Question

Write a Turing machine that efficiently adds 3 to any unary number.

Short answer

The Turing machine adds three '1's to the unary number by transitioning through states designated for adding unary digits.

Step-by-step solution

Understand Unary Representation

In unary representation, a number is represented by a series of the same symbols; for example, the number 3 is represented as '111'. Each '1' represents a count of one. Thus, to add 3 to a unary number means adding three more '1's to it.

Define Turing Machine States

We will define our Turing machine states: q0 will be the initial state, q1 will be the state where we read the end of the number, and q2 will be the state where we write the additional '1's. The halt state will be qh.

Reading the Unary Number

Start at q0. As long as the tape head reads '1', move right. This indicates moving over the unary encoded number to find the end of the input. When you encounter a blank, transition to state q1.

Transition to Writing Stage

At q1, write '1' to add the first '1' beyond the original number, then move the tape head to the right. Transition to state q2 for adding additional '1's.

Add Additional Ones

In state q2, write '1', move right two times to position the head correctly, and repeat this step two more times to add the remaining '1's. Each write transitions to the next step until we've added all three '1's.

Halt the Machine

Once all three '1's are added, transition to the halt state qh. This will stop the Turing machine, leaving the tape with 3 additional '1's appended to the initial unary input.

Key concepts

Unary Representation

Unary representation is one of the simplest forms of numerical representation. It uses the repetition of a single symbol to denote quantities. In the case of unary, this symbol is usually '1'. For example, the number 5 is expressed as '11111', where each '1' represents a single unit. This approach, although not space-efficient for very large numbers, simplifies arithmetic operations like addition as it involves only counting and concatenating symbols.

In unary addition, to add two numbers, you simply concatenate their unary representations. For instance, adding two (represented as '11') and three (represented as '111') will result in five (represented as '11111'). Similarly, when a Turing machine is tasked with "adding 3", it adds three instances of '1' to the unary input.

Understanding unary representation is crucial for programming simple algorithms in theoretical computing models like Turing machines. It serves as a fundamental example to illustrate how computations can be performed mechanically.

State Transitions

State transitions in a Turing machine define how the machine moves between different states based on the current tape symbol. This is similar to a step-by-step set of instructions on how to proceed with the reading and writing processes.

In this exercise, the Turing machine starts at an initial state, commonly labeled as 'q0'. It reads through the unary encoded number on the tape, continuing right until it encounters a blank. At this point, it transitions to another state, 'q1', signaling the end of the input number.

This design illustrates the principle of a state transition diagram where each movement and action taken by the Turing machine, such as move left, move right, write, and read, is directed by the current state and tape symbol. The subsequent state, 'q2', is where additional conduct, like writing more '1's, takes place. Thus, state transitions effectively dictate the machine's operation and culminate in the halt state, 'qh', where the machine stops after completing its task.

Algorithm Development

Algorithm development in the context of Turing machines involves designing a sequence of well-defined operations. The goal is to solve a specific problem or perform a particular computation task.

For the problem of adding 3 to a unary number, the algorithm starts by defining clear states and transitions. First, the machine needs to recognize the end of the user’s input unary number as described previously. Once recognized, the machine proceeds to append three additional '1's to the number.

Key concepts of algorithm development include:

• Defining tasks: Identify what the machine must do, such as adding '1's.
• State definition: Set the primary states like q0, q1, and q2.
• Transition mapping: Specify transitions based on tape symbol inputs.
• Execution checking: Ensure the machine halts appropriately after completing its task.

Strong and efficient algorithm design is essential for leveraging a Turing machine's capabilities to simulate any computational process. Despite the simplicity of this unary addition task, it provides a fundamental insight into how complex computational algorithms can be structured and understood through Turing machines.