Question $5(G-13$ points$)$: Famous painting is stolen overnight. The surveillance cameras show that only $5$ museum custodians entered$/$left museum overnight, so police assume that one of them must be guilty. Police also assume that there is only one culprit who is lying because there is no :$20.$ cis re believe that there was conspiracy at work. Here is what custodians claimed at interrogation:

Bill: Ted is guilty:

Frank: I am innocent:

Ied: Diana is guilty:

Ijina: Ted lies if he says I am guilty;

Wike: Bill tells the truth and Ted lies;

Police believes that only one of them is guilty and only the person who is guilty is lying. So$,$ who stole the painting?

Write up the knowledge base $($KB$)$ and then formally prove $($i$.$e$.$ apply either MP$,$ or resolution $-$your choice, with axioms of KB $)$ the culprit.

Hints:

You need two Boolean variables per custodian: 'telling truth' and 'being guilty' to express the formula. For example, for Bill you need variable $B_T-$ for 'Bill is truthful', and variable $B_G-$ for 'Bill is guiltu':

Try to prove first $($and add to KB$)$ that

Diana and Ted cannot both be truthful

Bill. Frank. and Mike are all truthful