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Quiz 11-18-2008 Propositional Logic

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1. Let A,B,C be propositions, i.e. they can take values False (F) or True (T). a) How many possible worlds are there in this universe of 3 propostions? b) Define the Knowledge Base:. Convert this KB1 to CNF form (conjunctive normal form). c) In which worlds is KB1 true? (provide assignments to A,B,C, e.g. [T,T,T]). d) Consider the query B. In which worlds is B=T? (again provide assignments). e) Can we entail B from KB1? Prove your statement or give a counterexample. f) Consider the Knowledge Base: Prove that B is entailed by the KB2 using resolution. g) Is KB2 in Horn form? all question are worth 2 points

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a)How many possible worlds are there in this universe of 3 propostions 8 b) Define the Knowledge Base:. Convert this KB1 to CNF form (conjunctive normal form). c) In which worlds is KB1 true? (provide assignments to A,B,C, e.g. [T,T,T]). {[T,T,T],[F,T,T],[F,F,T]} d) Consider the query B. In which worlds is B=T? (again provide assignments). {[T,T,T],[F,T,T],[T,T,F],[F,T,F]} e) Can we entail B from KB1? Prove your statement or give a counterexample. No, e.g. [T,T,F] is True in the query but not in the KB f) Consider the Knowledge Base: Prove that B is entailed by the KB2 using resolution. Add not(B) to the KB. Then A cancels with (not(A) and B) to give B which then cancels with not(B) to give the empty clause. There are no solutions, hence unsatisfyable, hence we proved entailment. g)Is KB2 in Horn form? Yes, all clauses have at most 1 positive literal

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