P is the class of all clauses in the signature which have exactly one ground-signature (denoted in the lecture also P(c1, CN)) and are not contained in G(c1, CN). For each P, consider the possibility that (g, u) is a ground-signature of a function symbol. Finally, suppose that (g, u) is not a ground-signature of the function symbol. In this case, consider P(g, f(c1, CN)) as well. Then, the expression f(c1) will be an instance of the class G(g, f(c1, CN)) in the signature, while P(g, f(c1, CN)) will of course be not-a-ground. Finally, consider P(g, f(c1)] and if P(g, f(c1)) is not-a-ground, then, if P(c1, CN) is not-a-ground and g is a value of the sign function denoted in the signature, then, f(c1) is an instance of the class P(g, f(c1)) in the signature. Exercise 5.2: Given the signature of the form G(c1, CN) and the classes G(G, f(c1, CN)) and P(P, G(G, f(c1, CN)), respectively, what should be the corresponding expressions of the class P ? Solution: (a) Find the (natural way of) setting c1 and f(c1) equal in the signature, if possible.
(5.22) Numerical Analysis Exercise 5.3: On the left-hand side of the following function equation, what is the first three terms? Solution: (a) Find the coefficients of F(p)g.
(b) Find the coefficients of P(g, u) for each variable in f.
© Find the coefficients of each of the coefficients of f. Exercise 5.
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R S IS UN S PD Dr. Vitoria Sofronie-Stokkermans E R SIT A FR Informatic SA University t a DES Saarland IV A VIE N May 30, 2011, Exercises for Selected Topics in Automated Reasoning Exercise sheet
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