Turing Machine as an Integer Function

Turing Machines are theoretical models that can simulate any computer algorithm, so we can perform integer operations as well. In this chapter, we will cover the basics of the Turing machine, followed by demonstrating how it can perform operations on integers.

Turing Machines - A Brief Recap

Turing Machines have the following components −

• Tape − An infinitely long strip divided into cells, each of which can contain a symbol (such as 0 or 1) or be blank.
• Head − A device that reads and writes symbols on the tape and can move left or right.
• Finite Control − A state machine that decides what the machine does next based on the current state and the symbol under the head.

The Turing machine operates on an input, processes it according to a sequence of states, and produces an output. Despite its simplicity, it's as powerful as any modern computer in terms of the types of calculations it can perform.

Turing Machine and Integer Functions

An integer function takes integer inputs and produces an integer output. For example, functions like addition, subtraction, and multiplication take integers as input and return an integer as output. We can design a Turing machine to compute such functions.

The input for a Turing machine that computes an integer function is typically represented by a sequence of symbols on the tape. For instance, the number 3 might be represented as three 0s: "000". If we need to compute a function of two numbers, we can use a blank space (B) as a separator: "000B00" represents the numbers 3 and 2.

Addition Using Turing Machine

We will see the operations in detail in the next few chapters. As the numbers are represented in unary format, so the count of 0s could be the number, the result will also be in such form. Its head will track the numbers and put them in a separate space in the tape. So, by concatenating them, the addition could be found.

Subtraction Using Turing Machine

Just like the Addition operation, here too it will remove the 0s from the larger number by replacing with some other tape symbol. At the end, the tape will hold the remaining 0s which is nothing but the result of subtraction.

Multiplication Using Turing Machine

Multiplication is another similar example where it will copy the numbers into another spot of the tape and when a number is copied it will reduce the multiplier by removing one 0. In such a way it will form the result of multiplication.

Conclusion

In this brief chapter, we presented the basics of integer operations through Turing Machines. As we know, the Turing Machine can handle any task that is solvable with certain finite step algorithms. So, integer functions like addition, subtraction, multiplication can also be solved by Turing Machines.

Here, we represent the numbers through unary representation where the number of 0s represents the number itself. In the next a few chapters, we will see these in detail with examples and diagrams for a clear understanding.