GATE CS 2024 Set 1 — Question 50
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Theory of Computation → Finite Automata & Regular Languages → DFA Minimization
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Question
Consider the 5-state DFA accepting the language shown below. For any string let be the number of 0's in and be the number of 1's in .
Correct answer
(B) States 3 and 4 are distinguishable in M; (C) States 2 and 5 are distinguishable in M
Solution
To identify the FALSE statement(s), we analyze the given DFA :1. DFA Components:
Two states are distinguishable if there exists a string such that exactly one of is in .
- States
- Start state
- Final states (indicated by the double circle)
- Transitions :
Two states are distinguishable if there exists a string such that exactly one of is in .
- (A) States 2 and 4: and . Thus, 2 and 4 are distinguishable. Statement (A) is TRUE.
- (B) States 3 and 4: and . Thus, 3 and 4 are distinguishable. Statement (B) is TRUE.
- (C) States 2 and 5: and . Thus, 2 and 5 are distinguishable. Statement (C) is TRUE.
- Statement (D) claims that any string with an equal number of 0s and 1s is accepted.
- Consider the string . Here and .
- Trace in the DFA: .
- The final state reached is 3, which is not a final state ().
- Since , statement (D) is FALSE.
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