The Gateway to Computer Science Excellence
0 votes
171 views

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.

in Mathematical Logic by Veteran (105k points) | 171 views

1 Answer

+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...
by Veteran (425k points)
edited by
0

@Arjun sir

what is clausal form??

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. 

Related questions

Quick search syntax
tags tag:apple
author user:martin
title title:apple
content content:apple
exclude -tag:apple
force match +apple
views views:100
score score:10
answers answers:2
is accepted isaccepted:true
is closed isclosed:true
50,645 questions
56,563 answers
195,731 comments
101,651 users