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{\bf Logic 1, WS 2004.
Homework 5, given Nov 04, due Nov 11}

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\noindent
1. Prove that if the formula $P(a)$ is satisfiable, then the formula
$\exi_x P(x)$ is also satisfiable.

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\noindent
2. Prove that if the formula $\all_x \exi_y P(x,y)$ is satisfiable,
then the formula $\all_x P(x, f(x))$ is also satisfiable.


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\noindent
3. Explain how to construct the Herbrand universe for the formula
$$ \all_x (P(f(x), a) \impl Q(b, g(x))).$$

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\noindent
4. Bring into the Skolem normal form the negation of the formula:
$$((\all_x (P(x) \impl (Q(x) \and R(x))))\ \and\ (\exi_x (P(x) \and S(x))))
\impl (\exi_x (R(x) \and S(x))).$$


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\noindent
5. Use the resolution method in order to infer the empty clause from the
clauses of the formula obtained above.

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