Prove that it is undecidable whether a given LBA accepts a regular set

Question

I know for an LBA the emptiness problem is undecidable. However I am not clear on how to reduce the halting problem of Turing machines to this as LBAs are strictly computationally less powerful than Turing machines. Or should I approach with reductions using computational histories of a Turing machine on an input.

Answer 1

Thanks to @YuvalFilmus I managed to reduce the Halting problem to the given problem. We design an LBA B, that given an instance of the Halting problem, $\langle M,x\rangle$, accepts the valid computation histories of M on x. If M doesn't halt on x, $L(B)=\phi$ which is regular. However, if M does halt $L(B)=VAL-COMPS(M,x)$ which is not regular(not even CFL). Techinically, I have reduced the complement of the Halting problem, but as we know that it is also not decidable, so it's fine.