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 seman-tic properties of propositional Horn clauses. For any set of clauses S, define Is to be theinterpretation 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 satisfiablealso.• In the propositional case, the normal Prolog interpreter can be thought of as taking a setof positive Horn clauses S (the program) and a single negative clause c (the query) anddetermining whether or not SU{c} is satisfiable. Use part (d) to conclude that Prologcan be used to test the satisfiability of an arbitrary set of Horn Clauses.

(a) If S is a set of positive Horn clauses, then Is |= S. Proof: We need to show that if an…

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Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

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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.

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