User blog comment:LittlePeng9/Binary relations via Turing machines/@comment-10429372-20131006180926/@comment-1605058-20131006200819

On empty input yes, because 0>0 doesn't hold. If we take pair of numbers such that a>b then machine will halt on input 1^a_1^b (it might not work for 0).

I'm planning to make machines for w+1, w2, w3, w^2 and, if it works out, w^w.