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7. Show that the following formula is valid.
(A(X) ∨ B( Y) ) → C(Z) ⇒ (¬ A(X) ∧ ¬ B(Y) ) ∨ C(Z)
where X, Y and Z are variables and A, B and C are predicates.
8. List all the satisfiability relations in a tabular form with four columns A,
B, C and the entire formula (say, Y) for the last formula.
9. Given X ε {a1, a2}, Y ε {b1, b2} and Z ε {c1, c2}, and S = {A(X) ∨
B(Y) ) → C(Z) ⇒ (¬ A(X) ∧ ¬ B(Y) ) ∨ C(Z)}, find the Herbrand
universe and the Herbrand base.
10. Illustrate the lifting lemma with the following parameters.
C1 = P (X, f (X)) Λ Q( Y, c) → R (X, b)
C2 = W ( f (Y) , Z ) → Q (Y, Z )
C 1 = P (a, f (a)) Λ Q( b, c) → R (a, b)
C 2 = W( f (b) , c ) → Q (b, c )
References
[1] Anderson, D. and Ortiz, C., “AALPS: A knowledge-based system for
aircraft loading,” IEEE Expert, pp. 71-79, Winter 1987.
[2] Ben-Ari, M., Mathematical Logic for Computer Science,
Prentice-Hall, Englewood Cliffs, NJ, pp. 11-87, 1993.
[3] Bender, E. A., Mathematical Methods in Artificial Intelligence, IEEE
Computer Society Press, Los Alamitos, chapter 1, pp. 26, 1996.
[4] Dougherty, E. R. and Giardina, C. R., Mathematical Methods for
Artificial Intelligence and Autonomous Systems, Prentice-Hall,
Englewood Cliffs, NJ, 1988.
[5] Herbrand, J., Researches sur la Theorie de la Demonstration, Ph.D.
Thesis, University of Paris, 1930.