this is my first stackoverflow question, so be gentle. I apologize in advance if this has been beaten to death already... I read a few threads on NP but I haven't found a tantalizing answer to my question (if anything, I came up with some new ones). Briefly:
- Are there decision problems which are decidable but not in NP?
- If so, are problems which ask for a solution harder than the equivalent decision problem?
My suspicion is that the answer to the first question is a resounding 'yes' and that the answer to the second is a resounding 'no'.
In the first case, an example problem might be "given a set S, a subset T of S and a function f with domain 2^S, determine whether T maximizes f". For generic S, T and f, you can't even verify this without checking f(X) for all subsets X of S, right?
In the second case... well, I'll admit this is more of a hunch. For some reason, it doesn't seem like it should matter whether an answer contains one bit (for decision problems) or any (finite) number of bits... or, in other words, why you shouldn't be able to consider the symbols left on the tape after the TM halts as part of the "answer".
EDIT: Actually, I have a question... how precisely does your construction show that function problems are "no harder" than decision problems? If anything, you've shown that it's no easier to answer a function problem than a decision problem... which is trivial. Perhaps this is my fault for asking the question in a sloppy fashion.
Given a TM T1 in NP which solves the problem "Is X a solution to problem P" for variable X and (for the sake of argument) fixed P, is it guaranteed that there will be a TM T2 in NP which halts everywhere T1 halts, which ends in the "halt accept" state everywhere it halts, and leaves e.g. a binary representation of X on the tape?