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+16 votes
Let $L$ be the language of all binary strings in which the third symbol from the right is a $1$. Give a non-deterministic finite automaton that recognizes $L$. How many states does the minimized equivalent deterministic finite automaton have? Justify your answer briefly?
asked in Theory of Computation by Veteran (68.8k points)
retagged by | 1.3k views
If we convert any NFA to DFA is that the minimal DFA or the process is NFA -> DFA -> minimize the DFA
Minimal dfa is reducing the no of states of an Dfa

6 Answers

+13 votes
Best answer

Answer :-

Check following NFA. I've done subset construction too. 8 States are needed even after minimization..

Every state containing D is final state. 


Third symbol from the right is a '1'. So, we can also consider Myhill-Nerode theorem here. Intuitively we need to remember the last 3 bits of the string each of which forms a different equivalence class as per Myhill-Nerode theorem as shown by the following table. Here, for any set of strings (in a row), we disntiguishes only the rows above it - as the relation is symmetric. Further strings in the language and not in the language are disntinguished separately as $\epsilon$ distinguishes them.

  Last 3 bits Distinguishing string In the language?
1 000   N
2 001 "00" - disntinguishes from  strings in 1. N
3 010 "0" - disntinguishes from strings in 1 and 2. "00" distinguishes from strings in 4. N
4 011 "0" - distinguishes from strings in 1 and 2. "00" distinguishes from strings in 3. N
5 100   Y
6 101 "00" disntinguishes from strings in 5. Y
7 110 "0" distinguishes from strings in 5 and "00" distinguishes from strings in 6 Y
8 111 "00" distinguishes from strings in 5 and 7. "0" distinguishes from strings in 6. Y


answered by Veteran (48.5k points)
edited by
Subset construction algo does not always give minimal dfa.
Yup got it. Thank you.
any specific example?
@VS , Can you tell me where is it written that "subset construction algo does not always give minimal DFA" ? Is there any method which can directly give minimal DFA from NFA or do we have to check it anyway after applying any algo?


eg : (ooo+ooooo)*

Minimal nfa has= 5 states

DFA by subset construction = 13 states

Minimal DFA=9 states //In dfa obtained after subset construction if you check closely the last 5 states can be combined into a single state i.e. states 9 to 13 .

AFAIK there is no algo from which we could directly generate a minimal dfa

0's as inputs

btw we can find the minimal dfa using equivalence classes of states followed by transition table.
@Praveen sir

but,it is not easy to find equivalence classes always.
[ACD] on input 1 is going to [ABD] not in [ABC]. Kindly correct the answer.
It is easy to find equivalence class always. we can find minimum dfa also.
+9 votes

NFA =4 state 


P.S Here { AD,ABD,ACD,ABCD} these are final states i forgot to circle them

answered by Boss (9.5k points)
is the above diagram accepting the string 1100???
+2 votes

The answer is 8


To attempt these type of questions, the important thing is that since u are dealing with symbols from right when you consume string from the left u have to take care of all possible combinations 

So the answer wil always be 2where n is the position from the right 

answered by (487 points)
And for position from left, it is n+1
–1 vote
third symbol from right is 1..

bit setting from right so no trap/dead state..
total 4 states required..
answered by Veteran (49.1k points)
reshown by
8 states are needed ! (Even if you try to minimize)
i didnt understand, can sir  help?
–2 votes
Minimal no states in DFA : 8

no of states in NFA : 4
answered by Loyal (4.4k points)
edited by
–2 votes

4 states are enough for the language of all binary strings in which the third symbol from the right is a 1.




answered by Loyal (4.4k points)
NO, you're wrong here, you can not draw DFA like this for this question it will not give all valid sequence

e.g  1100 ,1101, 1111,1110 etc

these all are binary strings in which the third symbol from the right is a 1.which you can not draw from your DFA .see my explanation i have uploaded a pic for better clarity.

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