Question

Write a Turing machine that operates on the unary representation of any number and decides whether the number is 0 ; your machine should produce an output tape containing the unary representation of 1 if the number was 0 and the unary representation of 2 if the number was not 0 .

Short answer

The Turing machine writes "1" if the input is 0 and "11" if the input is not 0.

Step-by-step solution

Understanding Unary Representation

In unary representation, a number n is represented by n consecutive 1s. For example, the number 3 is represented as '111'. The number 0 is represented by an empty string, denoted as ' '.

Define Turing Machine States

We need a Turing machine with a starting state, say q0, a state to identify when the input is 0 (let's call it q_accept), and a state to identify when the input is not 0 (q_reject). There should also be a final state, q_final, for writing the output.

Transition for Input 0

In the state q0, check if the tape is blank (representing 0). If so, transition to q_accept, write '1' to the tape, and move to the q_final state.

Transition for Input Not 0

In the state q0, if there is a '1' on the tape, transition to q_reject. Replace the first '1' with '*', and move one cell to the right to ensure there are more '1's. Then, transition to a state that writes two '1's on the blank part of the tape, achieving a unary representation of 2, then halt in q_final.

Finalizing the Output

For both final states (q_accept for input 0 and q_final reached after q_reject for input not 0), make sure the output is left as "1" if the input was 0 or "11" if the input was not 0. The machine does not continue further from q_final.

Key concepts

Unary Representation in Turing Machine

A unary representation is one of the simplest ways to represent numbers, especially suitable for Turing machine exercises. In unary, a number is depicted by a series of consecutive symbols – typically 1s. For example:

• Number 1 is represented as '1'
• Number 2 is represented as '11'
• Number 3 is represented as '111'
• Number 0 is uniquely represented by an absence of symbols, an empty string: ''

This simplicity makes it straightforward for a Turing machine to decode and operate on since each increment is a uniform addition of a symbol. It's a linear representation, meaning each count doesn't involve complex base shifts, making it ideal for illustrating basic decision processes like zero-checking.

State Transitions of a Turing Machine

A Turing machine's operation is governed by its states, which dictate how it reads, processes, and writes information on a tape. For the discussed problem, the states are crucial:

• **q0 (Start State):** Where the machine initializes and inspects the current tape position.
• **q_accept:** If the input number is 0 (empty), the machine transitions here to write '1' as output.
• **q_reject:** If there's a '1' at the start, the indication that the number is not zero. Here, it redirects to process extending the input.
• **q_final:** The wrap-up state where the machine stops processing and halts.

State transitions act as decision points based on the input being read at any position on the tape, directing the flow of operations.

Tape Operations in Turing Machines

The tape in a Turing machine plays the role of both input and output storages. It functions similarly to memory, storing symbols that the machine reads and modifies. Specific operations addressed in the solution include:

• **Reading and Writing:** The tape head can read the current symbol (e.g., '1' or '') and overwrite it if necessary.
• **Moving Left or Right:** The head can shift one position at a time, either left ⟵ or right ⟶, facilitating navigation and inspection of the tape.
• **Erasing and Rewriting:** Particularly seen when transforming a '1' to '*' during processing, indicating an inspected or transformed unit.

Through these operations, a Turing machine can embody logical processes and calculate outputs effectively.

Output Computation in Turing Machines

Once a Turing machine completes processing, its final task is generating an output on the tape, representing its decision. For this exercise, the output computation is straightforward:

• **Unary '1' Output:** If the input was 0, the output becomes '1', indicating an acknowledgment of the initial empty input.
• **Unary '11' Output:** If the input was one or more '1's, means a non-zero, reflecting an incremented interpretation on the tape.

Each final state dictates how these symbols are arranged, with q_accept appending a single '1' and q_final post-processing appending '11'. This output stage ensures the machine fulfills its intended function of differentiation and produces a result usable for verification.