GATE CS 2025 Set 2 — Question 60
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Theory of Computation → Finite Automata & Regular Languages → DFA
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Question
Let . For , let be the product of symbols in modulo 7. We take , where is the null string.For example, .Define .The number of states in a minimum state DFA for is ___________. (Answer in integer)
Correct answer
6 to 6
Solution
To find the number of states in the minimum state DFA for the language , we use the Myhill-Nerode theorem, which states that the number of states in the minimal DFA is equal to the number of equivalence classes of the relation .
1.Define the Language: , where .
2.Identify Reachable Residues: The state of the DFA must track the current product modulo 7. The possible non-zero residues modulo 7 are .
3.Check Reachability: Starting from the initial product (for ), we can reach any residue if the set generates the multiplicative group .
- The powers of modulo 7 are: , , , , , .
- Since is a primitive root modulo 7 and , all 6 non-zero residues are reachable.
5.Equivalence Classes: Two strings and are equivalent () if for all , .
- .
- Because is a group, this condition is equivalent to .
- Thus, .
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