I'm a beginner in learning about computational complexity and this has stumped me. I've read that by the time hierarchy theorem, it's known that EXP-complete problems are not in P. (Wikipedia) It makes absolute sense intuitively that this is the case, as does P≠NP. From what I understand, the time hierarchy theorem states that given more time, a turing machine can solve harder problems.
I have 2 questions:
- How is the time hierarchy theorem (or anything else) used to prove that P≠EXPTIME?
- If we assume P=NP and NP=EXP, P=EXP and that contradicts P≠EXP. So one of those statements MUST be false. But we can only conjecture that both these are false, and can't prove it? Is that the case?