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SS 2013 July 17, 2013 In-class Exercises to the Lecture Logics

Sheet 7

Jun.-Prof. Dr. Roland Meyer Discussion on July 18 and 19, 2013 Exercise 7.1 [Resolution]

Using resolution, show that the formula

@z

1

rqpz

1

qs _ @xrpqpxq _ rpxqq ^ Dz

2

r ppz

2

q ^ pppz

2

q _ rpxqqss is a tautology. This amounts to

a) negating the formula,

b) bringing the result into clause form (Skolem + KNF), and c) applying resolution to the formula in clause form.

Exercise 7.2 [Calculating MGU]

For each of the following sets of literals, decide whether it is unifiable and if so, determine a most general unifier.

a) tqpx, zq, qphpy, zq, f paqq, qphpf pbq, zq, zqu . b) tppx, fpyqq, ppf paq, yqu.

Exercise 7.3 [An application to graphs]

By a graph, we mean an undirected (not necessarily finite) graph that may have loops.

a) Formalize the following statement as a formula in first order predicate logic: If every node has a loop or has at least one other node that it is connected to, then every node is connected with an edge.

b) Negate the formula and transform the result into an equisatisfiable formula without

“=” (see In-Class Exercise 5.3).

c) Using resolution, show that the obtained formula is unsatisfiable.

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