Example of Turing Machine

An example of a Turing machine can be explained using 7-tuples (Q, T, B, ∑, δ, q0, F), which show the mechanism of the Turning machine. Following are the 7-Tuples of the Turing machine.

  • Q contains a finite set of states.
  • T is the input tape that holds the Input and Special symbols. So, Both input and Special symbols are subsets of input tape T.
  • B is a Special symbol, also called a Blank symbol (every cell is filled with a special symbol except the input alphabet initially)
  • is the input symbols (input tape holds initial input alphabet)
  • δ is a transition function is Q × T → Q × T × {L,R}. At any state (Q) when the
    • If the input tape symbol (T) is applied, it will move to a new state, update the tape symbol (may or may not change), and move the head pointer to either right or left.
  • q0 is the initial state.
  • F is the set of final states. It is considered accepted when the input string reaches a state of F.

Example of Turing Machine

Let us construct a Turing machine for simple language L= “010”. The Input tape for language “010” will become as follows.

Turing Machine with Example - Input Tape

Special symbols are inserted after entering the language “010” into the input tape. This language is considered valid upon the arrival of the first special symbol.

Consider,

  • Q = {q0,q1,q2,q3} where q0 is initial state.
  • T = {0,1, X, Y,$} where $ represents blank.
  • ∑ = {0,1}
  • F = {q3}
Important: Symbols “X and Y” In input Tape are used to Update values of Sigma input ∑ (0,1). It is just identification that tells that the input symbols are read successfully.

(i.e., when reading Sigma input “0”, it should update with “X” in the input tap, and “1” should use “Y”).

 “0” for First Input Symol

  • Transition goes to state (q1) from initial state (q0)
  • The input tape’s first symbol, “0”, is updated to “X”.
  • Read/Write Head Goes to One Position Right.

Example of Turing Machine - Step 1

“1” for Second Input 

  • Transition goes to state (q2) from state (q1)
  • The input tape’s second symbol, “1”, is updated to “Y”.
  • Read/Write Head Goes to One Position Right.

Turing Machine with Example - Step 02

“0” for Third Input 

  • Transition goes to state (q3) from state (q2)
  • The third symbol on the input tape, which is currently “0,” is being changed to “X.”
  • Read/Write Head Goes to One Position Right.

Example of Turing Machine- Step 03

“$” for First Special Symbol 

  • The transition goes to state (q4) from state (q3), which is the accepting state.
  • The special symbol “$” on the input tape remains unchanged.
  • Read/Write Head Goes to One Position Right.

Example of Turing Machine - Step 04

Other than the “010” language, all languages are rejected. All possible paths for each input (0,1 and $) at each state are given below.

Turing Machine with Example - Step 05

Example of Turing Machine for Regular Languages

Let’s explain some examples of turning machines for regular languages

Important: We can design a Turing machine for all regular expressions where the R/W head always moves towards the right. However, we can design a Turing machine for some regular languages where the R/W head moves in both directions, either left or right.

The R/W head always moves towards the right until it reaches the final state, after which it stops moving.

Turing Machine Example 01:

Construct a TM for the language L = {0n} where n≥1

Solution

The following is the Regular expression for the given example

0(0)*

The following are the all possible strings created from L = {0n}

L= {0, 00, 000, 0000,…….}

Let’s suppose a string of “0000”. R/W head movement for input tap is given below

Turing Machine with Examples 1 (Head )

Finally, the turning machine to accept all possible strings over language L = {0n} where n≥1 is given below

Turing Machine with Examples 1 (Machine)

Turing Machine Example 02:

Construct a TM for the language L = {02n+1} where n≥0

Solution

The following is the Regular expression for the given example

0(00)*

The following are the all possible strings created from L = {02n+1}

L= {0, 000, 00000, 0000000,…….}

Let’s suppose a string of “000”. R/W head movement for input tap is given below

Turing Machine with Examples 2 (Head)

Finally, the turning machine to accept all possible strings over language L = {02n+1} where n≥0 is given below

Turing Machine with Examples 2 (Machine)

Turing Machine Example 03:

Construct a Turing machine that accepts all strings that start with 0.

Solution

The following is the Regular expression for the given example

0(0+1)*

The following are all the possible strings created

L= {0, 00, 01, 000, 011, …….}

Let’s suppose a string of “011”. R/W head movement for input tap is given below

Turing Machine with Examples 3 (Head)

Finally, the turning machine to accept all possible strings starts with “0” 

Turing Machine with Examples 3 (Machine)

Turing Machine Example 04:

Construct a Turing machine that accepts all strings that start with 01.

Solution

The following is the Regular expression for the given example

01(0+1)*

The following are all the possible strings created

L= {01, 010, 011, 0100, 0111, …….}

Finally, the turning machine to accept all possible strings starts with “01” and is

Turing Machine with Examples 4 (Machine)