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However, the machine can also have one of the following two forms, with D and G being the actions of simply moving the read head to the right or left without writing anything on the read–write tape:

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This machine is totally automatic, and it is the most elementary possible with regard to the calculations to be carried out. It is the most important universal machine in the history of computation. All the algorithms use sequences of instructions and are therefore sophisticated compositions of Turing machines, and the instructions can lead to the request after their execution to place themselves on another instruction to be executed by the famous “Go to” and this repeatedly until meeting the instruction “End” of the end of execution of the instructions of the algorithm.

The functions that the Turing machine computes are called recursive primitive functions and they operate on sequences of natural numbers. They are obtained from basic functions, like identity, projection, successor function, using composition, recurrence and minimization, and they are executed in associations. They define all the usual arithmetic functions by machine associations, like power functions, products and sorts, and they are thus the basic model of what can be defined in mathematics to operate on sequences of integers.