Question

Write a Turing machine that operates on any string of \(1 \mathrm{~s}\) and changes it to a string of alternating \(1 \mathrm{~s}\) and \(0 \mathrm{~s}\).

Short answer

Define states to alternate writing '1' and '0'; simulate for input '111'. Finalize when encountering blank.

Step-by-step solution

Understanding the Task

The task is to create a Turing machine that takes an input string consisting only of the symbol '1' and modifies it into a string that alternates between '1' and '0'. For example, the input '111' should become '101' or '010' based on design.

Define the Turing Machine Components

To develop our Turing machine, we need to define its components: states, tape alphabet, and transitions. - **States**: We will use at least two states, one to write '1' and another to write '0'. - **Tape Symbols**: Include '1', '0', and a blank symbol (B) for the tape. - **Start State**: The initial state where the machine begins processing. - **Accept/Reject States**: States indicating the end of the process.

Establish Initial Turing Machine Design

Define an initial design where the machine reads the first '1', writes a '1', moves right, and transitions to a state to write '0'. Upon writing '0', it moves right and returns to the '1' writing state. Do this repeatedly for the length of the string.

Define State Transitions

- **Start State (q0)**: Start reading. If '1', write '1', move right, and switch to state q1. - **State q1**: Writes '0', moves right, and switches back to q0. - **Finish**: When a blank is encountered, the transition to the accept state (halt) is triggered. The transition function can be illustrated as: - For q0: Read '1', Write '1', Move right, Go to q1. - For q1: Read '1', Write '0', Move right, Return to q0.

Validate Turing Machine Process

Simulate the machine using an example input, such as "1111". 1. **Initial (q0)**: Read '1', Write '1', move right. 2. **Switch to q1**: Read next '1', Write '0', move right. 3. **Return to q0**: Repeat the above until the end of string. 4. **Halt**: When the blank is encountered, the machine halts.

Key concepts

State Transitions

In a Turing machine, state transitions are crucial for dictating how the machine behaves at each step. A state is essentially a status that defines what to do next as the machine reads from the tape.

For this exercise, we have defined several states:

• Start State (q0): This is where the machine begins. It reads a '1', writes the same '1', moves right, and transitions to the next state, q1.
• State q1: In this state, the machine writes a '0' after reading '1', moves right again, and transitions back to q0.
• Accept State: This state is reached when a blank symbol is read. It halts the machine's operation, signifying the completion of the task.

Transitions are essentially rules that guide the machine from one state to another based on the current symbol being read. They ensure that each symbol is processed in the desired sequence, alternating between '1' and '0'.

Tape Symbols

Tape symbols are the characters that the Turing machine reads and writes on its tape. The tape and its symbols are foundational to the operation of the machine, as they store the input and output data while the machine processes the transitions.

In this context, we work with the following tape symbols:

• '1': The symbol from the initial input string. Each '1' is transformed to either remain a '1' or become a '0'.
• '0': The result of rewriting some '1' symbols as the Turing machine progresses in its operations.
• Blank (B): These indicate parts of the tape that are not used by the current input. The presence of a blank will eventually trigger the transition to the halt state.

Understanding the role of each symbol is key to designing the rules and transitions necessary to accomplish the task. The symbols dictate the logic that the machine follows to know which parts of the tape are modified and which parts signal completion.

Write and Move Operations

In a Turing machine, write and move operations modify the tape content and guide the machine's head along the tape, respectively. Together, they are central to transforming the input according to predefined rules.

Write Operations:
These involve changing a tape symbol at the machine's current position. In this exercise:

• Initially, if a '1' is read in state q0, the machine writes '1' and transitions to q1.
• In state q1, on reading '1', the machine writes '0'. This alternation continues until the entire input is processed.

Move Operations:
These operations determine in which direction the tape head moves after writing a symbol:

• The move right operation (\( R \)) moves the tape head to the next symbol on the tape.
• This sequence allows the machine to scan through the string systematically, executing the alternating sequence of writing '1' and '0'.

The combination of write and move operations allows the Turing machine to control the sequence in which it processes the string, ensuring that the resulting output alternates as required.