Resolution versus Search:Two Strategies for SAT

Brad DunbarShamik Roy Chowdhury

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

Backtracking Vs Resolution

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 )

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.

Directional Resolution – An ordering based algorithm

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

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

Complexity : Induced Width

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.

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 }

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 }

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 }

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.

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 .

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 )

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.

–

Directional Resolution and Tree Clustering

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 )

Directional Resolution and Tree Clustering

Directional Resolution and Tree Clustering

Directional Resolution and Tree Clustering

Directional Resolution and Tree Clustering

Backtracking (DP) Algorithm

Backtracking (DP) Algorithm

Comparison of Backtracking and Resolution

Random Problem Generators

DR vs DP, 3-cnf Chains

DR vs DP, > 5000 Dead-Ends

DP vs DR, Uniform Random 3-cnfs

DR and DP on 3-cnf Chains, Different Ordering

Numer of Deadends

DP vs Tableau (Uniform Random)

DP vs Tableau (Chains)

Bounded Directional Resolution - BDR(i)

Dynamic Conditioning + Directional Resolution - DCDR(b)

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

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)