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SS 2012 18.07.2012 In-class Exercises to the Lecture Logics Sheet 7 Jun.-Prof. Dr. Roland Meyer Discussion on 19./20.07.2012

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SS 2012 18.07.2012 In-class Exercises to the Lecture Logics

Sheet 7

Jun.-Prof. Dr. Roland Meyer Discussion on 19./20.07.2012 Exercise 7.1 [The Löwenheim-Skolem Theorem]

A variant of Herbrand’s Theorem reads: A formula in first order predicate logic without “=” is satisfiable if and only if it has a Herbrand model.

a) Present a satisfiable formula in first order predicate logic that has no Herbrand model.

b) Show that for each formula in first order predicate logic, there is an equisatisfiable formula A

1

without “=”.

c) Prove that for each formula in first order predicate logic, there is a formula A

1

without

“=” with the following property: A is satisfiable if and only if it has a model that is obtained by a quotient set construction from a model of A

1

.

d) Conclude that each satisfiable formula in first order predicate logic has a countable model. Note: This result is also known as the Löwenheim-Skolem Theorem.

Exercise 7.2 [Resolution]

Consider the formula

A @ x r p p x q ^ p q p z, b q Ñ D y p q p x, y q _ p p y qqqs ^ @ x @ yq p x, y q .

a) Determine a formula equisatisfiable in clause form which is equisatisfiable to A.

b) Using the resolution calculus, show that A is unsatisfiable.

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