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1. Create a Turing Machine that subtracts 1 from a binary number (remember that the first element in the tape is empty, and that we store binary numbers starting with the least significant bit).

1. Create a Turing Machine that subtracts 1 from a binary number (remember that the first element in the tape is empty, and that we store binary numbers starting with the least significant bit).

Computer Networking: A Top-Down Approach (7th Edition)
Computer Networking: A Top-Down Approach (7th Edition)
7th Edition
ISBN:9780133594140
Author:James Kurose, Keith Ross
Publisher:James Kurose, Keith Ross
Chapter1: Computer Networks And The Internet
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### Exercise 1: Creating a Turing Machine for Binary Subtraction **Objective:** Create a Turing Machine that subtracts 1 from a binary number. **Instructions:** - **Start Position:** The first element on the tape should be empty. - **Binary Number Storage:** Store binary numbers starting with the least significant bit (LSB). This exercise involves designing a Turing Machine, which is a theoretical device that can process and manipulate symbols on a strip of tape according to a set of rules. In this task, you will create a Turing Machine specifically for performing subtraction by 1 on binary numbers. **Example:** If the binary number is `1001` (representing the decimal number 9), the Turing Machine will transform it to `1000` (representing the decimal number 8). The following steps provide a high-level overview of creating such a machine: 1. **Initialize the Tape:** - Place an empty symbol (often a blank or `B`) at the first position on the tape. - Follow this with the binary digits of the number, starting with the least significant bit. For example, the decimal number 6 (in binary, `011`) would be placed as `B110`. 2. **Design the Transition Rules:** - The transition rules define the behavior of the Turing Machine—how it reads, writes, and moves the tape head. For subtraction, you will need rules to: - Identify the least significant bit. - Perform the subtraction operation. - Handle borrowing if necessary (similar to handling carry in addition). 3. **Execute the Subtraction:** - Starting from the LSB, if you encounter a `0`, change it to a `1` and move to the next position to handle the borrow. - If you encounter a `1`, change it to a `0` and halt the process, as no further borrowing is needed. 4. **Handle Special Cases:** - Ensure the Turing Machine can handle special cases, such as subtracting 1 from `0` or single-bit numbers gracefully. By following these steps and defining appropriate transition rules, you can create a Turing Machine that can accurately and efficiently subtract 1 from any binary number input provided on the tape.
Transcribed Image Text:### Exercise 1: Creating a Turing Machine for Binary Subtraction **Objective:** Create a Turing Machine that subtracts 1 from a binary number. **Instructions:** - **Start Position:** The first element on the tape should be empty. - **Binary Number Storage:** Store binary numbers starting with the least significant bit (LSB). This exercise involves designing a Turing Machine, which is a theoretical device that can process and manipulate symbols on a strip of tape according to a set of rules. In this task, you will create a Turing Machine specifically for performing subtraction by 1 on binary numbers. **Example:** If the binary number is `1001` (representing the decimal number 9), the Turing Machine will transform it to `1000` (representing the decimal number 8). The following steps provide a high-level overview of creating such a machine: 1. **Initialize the Tape:** - Place an empty symbol (often a blank or `B`) at the first position on the tape. - Follow this with the binary digits of the number, starting with the least significant bit. For example, the decimal number 6 (in binary, `011`) would be placed as `B110`. 2. **Design the Transition Rules:** - The transition rules define the behavior of the Turing Machine—how it reads, writes, and moves the tape head. For subtraction, you will need rules to: - Identify the least significant bit. - Perform the subtraction operation. - Handle borrowing if necessary (similar to handling carry in addition). 3. **Execute the Subtraction:** - Starting from the LSB, if you encounter a `0`, change it to a `1` and move to the next position to handle the borrow. - If you encounter a `1`, change it to a `0` and halt the process, as no further borrowing is needed. 4. **Handle Special Cases:** - Ensure the Turing Machine can handle special cases, such as subtracting 1 from `0` or single-bit numbers gracefully. By following these steps and defining appropriate transition rules, you can create a Turing Machine that can accurately and efficiently subtract 1 from any binary number input provided on the tape.
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