Question

Write a Turing machine to perform a unary decrement (the opposite of an increment). Assume that \(n>0\).

Short answer

The Turing machine reduces the input by one by erasing the first '1'.

Step-by-step solution

Define the Problem

In unary, a number \( n \) is represented by a sequence of \( n \) 1's. To perform a unary decrement, we need to remove one '1' from this sequence, effectively turning \( n \) into \( n-1 \). This assumes that \( n > 0 \).

Initialize the Turing Machine

Imagine a Turing machine tape with a sequence of 1s representing the number \( n \) (e.g., 111 for 3). Position the tape head over the leftmost '1'. Our task is to build a state machine that will remove one '1' from this sequence.

Define the States and Transition Functions

1. **State Q0:** The initial state, where we look for the first '1'. - If '1' is encountered, change state to Q1. 2. **State Q1:** Perform deletion of the first '1'. - Change '1' to 'B' (blank) and transition to the halt state because the decrement is complete.

Write the Transition Rules

- In state Q0, upon reading '1', write 'B', stay on the same position (as writing 'B' already deletes the '1'), and move to the halt state (H). This completes the decrement operation.

Test the Turing Machine (Example)

Consider the unary sequence '111' representing the number 3 on the tape: - Start in state Q0 on the first '1'. - Read '1', write 'B', and transition to halt. The tape now reads '11', representing the number 2. - The machine successfully decrements 3 to 2.

Key concepts

Unary Decrement

A unary decrement operation involves reducing a unary number by one unit. In unary representation, a number is shown as a sequence of repeated '1's. For example, the number 3 is represented as '111'.
To perform a unary decrement, one simply needs to remove a single '1' from this sequence. If we take '111' (which represents 3) and remove one '1', the resulting sequence is '11', which represents the number 2.
This task sounds simple for humans, but for a machine, especially a Turing machine, it requires precise instructions and steps to get right without errors.

State Machine

A state machine is a crucial concept in computing and it forms the backbone of how a Turing machine operates. It consists of various states which the machine can be in at any time, and based on the input, it transitions through these states to perform computations.
In the context of a unary decrement, our Turing machine follows a basic state machine model. We start with an initial state, move through various states based on input conditions, and end at a final halt state when the computation is complete. This means every input and action is defined by the machine's current state.

• State Q0: This is the starting state where the machine begins to analyze the input sequence.
• State Q1: Upon finding the first '1', the machine changes this state to delete that character.

The state machine will repeat these actions until the decrement operation is complete, finally halting when done.

Transition Functions

Transition functions define how a Turing machine moves from one state to another based on the inputs it reads. These functions are critical to executing tasks as they dictate how the machine reacts upon reading specific characters from the tape.
For our unary decrement task, transition functions work as follows:

• In State Q0, when the machine reads '1', it updates that position to 'B' (representing a blank), and instructs the machine to move to State Q1.
• In State Q1, the machine does not perform further operations as deleting a single '1' accomplishes the task, and it directs the machine to halt.

These precise function rules ensure that the machine manipulates the input in a controlled manner to achieve the desired decrement.

Tape Head

The tape head is an integral component of a Turing machine. It operates as the pointer that reads and writes data on the tape. Essentially, it is the active part of a Turing machine that interacts with the tape's content.
In unary decrement, the tape head performs critical tasks:

• Starts at the leftmost '1', which symbolizes the beginning of the number on the tape.
• Reads the current symbol on the tape, deciding based on the state machine whether to write a new symbol (like 'B') or move.

This process of reading and writing through the tape head is what allows the Turing machine to decrement the unary sequence correctly. Without precise movement of the tape head, the machine would not be able to execute the unary decrement effectively.