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sav08:normal_forms_for_first-order_logic [2009/05/14 15:59] vkuncak |
sav08:normal_forms_for_first-order_logic [2015/02/16 10:36] vkuncak |
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| Consider this formula in ${\cal L}$: | Consider this formula in ${\cal L}$: | ||
| - | \[ | + | \begin{equation*} |
| \begin{array}{l@{}l} | \begin{array}{l@{}l} | ||
| & (\forall x. \exists y.\ R(x,y))\ \land \\ | & (\forall x. \exists y.\ R(x,y))\ \land \\ | ||
| Line 17: | Line 17: | ||
| & \rightarrow \forall x. \exists y.\ R(x,y) \land P(y) | & \rightarrow \forall x. \exists y.\ R(x,y) \land P(y) | ||
| \end{array} | \end{array} | ||
| - | \] | + | \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): | ||
| - | \[ | + | \begin{equation*} |
| \begin{array}{l@{}l} | \begin{array}{l@{}l} | ||
| \lnot \bigg( \big( & (\forall x. \exists y.\ R(x,y))\ \land \\ | \lnot \bigg( \big( & (\forall x. \exists y.\ R(x,y))\ \land \\ | ||
| Line 26: | Line 26: | ||
| & \rightarrow \forall x. \exists y.\ R(x,y) \land P(y) \bigg) | & \rightarrow \forall x. \exists y.\ R(x,y) \land P(y) \bigg) | ||
| \end{array} | \end{array} | ||
| - | \] | + | \end{equation*} |
| We will first consider a range of techniques that allow us to convert such formula to simpler normal forms. | We will first consider a range of techniques that allow us to convert such formula to simpler normal forms. | ||
| Line 130: | Line 130: | ||
| \] | \] | ||
| Note: it is better to do PNF and SNF //for each conjunct independently//. | Note: it is better to do PNF and SNF //for each conjunct independently//. | ||
| + | |||
| ===== CNF and Sets of Clauses ===== | ===== CNF and Sets of Clauses ===== | ||
| Line 152: | Line 153: | ||
| *$C_2=\neg R(x,y) \lor R(x,f(y,z)))$ | *$C_2=\neg R(x,y) \lor R(x,f(y,z)))$ | ||
| *$C_3=P(x) \lor P(f(x,a))$ | *$C_3=P(x) \lor P(f(x,a))$ | ||
| - | *$C_4=\neg R(c,x) \vee \neg P(x)$ | + | *$C_4=\neg R(c,y) \vee \neg P(y)$ |
| === Another Example: Irreflexive Dense Linear Orders === | === Another Example: Irreflexive Dense Linear Orders === | ||