Question

Describe how Turing machines are used to compute number-theoretic functions.

Short answer

Turing machine T then computes some number-theoretic function when the input of \(n + 1\) ones result in the output with \(f\left( n \right) + 1\) ones.

Step-by-step solution

General form

Turing machine\(T = \left( {S,\,I,\,f,\,{s_0}} \right)\):It is a four-tuple made of a finite set S of states, an alphabet I containing the space symbol B, a sub-function f from \(S \times I\) to \(S \times I \times \left\{ {R,\,L} \right\}\) and an initial state \({s_0}\)

Step 2: Describe the Turing machine are used to compute number-theoretic functions

Referring to Turing machine:

A four-tuple made of a finite set S of states, an alphabet I having the space symbol B, a sub-function f from \(S \times I\) to \(S \times I \times \left\{ {R,\,L} \right\}\), and an initial state \({s_0}\).

Then, describe it.

Let \(f\left( n \right)\) be a function where n is a nonnegative integer.

The integer n is represented by \(n + 1\) ones.

The Turing machine T then computes some number-theoretic function when the input of \(n + 1\) ones result in the output with \(f\left( n \right) + 1\) ones.

Hence, the result is founded.