GATE CS 2019 Set 1 — Question 58
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Engineering Mathematics → Finite Automata & Regular Languages → DFA
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Question
Let be the set of all bijections from to , where denotes the identity function, i.e. . Let denote composition on functions. For a string , let . Consider the language The minimum number of states in any DFA accepting is ________.
Correct answer
120 to 120
Solution
The set of all bijections from to itself is the symmetric group . The size of this group is . The language consists of all strings of elements from whose composition (product) is the identity element . For any finite group , the language is regular and its minimal DFA has exactly states. In this DFA:
Suppose . Let (the inverse bijection). Then . However, because . Thus, and are distinguishable. Since all 120 states are reachable from the start state and any two distinct states are distinguishable, the minimal DFA has 120 states.
1.The states are the elements of the group . Here, there are states.
2.The start state is the identity element .
3.The transition function is defined as for any state and input symbol .
4.The only accepting state is the identity element .
To show this DFA is minimal, we check if any two states are equivalent. Two states are equivalent if for every string , . Suppose . Let (the inverse bijection). Then . However, because . Thus, and are distinguishable. Since all 120 states are reachable from the start state and any two distinct states are distinguishable, the minimal DFA has 120 states.
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