Question

Draw a state diagram for a Turing machine that takes any string of \(1 \mathrm{~s}\) and changes every third 1 to a 0 . Thus, for example, \(\ldots b 111111 b \ldots\) becomes \(\ldots b 110110 b \ldots\)

Short answer

Construct a Turing machine with states cycling between three roles, changing every third '1' encountered on the tape into a '0' while reading from left to right until a blank is reached.

Step-by-step solution

Define the Turing Machine States

Identify states of the Turing machine. Let's create three main states: \(q_0\), \(q_1\), and \(q_2\). Additional states may be added as needed, such as a halting state \(q_h\). Initially, the machine starts in state \(q_0\).

Define Transition for the First State

In state \(q_0\), when the machine reads a '1', it will transition to state \(q_1\) while leaving the '1' unchanged and moving the tape head right. If it reads a blank (\(b\)), it will halt in state \(q_h\).

Define Transition for the Second State

In state \(q_1\), when a '1' is read, the machine transitions to state \(q_2\), again leaving the tape unchanged and moving to the right. Any blank encountered will transition the machine into the halt state \(q_h\).

Change the Third '1' to '0'

When the Turing machine is in state \(q_2\) and reads a '1', it changes this '1' to '0', transitions back to state \(q_0\), and moves the head to the right. This marks every third '1' that has been encountered. Any blank encountered will also transition the machine to state \(q_h\).

Final State Diagram

Now the Turing machine has a looping pattern: \(q_0\) -> \(q_1\) -> \(q_2\) -> modifications -> back to \(q_0\). This pattern ensures every third '1' is changed to a '0' until the entire string is processed or a blank is read.

Key concepts

State Diagram

To understand how a Turing machine works, one needs to look at its state diagram. A state diagram visually represents a Turing machine's operation. Each node, or state, in the diagram corresponds to a condition of the machine. For example, in our Turing machine that flips every third '1' in a sequence, the states might include:

• Initial State (q₀): where the machine begins.
• Intermediate States (q₁ and q₂): that keep track of how many '1's have been encountered.
• Halting State (q_h): indicating the machine has finished processing the string.

The arrows between states show transitions, determined by the input on the tape. These diagrams are vital as they outline the logic the Turing machine follows to solve a problem, providing a clear blueprint of all possible state transitions depending on input characters.

Transition State

Turing machines transition between states based on the input read. This movement from one state to another is called a "transition state." Let's break this down using the example of altering every third '1' to '0':

• In state q₀, if a '1' is read, the machine does not change the input but shifts to state q₁ and moves right.
• Entering q₁, if another '1' is encountered, the state changes to q₂, again moving right.
• At q₂, the machine changes the third '1' to '0', transitions back to q₀, and proceeds right.

If a blank is encountered at any point, the machine will move to the halting state, stopping any further transitions. This transition system allows the machine to maintain a consistent check on its actions and inputs, ensuring it correctly patterns actions based on rules encoded in the state transitions.

Halting State

The halting state of a Turing machine is a crucial component as it signifies the machine's completion of its task. When a Turing machine enters this state, it stops processing the input tape entirely.
In the context of flipping every third '1' to '0', the halting state, often denoted as q_h, is reached when:

• The tape reads a blank (b) in state q₀, q₁, or q₂. This blank indicates the end of the string being processed.

The presence of a halting state is vital for determining when a task is complete. Without it, the Turing machine would continue indefinitely. Therefore, ensuring clear transition definitions towards this state is essential when designing a Turing machine to solve specific tasks or problems.