Construct a finite state automaton that recognizes all strings that end with 11.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Transcribed Image Text:## Constructing a Finite State Automaton (FSA)
### Problem Statement:
Construct a finite state automaton that recognizes all strings that end with 11.
#### Solution:
To construct a finite state automaton (FSA) that recognizes all strings ending with "11", follow these steps:
1. **Define the States:**
- **q0:** The initial state.
- **q1:** State representing that the last seen character is '1'.
- **q2:** Accepting state representing that the last two seen characters are "11".
2. **Transitions:**
- **From q0:**
- On input '0', remain in q0.
- On input '1', transition to q1.
- **From q1:**
- On input '0', transition back to q0.
- On input '1', transition to q2 (accepting state).
- **From q2:**
- On input '0', transition back to q0.
- On input '1', remain in q2.
3. **Accepting State:**
- The accepting state is q2, indicating that the last two characters of the string are "11".
#### Diagram:
Below is the state transition diagram for the FSA:
```
-----(0)------> q0 ----(1)------> q1 -----(1)-----> q2 (Accepting State)
| (1) | (0) |
| | |
| | |
| | |
-----(0)---------- (0)
| (1) |
-----------------------------------|
```
#### Explanation:
- Start at q0.
- Read the input string character by character.
- Transition between states according to the defined transitions.
- If the final state after reading the entire string is q2, the input string is accepted as it ends with "11".
This structure is particularly useful for pattern matching and string analysis in computer science applications.
---
This explanation and the FSA provide a comprehensive understanding of how to recognize strings that end with "11". This model serves as a foundational concept in automata theory and is widely applied in computational logic and formal language processing.
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