Question

Write a Turing machine that operates on any binary string and changes it to a string of the same length with all \(1 \mathrm{~s}\). It should, for example, change the tape to $$ \ldots b 011010 b \ldots $$ to \(\ldots b 111111 b \ldots\) However, you must write instructions that allow your Turing machine to work on any binary string, not just the one shown here.

Short answer

A Turing machine will change all 0s to 1s by defining four transition rules.

Step-by-step solution

Define the Turing Machine States

Identify the necessary states for the Turing machine. The machine will need at least two states: an initial state (\(q_0\)) and a halting state (\(q_h\)). The job is to read each bit and change \(0\) to \(1\), but \(1\) stays as it is.

Set Up Transition Functions

Define the transition functions for the Turing machine. For every \(0\) encountered by the tape head, change it to \(1\), move the head to the right, and remain in the initial state. If \(1\) is encountered, move the head to the right without changing the state. Once a blank (\(b\)) is encountered, transition to the halting state.

Write Transition Rules

Create the explicit rules based on the transitions:1. From \(q_0\), if the current symbol is \(0\), write \(1\), move right, remain in \(q_0\).2. From \(q_0\), if the current symbol is \(1\), move right, remain in \(q_0\).3. From \(q_0\), if the current symbol is \(b\), move to \(q_h\).

Example Execution

Walk through an example using the defined rules.- Start with state \(q_0\) and the tape containing \(b011010b\).- **q0, 0**: change to **1**: move right.- **q0, 1**: move right.- Repeat steps for each symbol until the blank is reached initiating a transition to \(q_h\).

Analyze for Completeness and Errors

Make sure the machine handles all possible inputs (any combination of binary strings) according to the rules defined and stops processing once a blank is encountered.

Key concepts

binary strings

In the context of Turing machines, binary strings are sequences that consist solely of the numbers 0 and 1. These strings are the simplest form of data representation used in computing and digital electronics. A Turing machine operates by reading and writing symbols on a tape, with binary strings being common inputs due to their straightforward, two-symbol nature.

In our exercise, the Turing machine processes any arbitrary binary string by converting each 0 in the string to a 1, while leaving the 1s unchanged. This transformation is crucial because it demonstrates the machine's capacity to perform systematic operations on binary data, highlighting the fundamental characteristics of Turing machines. By only interacting with binary strings, Turing machines can simulate any algorithmic process, making them a cornerstone of computer science theory.

This capability to manipulate binary strings serves as a basis for more complex data processing and computation models. Such operations on binary strings are foundational for understanding how computers interpret input data and produce output efficiently.

state transitions

State transitions in a Turing machine refer to changes in the machine's configuration as it processes symbols on the tape. Each state transition is defined by specific rules determining what the machine should do when encountering a particular symbol in a given state.

In the Turing machine designed for this exercise, we see state transitions occurring in these ways:

• If the machine is in the initial state (\(q_0\)) and reads a 0, it changes this to 1 and remains in state \(q_0\) while moving right.
• If the machine reads a 1 while in state \(q_0\), it simply moves right without altering the machine's current state.
• Upon reading a blank symbol (\(b\)), the machine transitions from state \(q_0\) to the halting state (\(q_h\)).

These transitions are crucial because they define the logical flow of the machine's operation, ensuring it processes the entire binary string before ceasing operation. State transitions make the Turing machine flexible, allowing it to handle various input configurations and perform desired tasks systematically.

halting state

The halting state in a Turing machine represents the end of its computation process. When the machine enters this state, it indicates that all operations are complete, and the machine stops running.

For the exercise, the halting state is reached when the tape head encounters a blank symbol (\(b\)) after processing the sequence of binary digits. This specific transition to the halting state is crucial because it ensures that the machine only stops functioning after processing every part of the input binary string.

Having a clear halting state is fundamental to determining the completion of a computation task by the Turing machine. It assures that the process is not left in an indefinite loop and that all operations are performed correctly. The halting state thus acts as a safeguard, confirming for programmers and engineers alike that their algorithms have effectively executed and can cease action without errors.

transition functions

Transition functions in a Turing machine define the rules that determine the machine's behavior during its operation. They specify what the machine does when it is in a particular state and reads a specific symbol on the tape.

For the task in question, the transition functions can be concisely represented by rules like:

• If in state \(q_0\) and the symbol is 0, write 1, move right, remain in \(q_0\).
• If in state \(q_0\) and the symbol is 1, move right, remain in \(q_0\).
• If the symbol is a blank (\(b\)), transition to the halting state \(q_h\).

These transition functions are pivotal because they outline the logic necessary for the Turing machine to execute its task, step by step. They ensure every possible scenario is covered, and the correct actions are taken given any symbol's presence. In doing so, transition functions are the backbone of the Turing machine's logic, meticulously guiding it through each input it processes. Understanding these functions is essential for developing efficient algorithms and learning how complex computational operations are designed.