Induced Width with Variable Ordering
BUCKET CLAUSES
B ( A V B V C ) , ( ~A V B V E), ( ~B V C V D)
A ( ~A V C V D V E ), ( A V C V D )
C ~C, ( C V D V E )
D D V E
E
Theory ( ~C ), ( A V B V C ), ( ~A V B V E ), ( ~B V C V D)
Ordering { E, D, C, A, B }
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Change of Induced Width with Variable Ordering
BUCKET CLAUSES
A ( A V B V C ) , (~A V B V E),
B ( ~B V C V D )( B V C V E )
C ~C, ( C V D V E )
D D V E
E
Theory ( ~C ), ( A V B V C ), ( ~A V B V E ), ( ~B V C V D)
Ordering { E, D, C, B, A }
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Change of Induced Width with Variable Ordering
BUCKET CLAUSES
E (~A V B V E),
D ( ~B V C V D )
C ~C, ( A V B V C )
B A V B
A
Theory ( ~C ), ( A V B V C ), ( ~A V B V E ), ( ~B V C V D)
Ordering { A, B, C, D, E }
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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.
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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 .
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Diversity computation Bucket CLAUSES
G (G V E V ~F),(GV~EVD) 2 0 0
F ( ~A V F ) 1 0 0
E ( A V ~E ), (~B V C V~E) 0 2 0
D B V C V D 1 0 0
C
B
A
Theory {(G V E V ~F), (G V ~E V D), (~A V F), (A V ~E),(~B V C V ~E)}
Diversity :div(o) = 0
Ordering ( A, B, C, D, E, F, G )
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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.
–
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Directional Resolution and Tree Clustering
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Directional Resolution and Tree Clustering
BUCKET CLAUSES
E C V D V E
D ~B V D
C A V ~C
B ~A V B
A
Theory { ( ~A V B ), ( A V ~C), (~B V D), ( C V D V E) }
Ordering ( A, B, C, D, E )
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Directional Resolution and Tree Clustering
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Directional Resolution and Tree Clustering
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Directional Resolution and Tree Clustering
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Directional Resolution and Tree Clustering
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Backtracking (DP) Algorithm
Backtracking (DP) Algorithm
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Comparison of Backtracking and Resolution
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Random Problem Generators
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DR vs DP, 3-cnf Chains
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DR vs DP, > 5000 Dead-Ends
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DP vs DR, Uniform Random 3-cnfs
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DR and DP on 3-cnf Chains, Different Ordering
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Numer of Deadends
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DP vs Tableau (Uniform Random)
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DP vs Tableau (Chains)
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Bounded Directional Resolution - BDR(i)
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Dynamic Conditioning + Directional Resolution - DCDR(b)
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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
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References• Rish and Dechter (Irina Rish and Rina Dechter.
"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)