Please Answer Part C

An especially simple class of CNF formulas are those built from Horn clauses. A Horn clause is a clause containing, at most, one positive literal. (A pure Horn clause is a clause containing exactly one positive literal.) Horn clauses form the basis for the computer language Prolog, which allows the programmer to input a set of requirements (specified in formal logic) and to ask the computer to find how to satisfy them all (if possible)-as opposed to the user's having to write out the case analysis. (a) Using the atomic formulas a = "Tweety is a penguin." b = "Opus is a penguin." c = "Phoenix is a penguin." d = "Elvis lives!" express "If Tweety is a penguin and Opus is a penguin, and Phoenix is a penguin, then Elvis lives" as a Horn clause. (b) Find a set of Horn clauses logically equivalent to (a b c rightarrow d e) (a e). Find the shortest such set of clauses. (c) Find all satisfying truth assignments for the following set of Horn clauses: {p_1, p_1 p_2 p_1 p_2 p_3, p_1 p_2 p_4, p_3 p_4 p_5} Now, show that the following set of Horn clauses is not satisfiable: {p_1, p_1 p_2, p_1 p_2 p_3, p_1 p_2 p_4, p_3 p_4 p_5}