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 S = p. Show that if S is a set of positive Horn clauses, then Is |= S. 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 Is \|= c then SU{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 SU{c} is satisfiable for every c E T, then SUT 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 SU{c} is satisfiable. Use part (d) to conclude that Prolog can be used to test the satisfiability of an arbitrary set of Horn Clauses.

In this question, we will explore the semantic properties of propositional Horn clauses. For any set of clauses S, define I_s to be the interpretation that satisfies an atom p if and only if S = p. 

• Show that if S is a set of positive Horn clauses, then I_s |= S.
• Give an example of a set of clauses S where I_s  \|= S.
• Suppose that S is a set of positive Horn clauses and that c is a negative Horn clause. Show that if I_s  \|= c then SU{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 SU{c} is satisfiable for every c E T, then SUT 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 SU{c} is satisfiable. Use part (d) to conclude that Prolog can be used to test the satisfiability of an arbitrary set of Horn Clauses.