Design a Turing Machine with
One infinitely long tape containing 0, 1, or blank The minimal number of states in its transition An input number written on the tape initially
that computes the function of
f(n)=n×4
for any natural number n.
You are expected to submit
A definition of the encoding of numbers with a brief justification
The transition table/diagram for the Turing Machine
A sample input/output of your function on the tape
To design a Turing Machine that computes the function of f(n) = n x 4 for any natural number n, let's break it down into the following steps:
1. Encoding of numbers:
We can use a unary encoding system to represent numbers on the tape. In this system, the number n is represented by n consecutive '1' symbols, followed by a '0' symbol to separate the encoded number. For example, the number 3 would be represented as "1110" and the number 4 would be represented as "11110". This encoding ensures that we can easily count and manipulate the numbers using the Turing Machine.
2. Transition table/diagram:
The Turing Machine will have a Finite Control Unit (FCU) with states and a tape with symbols. The transition table/diagram specifies the behavior of the Turing Machine as it reads symbols from the tape and moves between states. Here's the transition table for our Turing Machine:
Current State Current Symbol New Symbol Move New State
q0 1 1 R q0
q0 0 0 R q1
q0 B B L q5 (to signify the end of input)
q1 1 1 R q1
q1 0 0 R q2
q2 1 1 R q3
q2 0 0 R q4
q3 1 1 R q3
q3 0 0 R q4
q4 1 1 R q4
q4 0 0 R q4
q5 1 0 L q5
q5 0 0 L q5
q5 B B R q6 (halt state)
In this transition table, 'B' represents a blank symbol on the tape. Each row represents a possible combination of current state and current symbol, and specifies the new symbol to write, the direction to move the tape (R for right and L for left), and the new state to transition to.
3. Sample input/output:
Let's take an example where the input number n is 3. In unary encoding, 3 is represented as "1110". We start the Turing Machine from the initial state q0 and read the symbols from the tape:
q0 1110
- In state q0, we read '1', write '1', move right, and transition to state q0.
- In state q0, we read '1', write '1', move right, and transition to state q0.
- In state q0, we read '1', write '1', move right, and transition to state q0.
- In state q0, we read '0', write '0', move right, and transition to state q1.
- In state q1, we read 'B' (blank symbol), write 'B', move left, and transition to state q5.
At this point, the Turing Machine has multiplied the unary encoded number 3 by 4, and the final tape configuration would be:
q5 BB0000
The output can be read as '0000' in unary encoding, which represents the number 12. Therefore, the Turing Machine has successfully computed the function f(3) = 3 x 4 = 12.
Note: This is a simplified example, and the actual implementation of the Turing Machine may require additional states and transitions to handle all possible inputs.