Deterministic Turing machines

The Turing machine is certainly the most powerful of the machines that we have considered and, in a sense, is the most powerful machine that we can consider. It is believed that every well-defined algorithm that people can be taught to perform or that can be performed by any computer can be performed on a Turing machine. This is essentially the statement made by Alonzo Church in 1936 and is known as Church's Thesis. This is not a theorem. It has not been mathematically proven. However, no one has found any reason for doubting it.

It is interesting that although the computer, as we know it, had not yet been invented when the Turing machine was created, the Turing machine contains the theory on which computers are based. Many students have been amazed to find that, using a Turing machine, they are actually writing computer programs. Thus computer programs preceded the computer.

We warn the reader in advance that if they look at different books on Turing machines, they will find the descriptions to be quite different. One author will state a certain property to be required of their machine. Another author will strictly prohibit the same property on their machine. Nevertheless, the machines, although different, have the same capabilities.

The Turing machine has an input alphabet Σ, a set of tape symbols, Γ containing Σ, and a set of states Q, similar to the automaton. The Turing machine has two special states, the start state s0 and the halt state h. When the machine reaches the halt state it shuts down. It also has a tape which is infinitely long on the right.