Of si 0 fqg not in any one of qn qi1 subsumed clause

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Unformatted text preview: assumption. 2 Theorem 11: The complexity of the algorithm recognize zero-diversity is O(n2 1 j'j3), where j'j is the number of clauses in '. by resolving over this variable, and each clause is checked for subsumption in O(j'j) time, Proof: For each variable Q at the i-th step of the algorithm O(j'j2 ) clauses are generated which yields O(j'j3 ) complexity of checking a variable for extended zero diversity. The algorithm performs n steps, checking no more than n variables at each step, i.e. the total bound on algorithm's complexity is O(n2 1 j'j3). 2 16 DP-backtracking(') Input: A cnf theory '. Output: A decision of whether ' is satis able. 1. Unit propagate('); 2. If the empty clause generated return(false); 3. else if all variables are assigned return(true); 4. else 5. 6. Q = some unassigned variable; return( DP-backtracking( ' ^ Q) _ DP-backtracking(' ^ :Q) ) Figure 6: Davis-Putnam procedure 5 Bounded directional resolution Since the algorithm directional resolution is time and space exponential in the worst case, we propose an approximate algorithm called bounded directional resolution (BDR). The algorithm records clauses of size k or less when k is a constant. Consequently, its complexity is polynomial in k . Algorithm bounded directional resolution parallels algorithms for directional k -consistency in constraint satisfaction problems [7]. 6 Experimental evaluation In this section we report experimental results demonstrating advantages and drawbacks of the algorithms directional resolution (DR), bounded directional resolution (BDR), and DPbacktracking on problems with di erent structures. A combination of the last two algorithms called BDR-DP is proposed as an overall most ecient algorithm among them. Directional resolution has been implemented in accordance with the algorithm described in section 3. DP-backtracking is a version of the Davis-Putnam procedure (see Figure 6) 17 ~A B ~C A ~B ~C A ~G EF E ~F G G ~F I ~H ~ I H I ~J ~J Figure 7: An example of a theory with the chain structure that uses a loop construction instead of recursion in order to increase space eciency. The algorithm has been also augmented with the 2-literal clause heuristic proposed in [4]. The heuristic suggests to instantiate next a variable that would cause the largest number of unit propagations. The number of possible unit propagations is approximated by the number of 2literal clauses in which the variable appears. The modi ed version signi cantly outperforms DP-backtracking without this heuristic [4]. In order to nd a solution, DR was followed by DP-backtracking without the 2-literal clause heuristic so that the order of variables was xed. As the theory dictates, no deadends occur when DP-backtracking runs after DR on the same ordering, and the time it takes is linear in the size of DR's output theory. Algorithm BDR, since it is incomplete for satis ability, was followed by DP-backtracking augmented with the 2-literal clause heuristic. We call this combination BDR...
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