P, NP, NP Hard & NP Complete Decision Problems

Decision problems are problems solved by boolean functions of inputs.  Polynomial time Turing machines are Turning machines with the number of required steps always bounded by some polynomial function of input size.  Polynomial time decision problems (P) are decision problems that can always be solved by some polynomial time Turing machine.  Nondeterministic polynomial time decision problems (NP) are decision problems that always have answers that can be verified by some polynomial time Turing machine.  NP hard decision problems are decision problems such that, if any can be solved by some polynomial time Turing machine, then so can all NP decision problems.  NP complete decision problems are decision problems that are both NP hard and NP decision problems.  An example of an NP hard decision problem is the halting problem. An example of an NP complete decision problem is the traveling salesman problem.  An example of a P decision problem is primality determination.