I am new to complexity theory and I have some maybe stupid questions:
Are "decision problems" with the characteristic functions x(t)=1 or x(t)=0 "decision problems" or not?
If yes: Are the corresponding formal languages members of P or not?
(The calculation time needed is constant.)
If the answer is yes in both cases:
If $P\neq NP$ no P decision problem can be NP-complete. This is easy to prove.
I have read that if $P=NP$ all P decision problems would be NP-complete. This is easy to prove for all other problems however the problems described above cannot be complete in any way.
Are these problems an exception?