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The FSA shown below has E = {0, 1} and determines whether or not a binary number is divisible by three. Assuming that the input to this FSA is a binary number n, prove that the FSA can determine whether n is divisible by 3 in a number of steps which is O(lg(n)). 1

The FSA shown below has E = {0, 1} and determines whether or not a binary number is divisible by three. Assuming that the input to this FSA is a binary number n, prove that the FSA can determine whether n is divisible by 3 in a number of steps which is O(lg(n)). 1

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
Section: Chapter Questions
Problem R1RQ: What is the difference between a host and an end system? List several different types of end...
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**Automata Theory - Finite State Automata and Divisibility by 3** **Description:** The Finite State Automaton (FSA) depicted below is designed to determine if a binary number is divisible by three. Here, \(\Sigma = \{0, 1\}\), representing the binary digits. The problem can be formalized by proving that the FSA, given a binary number \(n\) as input, can ascertain whether \(n\) is divisible by 3 in a number of steps that is \(O(\log n)\). **Diagram Explanation:** The FSA consists of three states, including one start state and one accepting state: - **State 0**: This state doubles as the start state and the accepting state. It is represented as a circle with a double outline and an arrow indicating the start. - **State 1**: This state transitions to and from the other states based on the input binary digit. - **State 2**: This state also transitions based on input binary digits. **Transitions:** - From **State 0**: - Input 1 leads to **State 1**. - Input 0 leads back to **State 0**. - From **State 1**: - Input 1 leads to **State 2**. - Input 0 leads back to **State 1**. - From **State 2**: - Input 1 leads back to **State 0**. - Input 0 leads back to **State 2**. **Conclusion:** The FSA transitions through these states based on the binary input. When the input is completely processed, if the FSA ends in the accepting state (State 0), the binary number is divisible by 3. The efficiency of this process can be shown to be \(O(\log n)\) due to the logarithmic relationship between the length of the binary input and the numerical value of \(n\). This proves the FSA’s capability to verify divisibility by 3 efficiently for binary numbers.
Transcribed Image Text:**Automata Theory - Finite State Automata and Divisibility by 3** **Description:** The Finite State Automaton (FSA) depicted below is designed to determine if a binary number is divisible by three. Here, \(\Sigma = \{0, 1\}\), representing the binary digits. The problem can be formalized by proving that the FSA, given a binary number \(n\) as input, can ascertain whether \(n\) is divisible by 3 in a number of steps that is \(O(\log n)\). **Diagram Explanation:** The FSA consists of three states, including one start state and one accepting state: - **State 0**: This state doubles as the start state and the accepting state. It is represented as a circle with a double outline and an arrow indicating the start. - **State 1**: This state transitions to and from the other states based on the input binary digit. - **State 2**: This state also transitions based on input binary digits. **Transitions:** - From **State 0**: - Input 1 leads to **State 1**. - Input 0 leads back to **State 0**. - From **State 1**: - Input 1 leads to **State 2**. - Input 0 leads back to **State 1**. - From **State 2**: - Input 1 leads back to **State 0**. - Input 0 leads back to **State 2**. **Conclusion:** The FSA transitions through these states based on the binary input. When the input is completely processed, if the FSA ends in the accepting state (State 0), the binary number is divisible by 3. The efficiency of this process can be shown to be \(O(\log n)\) due to the logarithmic relationship between the length of the binary input and the numerical value of \(n\). This proves the FSA’s capability to verify divisibility by 3 efficiently for binary numbers.
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