Please help me get this right,
The question asks Is L Turing decidable ?
where L ={<M> : M is a TM, which accepts some palindrome
My thoughts on the poof to show that it is undecidable are the following. Assume L is decidable and let R be a decider for that language. I will construct a decider S for A_tm using R as a subroutine.
S = on input <M,w>
1. constructs TM M_w = on input x
if x != (some palindrome ex 1^n 0 1^n) reject
else run M on w if M accepts => accept; if M rejects => reject.
2. Run R on M_w
3. If R accepts => accept, if R rejects => reject.
Note: Assume M accepts w then L(M_w) = some palindrome, Then R(M_w) accepts it => S accepts. Assume M rejects or loops then L(M_w) = empty set. Then R(M_w) rejects => S rejects.
Now since we know that A_tm is undecidable , assumption was wrong => L is undecidable.