Consider the following deterministic finite automaton (DFA) defined over the alphabet, \( \Sigma = \{a, b\} \). Identify which of the following language(s) is/are accepted by the given DFA.
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When analyzing DFAs, focus on the transitions between states and how they determine whether a string is accepted. In this case, the DFA accepts strings that end with the pattern "bab".
The set of all strings containing an even number of b's.
The set of all strings containing the pattern "bab".
The set of all strings ending with the pattern "bab".
The set of all strings not containing the pattern "aba".
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The Correct Option isC
Solution and Explanation
Let’s analyze the DFA:
DFA Structure:
The DFA has 3 states, and it accepts strings based on transitions for \( a \) and \( b \).
Start state (S0): On reading \( a \), the DFA stays in the same state, and on reading \( b \), it transitions to state \( S1 \).
State S1: On reading \( a \), the DFA transitions to state \( S0 \), and on reading \( b \), it transitions to the accepting state \( S2 \).
State S2 (accepting state): On reading \( a \), the DFA transitions to state \( S1 \), and on reading \( b \), it stays in state \( S2 \).
Key observation:
The DFA accepts a string if it ends in state \( S2 \), which means it ends with the pattern "bab" (since "bab" is the only transition path that leads to the accepting state).
Conclusion: Option (C) is correct because the DFA accepts all strings that end with the pattern "bab", as it ends in the accepting state after reading the pattern "bab".
Option (A) is incorrect because the DFA doesn't count the number of b's; instead, it focuses on the pattern "bab".
Option (B) is incorrect because the DFA does not specifically check for the "bab" pattern within the string, but instead looks for the string ending with "bab".
Option (D) is incorrect because the DFA does not explicitly reject strings that contain "aba"; it accepts strings that end with "bab".
Thus, the correct answer is (C): The set of all strings ending with the pattern "bab".
