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Let L be the language generated by regular expression 0*10* and accepted by the deterministic finite automata M. Consider the relation $R_M$ defined by M as all states that are reachable from the start state. $R_M$ has ___ equivalence classes.

1. 2
2. 4
3. 5
4. 6

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Here $M$ is a DFA for $L$. It is not mentioned that $M$ is a minimal DFA.

$R_M$ is defined by $M$ as the set of states reachable from the start state.

What does this mean? All reachable states form 1 class and other form another class? In this case answer must be 2.

PS: This question is not in any way related to Myhill-Nerode Theorem as first of all DFA is not minimal, and then Myhill-Nerode theorem talks about equivalence relation which is defined using a specific RELATION based on appending a string and reaching final state or not. It is not applicable on any other equivalence class/relation.
by Veteran (425k points)

Myhill Nerode equivalence relation = No of States in Minimal DFA , by Boss (35.7k points)
No of equivalence classes is the no.of states in minimum dfa..

Given language is " SRT of strings which contains exactly one 1"

For that we need 3 states so No.of equivalence classes are 3 with dead state.
by Boss (25.5k points)
edited by
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minimum dfa for regular expression 0*10* need 3 states then how No.of equivalence classes are 4.
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yes...i missed...
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No of equivalence classes as per  Myhill Nerode equivalence relation = No of States in Minimal DFA ,

then why it does not follow here

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i think no. of equivalence classes here is 4 only

string  starting with 0 , 0 0.........

string  starting with 0 , 0 0........and  end with exactly one 1 , exactly one 1 can also come here .

string  starting with 1 and followed by 0's.

so 4 equivalence class but 3 states in min. dfa

Tell me where i am wrong ?
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No 3 is correct
+1
@Akansha classes 2 and 3 are the same as far as equivalence class as per Myhill-Nerode theorem is concerned. That relation consider only the "remaining part of string".
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Arjun Sir , Am i right now ?

Equivalance class -

1st --> string starting with 0  and 0000........

2nd --> string having exactly one 1 ,  it can be of the form -- 0001 , 10000 , 1 etc

Sir 1 doubt

No of equivalence classes as per  Myhill Nerode equivalence relation = No of States in Minimal DFA

is always true na..

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That statement is always true- and hence called a Theorem. But there is something wrong with the question statement here.
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Is this Myhill Nerode equivalence relation is in GATE syllabus?