Question

Write a Turing machine to perform a unary decrement. Assume that \(n\) may be 0 , in which case a single 0 should be output on the tape to signify that the operation results in a negative number.

Short answer

The Turing machine decrements a unary number or writes '0' if the input is 0 to indicate negative.

Step-by-step solution

Understand the Problem

A unary number system uses a sequence of 1s to represent numbers, e.g., 3 is represented as '111'. When we perform a unary decrement, we reduce the number by one, represented by removing one '1'. If the number is 0, represented by a blank or a different symbol, we need to output a single '0' to indicate a negative number.

Set Up the Turing Machine States

Design the Turing machine with several states. We'll use state Q1 for identifying the first '1' in non-zero input, Q2 for replacing '1' with a blank (effectively decrements the number), and Q3 as the halt state for non-zero inputs. If no '1' is found, we'll transition to a state where '0' is written and halt.

Define Transitions for Non-zero Inputs

From the initial state (Q0), if the head reads '1', move to state Q1, write a blank, then to Q2, where the machine moves left and checks if there are more '1's. If there's another '1', it remains in Q2, if not, transitions to Q3 and halts.

Handle Zero Input Case

From Q0, if the head reads a blank (indicating zero), move to a distinct state (Q4), where '0' is written, and immediately move to halt state indicating the negative result.

Test the Turing Machine

Run tests: (1) Input of '111', move through Q0 to Q1 to Q2 giving '11' and halting at Q3, (2) Input of blank, transitioning from Q0 to Q4 and writing '0', halting to signify the operation resulted in negative.

Key concepts

Unary Number System

The unary number system is a simple mathematical system where numbers are represented by repeated instances of a single digit or symbol. In the unary system, numbers are often expressed using the digit '1', repeated as necessary to represent a given magnitude. For example, the number three would be expressed as '111'.
This method of representation is quite different from more familiar systems, such as the decimal or binary systems, because each numeral represents a singular unit rather than a power of a base.
The unary system is advantageous for its simplicity and is used in theoretical computer science, specifically with systems like Turing machines, where operations like increments and decrements can be visually and functionally intuitive.
When translating the concept of unary numbers into a Turing machine, these systems offer a straightforward way to conceptualize numbers, which can be beneficial for certain computational processes.

Turing Machine States

Turing machines are theoretical devices that manipulate symbols on a strip of tape according to a set of rules. The different conditions a Turing machine can be in during computation are referred to as states.
These states guide the machine's actions based on the current input. For instance:

• **Q0** is often the initial state, indicating where the Turing machine starts processing.
• **Q1, Q2, ...** are intermediary states used for various steps of an operation, like identifying inputs or modifying them.
• **Q3** could symbolize a halt state, meaning the process is complete.

The states ensure that a Turing machine performs operations accurately and transitions correctly between processes.
Each state plays a specific role in processing or halting based on certain conditions and defined transitions. Designing these states correctly is essential in achieving the desired computation for tasks like unary decrements.

Unary Decrement

A unary decrement operation reduces the representation of a number by one in a unary system. Visually, this translates to removing a single '1' from the sequence of '1's that denote the number.
In simpler terms, if we decrement '111', the result would be '11'. This operation is straightforward in unary form because it involves eliminating a symbol rather than manipulating multiple, as in other numeric systems.
When programming a Turing machine for unary decrement, if the input number is zero (often indicated by a blank), the operation should recognize this and respond by writing a '0' to signify a negative result.
This handling of zero ensures that the machine's output is meaningful, maintaining system integrity even when the operation may conceptually result in a negative number.

Turing Machine Transitions

Transitions in a Turing machine dictate how it moves from one state to another. These are crucial because they define the machine's behavior based on the current symbol being read and the current state.
Each transition involves:

• Reading a symbol on the tape.
• Writing a new symbol in its place (or possibly leaving it unchanged).
• Moving the tape either left or right.
• Switching to a new state.

For a unary decrement operation, we setup transitions: starting from the initial state, checks occur as the machine reads symbols. If a '1' is found, it replaces it with a blank and transitions to another state to check for more '1's.
These ensure that the machine reduces the number effectively by removing '1's until the correct decrement is achieved. For a blank input indicating zero, the transitions direct the machine to write a '0', capturing the unary logic of a negative number. Effective transition design is essential for the Turing machine to perform tasks accurately and reliably.