GATE CS 2021 Set 2 — Question 19
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Theory of Computation → Finite Automata & Regular Languages → Closure Properties (Regular)
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Question
Let be an arbitrary regular language accepted by a minimal DFA with states. Which one of the following languages must necessarily be accepted by a minimal DFA with states?
Correct answer
(C) \0, 1\^ - L
Solution
A fundamental property of regular languages is that if a language is regular, its complement is also regular. If is the minimal DFA for , it has states. The minimal DFA for is obtained by swapping the final and non-final states: . This resulting DFA is also minimal and has the same states. This is because the Myhill-Nerode equivalence relation for and is identical:
.
Since the number of states in a minimal DFA is equal to the number of equivalence classes of this relation, both and must have minimal DFAs with the same number of states .For the other options:
.
Since the number of states in a minimal DFA is equal to the number of equivalence classes of this relation, both and must have minimal DFAs with the same number of states .For the other options:
- (A) and (B) involve adding or removing a specific string, which can change the number of states in the minimal DFA.
- (D) (concatenation) typically increases the number of states significantly.
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