Question

Write a Turing machine that takes as input the unary representation of any two different numbers, separated by a blank, and halts with the representation of the larger of the two numbers on the tape. (Hint: You may need to use a "marker" symbol such as \(X\) or \(Y\) to replace temporarily any input symbols you have already processed and do not want to process again; at the end, your program must "clean up" any marker symbols.)

Short answer

Implement a Turing machine to mark and compare unary numbers, outputting the larger one on the tape.

Step-by-step solution

Understand the Input and Output

We need to design a Turing machine that compares two numbers in unary format on a tape. The input is given in the form of two numbers separated by a blank space (e.g., '1110 1111' represents the numbers 3 and 4). The Turing machine should output the larger number in unary form, clearing any marker symbols used during its operation.

Initialize the Turing Machine

Start by setting the tape's initial state to the input state. The Turing machine reads the tape from left to right. We'll assume 'B' represents the blank separating the numbers. Mark the beginning of the first number with an 'X' upon reading each '1'. After processing the first number, move to the right past the blank space to begin processing the second number.

Compare Unary Numbers

While reading the second number, for each '1' encountered, if there is an 'X' available leftward from the current position, replace 'X' with '0' to mark they cancel each other (indicating both numbers have equal parts so far). If you attempt to cancel a '1' in the second number and find no 'X', it means the second number is larger.

Determine and Record the Larger Number

If all '1's in the second number can be matched with 'X's from the first number and there are no remaining '1's in the first set of markers, the two numbers are equal or the first number is larger. Otherwise, if any extra '1's remain in the second number unmarked by 'X', the second number is larger. Record accordingly by writing the larger number in unary on the tape starting from the first position, replacing all previous symbols.

Clean Up the Tape

Remove all marker symbols ('X' or '0') from the tape. This involves making multiple passes to clear any markers left from comparison, leaving only the correct unary number representing the larger input number.

Key concepts

Unary Representation

The unary representation is a way of expressing numbers using only one type of symbol, typically '1'. In unary, the number of '1's corresponds directly to the value of the integer. For example, the number 3 is written as '111', and the number 4 is '1111'. This is in contrast to binary or decimal, where multiple symbols represent numbers.
Unary representation is straightforward and often used in theoretical computer science, including Turing machines. It simplifies operations like addition and comparison, because you can determine the size of a number simply by counting the '1's.
A Turing machine using unary numbers can effectively compare sizes by looking at how many '1's there are in each segment. In particular, our exercise uses this representation to compare the first and second numbers entered on the tape.

Marker Symbols

Marker symbols are temporary symbols used within a Turing machine's operation to assist in processing the tape. These markers are critical for keeping track of changes and ensuring symbols are not reprocessed.
In our exercise, markers such as 'X' are used to mark '1's in the first number as they are processed. This helps manage comparisons by keeping a visual record of accounted symbols. 'X' markers avoid redundant comparisons and track equality between '1's in the first and second numbers.
Once the comparison task is complete, a cleanup operation removes these temporary symbols, ensuring only relevant numbers remain. Marking ensures efficiency and accuracy in the Turing machine's algorithms.

Tape Cleaning

Tape cleaning refers to the process of removing any temporary markings made during the Turing machine's operations. After markers like 'X' or '0' are used to assist in processing, tape cleaning ensures these markers do not remain, which would disrupt the final result.
For this specific machine, once the comparison is complete and the larger number is recorded, markers no longer serve a purpose. The cleanup step involves systematically replacing all temporary symbols with blank spaces, leaving only the desired representation of the larger number.
This process resembles resetting the tape to a state where it only holds the useful information without any excess symbols, ensuring the output is clean and readable.

Comparison Algorithm

The comparison algorithm in a Turing machine involves evaluating two unary numbers input on the tape. The process entails:

• Reading the first unary number and marking each '1' with an 'X'.
• Proceeding to the second number and for every '1' encountered, checking if there's a corresponding 'X' to match it, signifying equality at that stage.
• If a '1' in the second number cannot find an 'X', the second number is larger.

Upon completing this comparison, the algorithm records which number is greater. If '1's in both numbers match perfectly, they are equal. If any extra '1's remain unmarked in the second number, this indicates a larger value.
Ultimately, the goal is to rewrite the tape with only the unary number representing the larger input, after which the cleaning phase removes any temporary symbols.