Universality of Turing Machine and Its Limitations
The universal Turing machine is the one that, when given a description of Turing machine M and input w, can simulate the behavior of M with input w. In this chapter we construct the universal Turing machine. This means that the universal Turing machine can behave like any Turing machine with any input if the description of the Turing machine to be simulated is given together with its input. On the other hand, there exists a limit to the computational power of Turing machines. This limitation is shown by giving the halting problem, which any Turing machine can never solve. The halting problem asks whether, when a Turing machine M and an input w are given, M with input w will eventually halt or continue to run forever. As another example of a problem that any Turing machine cannot solve, we give the Post correspondence problem. The Post correspondence problem asks whether or not, when a collection of pairs of strings is given, there exists a sequence of the pairs (repetitions permitted) that has certain properties of a match.