Question text is in black, my answers in blue.
I'm having trouble getting started on 6.23 -- should I try to reduce ATM to this problem?
That's often a good idea, if your target problem is TR, but actually the
language of incompressible strings is not TR, it's co-TR. (You could prove a
string is not incompressible by giving a short description of it, but
there's no clear way to prove that it is incompressible.) So trying
to reduce ATM-bar to this language might make sense.
But it's actually easier here to in effect use Turing reductions and derive
a contradiction from the assumption that this set is TD. Here is a big hint:
the Richard Paradox
refers to a particular positive integer, "the least positive integer that
cannot be defined in fewer than thirteen words". If this definition made sense,
I would have just described the number in only twelve words, a contradiction.
So I could describe a string as being "the n'th incompressible string", if the set of incompressibles were TD. But I don't see a contradiction yet -- you said that most strings were incompressible, so the number n might not necessarily be any shorter than the n'th string itself.
Yeah, that's the right idea, but you need something more.
In 7.11, here's my idea for a certificate to show that G and H are isomorphic. I'll take G and renumber its vertices until it is identical to H. Then the verifier just has to check that my reordering is identical to H, which it can do in P. But this seems too easy...
It is too easy. A good way to think of this is an adversary situation, where I want to prove to you that G and H are isomorphic but you don't trust me. Your scheme amounts to my saying "Here is this graph. I made it by renumbering the vertices of G (trust me). You can see that it's identical to H, so you must believe G and H are isomorphic." I hope this does not convince you. But is there more evidence I could give you that would be convincing?
Last modified 24 April 2008