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Unformatted text preview: mples with high w3 having div 3  1. We next extend the class of causal theories by attributing a wider meaning to \zero" diversity. For the class of zero diversity theories de ned earlier directional resolution will not generate any new clauses. Sometime, however, directional resolution generate clauses which are tautologies, or which are all subsumed in the original set of clauses. 14 Example 5: The odd-parity relation over n propositional symbols P1 ; :::; Pn allows all models with an odd number of \true" assignments. This theory has a nice representation using an additional set of n symbols Q1 ; :::; Qn. Symbol Qi denotes the parity of the rst i propositions P1 ; :::; Pi. The general de nition of the theory is given by: P1 ! Q1 :P ! :Q 1 1 The ith de nitions is Pi ; Qi01 ! :Qi Pi ; :Qi01 ! Qi :Pi; Qi0 ! Qi 1 :Pi ; :Qi0 ! :Qi 1 If we want to state that the relation is odd-parity we add the proposition, Qn . Consider now the ordering: d = (P1 ; Q1; P2 ; Q2; :::; Pn ; Qn ). Clearly this ordering has diversity equal 2. Nevertheless its diversity equals to its induced diversity if we eliminate tautologies in resolvents. This leads to the following de nition: De nition 6: [Extended zero diversity] Given a theory ' and an ordering d, Qi is said to be of extended zero diversity i all the resolvents generated when bucketi is processed are either tautologies, or they are subsumed by clauses in bucket1; :::; bucketi01. An ordering has extended zero diversity i all its buckets have extended zero diversity. A theory has extended zero diversity i there exists an extended-zero-diversity ordering of its variables. Theorem 10: Algorithm recognize-extended-zero-div (see Figure 5) is guaranteed to recognize extended zero diversity. 15 recognize-extended-zero-div(') 1. For i = n to 1 do 2. Find a letter Q such that all its resolvents over the theory 'i = ' 0 [n=i+1 bucketj are j subsumed by 'i . If no such letter exists declare failure. Else put Q as ith in the ordering. Figure 5: Algorithm recognizing extended zero diversity Proof: Let S denote the set of variables of a theory '. Assume that there exists an extended-zero-diversity ordering d of variables in S , but the algorithm could not nd any such ordering. In other words, at some step i, after Qn ; :::; Qi+1 were selected, the algorithm was not able to nd any extended-zero-diversity variable among Si = S 0 [n=i+1 Qj . Now, j let Q be the highest variable in S w.r.t. d. Then any resolvent obtained in Q's bucket is subsumed by clauses in lower buckets of d. Those buckets correspond to the variables in Si 0 fQg and, possible, some of Qn ; :::; Qi+1. Now we delete all the clauses containing Qn ; :::; Qi+1 from those buckets, and the resulting theory will be exactly 'i . But then then all resolvents in Q's bucket are subsumed in the buckets of Si 0 fQg, not in any one of Qn ; :::; Qi+1 (subsumed clause should include all the variables of the subsuming clause), i.e. subsumed by 'i . This contradicts our...
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