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Prerita Agarwal

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23 Jul, 2024 @ 01:30 PM

Introduction

The Multitape Turing machine is one of the variants of the Turing machine that uses various tapes. The main feature is that each tape has its own head for reading and writing, which can move independently of the other heads.

A Turing Machineis an accepting device used to accept recursive Enumerable Language generated by type 0 grammar. There are Three Types of Turing Machines. In the following article, we will discuss the Multi-tape Turing machine and its formal definition. Before getting to it, let's briefly recap what Turing Machine exactly is.

What is Turing Machine?

It consists of an infinite-length tape on which read and write operations can be performed. There is an endless number of cells on the tape, where each cell contains an input symbol or a special symbol called a blank. A head pointer points to the cell currently being read, moving in both directions.

Let us discuss a variation of the Turing machine, a Multi-tape Turing machine.

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What is Multitape Turing Machine?

Multi-tape Turing Machines have many tapes, each with its own head. Each of the heads can move independently of the others.

Initially, the first tape is consumed as input, and other tapes are kept blank.

Then the consecutive symbols under its head are read by the machine, the symbols are printed on each tape and its head is moved by the TM.

The expressive power of a multi-tape Turing computer is similar to that of a k-track Turing machine. Single-tape Turing machines can be used to simulate multi-tape Turing machines.

Note: There is a single-tape Turing machine for every multi-tape Turing machine.

Multi-Tape Turing Machine Formal theorem

We can define a Multi-tape Turing machine formally as a 6-tuple M = (Q, X, B, δ, q0, F) where,

Q is the finite set of states

X is the tape alphabet/symbol

∑ is the finite set of symbols(the tape alphabet)

δ is a transition function; δ : Q × X^{K} → Q × (X × {Left_shift, Right_shift})^{K}. where k is the number of tapes

q0 ∈ Q is the initial state

B ∈ ∑ is the blank symbol

F ⊆ Q is the set of accepting states Also read, Arden's theorem

Explanation of Multitape Turing Machine

A multitape Turing machine follows certain steps for computation. Initially, the machine receives the input from the leftmost positions of tape 1, and the rest of tape 1 and the other tapes are blank(Blank symbols). All the heads begin from the leftmost positions of the tapes, and when the machine starts the computation process according to the rules described by our transition function(δ). The computation process will continue until it enters the accept states, i.e., the point at which it halts.

The transition function for a multitape Turing Machine has the form δ : Q × X^K → Q × (X × {Left_shift, Right_shift})^K

We will consider a 3-tape Turing machine ‘T’ for binary addition where the input string will have the form of w_{1}#w_{2}, where w_{1} and w_{2} are the binary numbers for the addition. The result will be printed on tape 3 in reverse order.

Steps of the Multitape Turing Machine theorem:

T = “Input String w_{1}#w_{2}”:

The first step is to copy the string w_{2} to tape 2.

Erase w_{1} string from tape 1.

Next is to initialize the carry bit ‘c’ to 0. The carry bit is recorded in the finite control.

The heads of all three tapes are to be positioned at the beginning of each tape.

As you know, each tape contains infinite cells, so let us assume that ‘s_{1}’ is the contents of the current cell of tape 1 and ‘s_{2} ’ is the contents of the current cell of tape 2.

Now there are two chances. First is that if s_{1} and s_{2} both are not blank:

Perform full-adder operation(Full-adder takes three inputs and produces two outputs) with s_{1}, s_{2}, and c, writing the sum bit to tape 3(Result will be printed), and updated carry bit c will be recorded in the finite control. Then the heads of all three tapes are moved one cell to the right.

What if either of the s_{1} or s_{2} is blank, in this case, consider it as 0 and perform the addition.

Go back to step 5.

Now if both s_{1} and s_{2} are blank:

Check the carry bit, if c = 1 then write 1 on tape 3.

Halt.

Frequently Asked Questions

What is Multitape Turing Machine?

Multitape Turing Machine is one of the variants of the Turing machine that have many tapes, each with its head, and each can move independently of the other heads.

What is multidimensional Turing machine?

In a multidimensional turing machine, the header can move up and down and in the left or right direction. Whereas in a normal Turing machine, the header can only move in Left or Right. The multidimensional turing machine is defined by 7 tuples.

Are multi tape and multi-track Turing machine same?

A multitape Turing machine has many tapes, and each one of them has its own Read/Write head that moves independently of the others, whereas a multi-track Turing machine has multiple tracks where only one tape will perform read/write on all tracks.

How does a multi-tape Turing machine work?

A multitape Turing machine follows certain steps to get the output symbol. Initially, tape 1 contains the input and the others are blank; after this, the machine reads the consecutive symbols on its way under its heads, and the Turing machine prints the symbol on each tape and moves further its heads.

What are tapes in Turing machine?

A tape in a Turing machine is of infinite length on which Read/Write operations are performed. It has infinite cells where each cells contain input or special blank symbol.

What is the transition function of a multitape Turing machine?

A multitape Turing machine's transition function specifies how the machine transitions between states based on the current state, the symbols read from each tape, and the direction to move the tapes.

What are the three types of Turing machines?

The three types of Turing machines are single-tape Turing machines, multitape Turing machines, and nondeterministic Turing machines. Each type varies in the number of tapes and the computational model's complexity.

Conclusion

In this article, we have discussed the Multitape Turing Machine. The multitape Turing machine represents a significant advancement in computational theory, offering enhanced computational power compared to single-tape machines. Its ability to utilize multiple tapes and transition between states efficiently opens up new possibilities for solving complex computational problems.