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The formula F is false under the interpretation I2 . Obviously, the truth of a formula in PL1 depends on the interpretation. Now, after this preview, we define truth. 4 • An atomic formula p(t1 , . . , tn ) is true (or valid) under the interpretation I if, after interpretation and evaluation of all terms t1 , . . , tn and interpretation of the predicate p through the n-place relation r, it holds that (I(t1 ), . . , I(tn )) ∈ r. 1 on page 17. • A formula ∀x F is true under the interpretation I exactly when it is true given an arbitrary change of the interpretation for the variable x (and only for x) • A formula ∃x F is true under the interpretation I exactly when there is an interpretation for x which makes the formula true.
Res(1, 2) : (N )5 Res(3, 4) : (P )6 Res(1, 4) : (S ∨ P )7 We could have derived clause P also from Res(4, 5) or Res(2, 7). Every further resolution step would lead to the derivation of clauses that are already available. Because it does not allow the derivation of the empty clause, it has therefore been shown that the knowledge base is non-contradictory. So far we have derived N and P . To show that ¬S holds, we add the clause (S)8 to the set of clauses as a negated query. With the resolution step Res(2, 8) : ()9 the proof is complete.
Today, resolution represents just one of many calculi used in high-performance provers. 2 into a resolution proof. First the formulas are transformed into conjunctive normal form and the negated query ¬Q ≡ ¬child(eve, oscar, anne) is added to the knowledge base, which gives KB ∧ ¬Q ≡ (child(eve, anne, oscar))1 ∧ (¬child(x, y, z) ∨ child(x, z, y))2 ∧ (¬child(eve, oscar, anne))3 . The proof could then look something like (2) x/eve, y/anne, z/oscar : (¬child(eve, anne, oscar) ∨ child(eve, oscar, anne))4 Res(3, 4) : (¬child(eve, anne, oscar))5 Res(1, 5) : ()6 , where, in the first step, the variables x, y, z are replaced by constants.