@Arjun sir

what is clausal form??

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0 votes

Consider the following well-formed formula:

- $\exists x \forall y [ \neg \: \exists z [ p (y, z) \wedge p (z, y) ] \equiv p(x,y)]$

Express the above well-formed formula in clausal form.

+3 votes

$$\begin{align}

&\quad \exists x \forall y \Biggl[ \color{blue}{\neg \: \exists z \Bigl [ p (y, z) \land p (z, y) \Bigr ]} \; \equiv \; \color{green}{p(x,y)} \Biggr ] \\[1em]

\equiv&\quad \exists x \forall y \Biggl [ \color{blue}{\neg \: \exists z \Bigl [ p (y, z) \land p (z, y) \Bigr ]} \longleftrightarrow \color{green}{p(x,y)} \Biggr ] \\[1em]

\equiv&\quad \exists x \forall y \Biggl [ \Bigl [\color{blue}{\neg \exists z \bigl [ p (y, z) \land p (z, y) \bigr ]} \land \color{green}{p(x,y)} \Bigr ] \lor \Bigl [ \color{red}{\neg} \color{blue}{\neg \exists z \bigl [ p (y, z) \lor \neg p (z, y) \bigr ]} \land \color{red}{\neg} \color{green}{p(x,y)} \Bigr ] \Biggr ] \\[1em]

\equiv&\quad \exists x \forall y \Biggl [ \Bigl [\color{blue}{\neg \: \exists z \bigl [ p (y, z) \land p (z, y) \bigr ]} \land \color{green}{p(x,y)} \Bigr ] \; \lor \; \Bigl [ \exists z \bigl [ p (y, z) \lor \neg p (z, y) \bigr ] \land \neg p(x,y) \Bigr ] \Biggr ]

\end{align}$$

Conversion not complete...

&\quad \exists x \forall y \Biggl[ \color{blue}{\neg \: \exists z \Bigl [ p (y, z) \land p (z, y) \Bigr ]} \; \equiv \; \color{green}{p(x,y)} \Biggr ] \\[1em]

\equiv&\quad \exists x \forall y \Biggl [ \color{blue}{\neg \: \exists z \Bigl [ p (y, z) \land p (z, y) \Bigr ]} \longleftrightarrow \color{green}{p(x,y)} \Biggr ] \\[1em]

\equiv&\quad \exists x \forall y \Biggl [ \Bigl [\color{blue}{\neg \exists z \bigl [ p (y, z) \land p (z, y) \bigr ]} \land \color{green}{p(x,y)} \Bigr ] \lor \Bigl [ \color{red}{\neg} \color{blue}{\neg \exists z \bigl [ p (y, z) \lor \neg p (z, y) \bigr ]} \land \color{red}{\neg} \color{green}{p(x,y)} \Bigr ] \Biggr ] \\[1em]

\equiv&\quad \exists x \forall y \Biggl [ \Bigl [\color{blue}{\neg \: \exists z \bigl [ p (y, z) \land p (z, y) \bigr ]} \land \color{green}{p(x,y)} \Bigr ] \; \lor \; \Bigl [ \exists z \bigl [ p (y, z) \lor \neg p (z, y) \bigr ] \land \neg p(x,y) \Bigr ] \Biggr ]

\end{align}$$

Conversion not complete...

0

You can see here: https://imada.sdu.dk/~felhar07/dm509e13/notes/convert_clausal.pdf

This answer is not complete; but is out of GATE syllabus -- was part of logic programming I believe.

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