3.1.10: Summary

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A Turing machine is a kind of idealized computation mechanism. It consists of a one-way infinite tape, divided into squares, each of which can contain a symbol from a pre-determined alphabet. The machine operates by moving a read-write head along the tape. It may also be in one of a pre-determined number of states. The actions of the read-write head are determined by a set of instructions; each instruction is conditional on the machine being in a certain state and reading a certain symbol, and specifies which symbol the machine will write onto the current square, whether it will move the read-write head one square left, right, or stay put, and which state it will switch to. If the tape contains a certain input, represented as a sequence of symbols on the tape, and the machine is put into the designated start state with the read-write head reading the leftmost square of the input, the instruction set will step-wise determine a sequence of configurations of the machine: content of tape, position of read-write head, and state of the machine. Should the machine encounter a configuration in which the instruction set does not contain an instruction for the current symbol read/state combination, the machine halts, and the content of the tape is the output.

Numbers can very easily be represented as sequences of strokes on the Tape of a Turing machine. We say a function \(\Nat \to \Nat\) is Turing computable if there is a Turing machine which, whenever it is started on the unary representation of \(n\) as input, eventually halts with its tape containing the unary representation of \(f(n)\) as output. Many familiar arithmetical functions are easily (or not-so-easily) shown to be Turing computable. Many other models of computation other than Turing machines have been proposed; and it has always turned out that the arithmetical functions computable there are also Turing computable. This is seen as support for the Church-Turing Thesis, that every arithmetical function that can effectively be computed is Turing computable.