$7.[4$ marks$]$

$($a$)$ A weak$-$tape Turing machine is a single tape TM that can over$-$write each tape square at most twice $($including the input portion of the tape$).$ Show that this variant is

equivalent to the ordinary Turing machine model.

$($b$)$ An extremeley weak$-$tape Turing machine is a single tape TM that can over$-$write each tape square at most once $($including the input portion of the tape$).$ Show that this

variant is equivalent to the ordinary Turing machine model by describing how you can modify your answer to the previous question.

$8.[3$ marks$]$ Prove that decidable languages are closed under intersection using a high$-$level algorithm.