GATE CS 2026 Set 1 — Question 26
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Theory of Computation → Finite Automata & Regular Languages → DFA Minimization
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Question
Let be a nondeterministic finite automaton (NFA) with 6 states over a finite alphabet.Which of the following options CANNOT be the number of states in the minimal deterministic finite automaton (DFA) that is equivalent to ?
Correct answer
(B) 65; (D) 128
Solution
The number of states in the given NFA is .The subset construction algorithm converts an NFA with states into an equivalent DFA with at most states. The minimal DFA is unique and has a number of states less than or equal to any other equivalent DFA. Therefore, the number of states in the minimal DFA, denoted as , is bounded by:Substituting :We analyze the given options:
(A) (Possible)
(B) (Impossible)
(C) (Possible, e.g., for a language accepting all strings or no strings)
(D) (Impossible)The question asks for the number of states that CANNOT be in the minimal DFA. Both 65 and 128 exceed the maximum possible states (64).
(A) (Possible)
(B) (Impossible)
(C) (Possible, e.g., for a language accepting all strings or no strings)
(D) (Impossible)The question asks for the number of states that CANNOT be in the minimal DFA. Both 65 and 128 exceed the maximum possible states (64).
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