$1.$ We previously proved that all finite languages are regular. In Syntactic Structures, Chomsky argues that english cannot have regular structure. We did not directly address the question if the set of grammatical english sentences is finite or infinite. Why is Chomsky's argument still correct, that english does not have regular structure, even if you suppose that english was finite?

$2.$ Suppose we have a push down automata with two stacks. It has nondeterministic transitions of the form read, $($pop$1,$ pop$2)($push$1,$ push$2).$ It reads from the input, pops from both stacks, and pushes to both stacks simultaneously. Give the state

diagram of this PDA with two stacks to decide $\{ww|win{}^{*}\}.($Hint: What allowed a single stack PDA to decide $w{w}^R$ but not $ww?)$