Question

Design a Turing machine that places 0 s and \(1 \mathrm{~s}\) in all the cells to the right of the current cell until it reaches a cell containing a plus symbol.

Short answer

Design uses three states: Start (q0), Fill (q1), and Halt (q2), filling cells with '0's and '1's until encountering '+'. Halt at '+'. State transitions alternate between writing '0' and '1' in q1.

Step-by-step solution

Understand the Input

The Turing Machine starts at a certain cell on the tape, which contains a symbol that is neither '0' nor '1'. The task is to fill subsequent cells with '0's and '1's until encountering a cell with a plus symbol ('+').

Define the States

Create necessary states for the task: Start state (q0), Fill state (q1), and Halt state (q2). The Turing machine transitions to the Fill state to start writing '0's and '1's.

Transition Rules for Filling Zeros and Ones

In state q0, upon encountering any symbol (except '+'), move to state q1 and start writing either '0' or '1'. In state q1, alternate writing '0' and '1' for each move to the right across blank spaces.

Stop at Plus Symbol

When the machine in state q1 reads a '+', it transitions to the Halt state q2, stopping further operations. This halting ensures the machine ceases to write when reaching a plus symbol.

Example Execution

If the machine starts at a cell with an 'a' followed by blanks, you'd transition from q0 (reading 'a', writing '0'/no change) to q1 (write '0', move right, state q1, then write '1', move right, state q1), continuing to alternate until reaching '+', upon which it switches to state q2 and halts.

Key concepts

State Transition

The concept of state transition is essential in understanding how a Turing machine operates. A state in a Turing machine is akin to a specific mode of operation, determining what the machine should do next based on the current input.
Different states are designed for different tasks:

• The **Start State** (often labeled as \(q_0\)) is where the machine begins its process. It defines the initial instructions.
• The **Fill State** (such as \(q_1\)) involves performing actions like writing on the tape and moving the tape head. It's operational, meaning it keeps the machine going until a specific condition is met.
• The **Halt State** (denoted as \(q_2\)) is a terminal state where the machine stops its processing. It signifies completion of the task.

The transitions follow a set of rules based on the symbol currently under the tape head. For example, if in state \(q_0\), the transition rule might be to move to \(q_1\) and start writing if a blank space is found.
These rules are critical as they ensure the machine performs the desired operations precisely, seamlessly moving from one state to another.

Tape Operations

Tape operations in a Turing machine are fundamental, as they define how data is read and written during the computation process. Imagine the tape as an infinite strip of cells, where each cell can hold one symbol at a time.
The operations include:

• **Reading**: The machine reads the current symbol where the tape head is positioned. This action guides subsequent operations.
• **Writing**: When instructed by the state transition rules, the machine can write a new symbol onto the current cell. In our exercise, it writes '0' or '1'.
• **Moving**: After reading or writing, the tape head moves either left or right to the adjacent cell, based on instructions.

Tape operations are executed repeatedly. For example, the Turing machine may start by writing a '0', moving right, then writing a '1' in the next cell, continuing this alternate pattern.
The capability of altering what’s on the tape while moving across it allows Turing machines to solve complex problems by performing a series of simple, clear steps, one at a time.

Halt State

The halt state is a crucial aspect of Turing machines, marking the endpoint of the computation. Reaching this state means the Turing machine has executed its instructions and there are no further actions to perform.
Implementing a halt state involves precise definition within the state transition rules:

• It is usually denoted by a special state, like \(q_2\), which has no further transition rules beyond its activation condition.
• The machine transitions to the halt state when it encounters certain symbols or conditions, like the plus symbol ('+') in our exercise.

Once in the halt state, the Turing machine stops all operations, ensuring no further changes occur on the tape. This predictability is vital, as it provides a clear signal that the task is complete.
By having such a decisive conclusion point, the halt state prevents the machine from running indefinitely, enabling it to solve problems efficiently and effectively.