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sav08:herbrand_model_for_an_example [2009/05/14 12:30] vkuncak |
sav08:herbrand_model_for_an_example [2015/04/21 17:30] (current) |
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====== Herbrand Model and Unsat Proof for an Example ====== | ====== Herbrand Model and Unsat Proof for an Example ====== | ||
+ | |||
+ | [[wp>Herbrand]] | ||
We will look at the language ${\cal L} = \{P, R, a, f\}$ where | We will look at the language ${\cal L} = \{P, R, a, f\}$ where | ||
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Consider this formula in ${\cal L}$: | Consider this formula in ${\cal L}$: | ||
- | \[ | + | \begin{equation*} |
\begin{array}{l@{}l} | \begin{array}{l@{}l} | ||
& (\forall x. \exists y.\ R(x,y))\ \land \\ | & (\forall x. \exists y.\ R(x,y))\ \land \\ | ||
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& \rightarrow \forall x. \exists y.\ R(x,y) \land P(y) | & \rightarrow \forall x. \exists y.\ R(x,y) \land P(y) | ||
\end{array} | \end{array} | ||
- | \] | + | \end{equation*} |
We are interested in checking the //validity// of this formula (is it true in all interpretations). We will check the //satisfiability// of the negation of this formula (does it have a model): | We are interested in checking the //validity// of this formula (is it true in all interpretations). We will check the //satisfiability// of the negation of this formula (does it have a model): | ||
- | \[ | + | \begin{equation*} |
\begin{array}{l@{}l} | \begin{array}{l@{}l} | ||
\lnot \bigg( \big( & (\forall x. \exists y.\ R(x,y))\ \land \\ | \lnot \bigg( \big( & (\forall x. \exists y.\ R(x,y))\ \land \\ | ||
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& \rightarrow \forall x. \exists y.\ R(x,y) \land P(y) \bigg) | & \rightarrow \forall x. \exists y.\ R(x,y) \land P(y) \bigg) | ||
\end{array} | \end{array} | ||
- | \] | + | \end{equation*} |
Parsing the formula. | Parsing the formula. |