# Questions and Answers on Homework Assignment #5

#### HW#5 is due on paper in class, Monday 28 April 2008

Question text is in black, my answers in blue.

• Question 4.1, posted 24 April:

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.

• Question 4.2, 24 April 2008

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?