5. :. : -. (Ex.4 in Ch5 of the book KRR) In this question, we will explore the semantic properties of propositional Horn clauses. For any set of clauses S, define IS to be the interpretation that satisfies an atom p if and only if Sp. - Show that if S is a set of positive Horn clauses, then ISS. - Give an example of a set of clauses S where IS=S. - Suppose that S is a set of positive Horn clauses and that c is a negative Horn clause. Show that if ISc then S{c} is unsatisfiable. - Suppose that S is a set of positive Horn clauses and that T is a set of negative ones. Using part (c), show that if S{c} is satisfiable for every cT, then ST is satisfiable also. - In the propositional case, the normal Prolog interpreter can be thought of as taking a set of positive Horn clauses S (the program) and a single negative clause c (the query) and determining whether or not S{c} is satisfiable. Use part (d) to conclude that Prolog can be used to test the satisfiability of an arbitrary set of Horn Clauses