Halting problem for turing machines with one input
What you call a &quot;liar&quot; program is typically called a diagonalization argument, since it somehow resembles Cantor's diagonalization argument.
Usually what we mean by that is that if we try to enumerate all elements with a certain property, then we can point out, by construction, an element that is necessarily &quot;incorrect&quot;, and we do so by showing that it does not satisfy the property with regards to itself (this is completely hand-wavy, but if you've seen the argument, you'll hopefully understand what I mean).
Now, what you're asking is whether we can do the same in order to show that ALL_TM (TM universality) is not decidable by a TM with oracle to HALT_TM.
Well, not really. At least not in a natural way: the argument would have to start by assuming there exists an oracle machine M that decides ALL_TM, and then constructing from M some other machine N (what you call a &quot;liar&quot; machine), which would yield a contradiction when we run N on N.
However, the machine N would have to be an oracle machine, not a standard TM (or you would essentially lose the power of the oracle in M). But since the oracle in N takes as input standard TMs, then the inner working of N would somehow need to create a standard TM to feed to the oracle. This doesn't seem like a natural construction, and it would certainly not be simple.
Now, perhaps there is some convoluted way of giving a direct diagonalization argument for this claim, but it's not well motivated, given that proving the claim directly (namely that ALL_TM is not decidable by a TM with oracle to HALT) is not difficult using standard tools, such as reductions.