Quiz 11-18-2008 Propositional Logic

1. Let A,B,C be propositions, i. e 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

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]}. Also correct if they if they used included KB1=T AND B=T e) Can we entail B from KB1? Prove your statement or give a counterexample. No, e.g. [F,F,T] 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. Is KB2 in Horn form?  Yes, all clauses have at most 1 positive literal