Turing Machines , P , NP and NP - completeness

I assume that most students have encountered Turing machines before. (Students who have not may want to look at Sipser’s book [3].) A Turing machine is defined by an integer k ≥ 1, a finite set of states Q, an alphabet Γ, and a transition function δ : Q×Γk → Q×Γk−1×{L, S,R}k where: • k is the number of (infinite, one-dimensional) tapes used by the machine. In the general case we have k ≥ 3 and the first tape is a read-only input tape, the last is a write-once output tape, and the remaining k−2 tapes are work tapes. For Turing machines with boolean output (which is what we will mostly be concerned with in this course), an output tape is unnecessary since the output can be encoded into the final state of the Turing machine when it halts.