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ASSIGNMENT -1
OF
AUTOMATA THEORY
SUBMITTED TO: SUBMITTED BY:
Ms. MANJIT KAUR GAGANPREET KAUR
ROLL NO: 42
SECTION: B1803
1
PART A:
1. The following are the state diagrams of two DFAs, M1 and M2. Answer the following questions about each of these machines.
a. What is the start state?Ans: 1) M1 start state is q12) M2 start state is q1
b. What is the set of accept states?Ans: accept states are the final states. So in this case, set of accept states are:-
1) M1 it is q22) M2 it is q1,q4
c. What sequence of states does the machine go through on input aabb?Ans:
For M1 :-
q1q2q3q1q1 For M2 :-
q1q1q1q2q4
D.Does the machine accept the string aabb?Ans:
1) For M1:-
2
q1q2q3q1q1
Since it does not end at any final state. So this string is not acceptable.
2) For M2:- q1q1q1q2q4
Since it ends at q4 which is one of the final state. So this string is acceptable.
e. Does the machine accept the string e?
Ans: No both the machines will not accept the string e because inputs given to the machine are a and b. rest inputs will not be acceptable. There exists no node namely e.
Ques 2. For ∑= {0, 1}, Construct DFA’s that accepts the set consisting of (a) All strings with at least one 0 and exactly two 1’s.
Ans: - 01010
(b) All strings ending in 010.
3
Ans :-
(c) All strings with no more than three 0’s.
Ans :- 11010
4
Ques 3. Construct DFA equivalent to the NDFA described by fig below: -
0, 1 1
0, 1 1
0
^ 0, 1
0
The transition table as per given diagram of the NDFA is as shown in figure;
States 0 1
Q0 q0,q3 q0,q1
Q1 q2
Q2 q2 q2
Q3 q4
Q4 q4 q4
And the corresponding transition table in DFA is as given…
States 0 1
[q0] [q0,q3] [q0,q1]
[q0,q3] [q0,q3,q4] [q0,q1]
[q0,q1] [q0,q3] [q0,q1,q2]
[q0,q3,q4] [q0,q3,q4] [q0,q1,q4]
[q0,q1,q2] [q0,q3,q2] [q0,q1,q2]
[q0,q1,q4] [q0,q3,q4] [q0,q1,q2,q4]
[q0,q2,q3] [q0,q3,q2,q4] [q0,q1,q2]
[q0,q1,q2,q4] [q0,q3,q2,q4] [q0,q1,q2,q4]
5
q1
q0
q4
q3
q2
[q0,q2,q3,q4] [q0,q3,q2,q4] [q0,q1,q2,q4]
PART B:
Ques 4. Construct a Mealy machine which is equivalent to the Moore machine defined by table given below.
Present states Next state Outputa=0 a=1
q0 q1 q1 1
q1 q3 q2 0q2 q2 q0 0q3 q0 q1 1
Ans: The corresponding Mealy machine is as shown below:
Present states Next states
A=0 Output A=1 output
->q0 Q1 0 Q1 0
Q1 Q3 1 Q2 0
Q2 Q2 0 Q0 1
Q3 Q0 1 Q1 0
Ques 5. Construct a Moore machine which is equivalent to the Mealy machine defined by table given below.
6
Present states Next state a=0state output
a=1state output
q1 q1 1 q1 0q2 q2 1 q4 1
q3 q4 1 q2 1
q4 q3 0 q3 0
Ans: now rewriting states, we do get what follows next….
q11 (q10 for 0)
0 (q11 for 1)
q21
q30
q41
The table corresponding table according to the Mealy machine is as shown :-
Present state Next state
A=0 output A=1 output
Q10 Q11 1 Q10 0
Q11 Q11 1 Q1 0
Q2 Q2 1 Q4 1
Q3 Q4 1 Q2 1
Q4 Q3 0 Q3 0
And finally the Moore machine is constructed as follows.
Present states Next state Output
7
a=0 a=1q0 q11 q10 0
q10 Q11 Q10 0Q11 Q11 Q10 1Q2 Q2 Q4 1Q3 Q4 Q2 0Q4 Q3 Q3 1
Ques 6. Construct a minimum state automaton equivalent to given automaton M whose transition table is defined by table given below.
Ans:
Step 1:
Q1= {q3}, Q2 = { q1,q2, q4,q5,q6,q7}
Step2.
π0={{q3},{q0,q1,q2,q4,q5,q6,q7}}
8
State Input
a b
q0 q0 q1
q1 q2 q0
q2 q1 q3
q0 q3
q4 q5 q3
q5 q4 q6
q6 q6 q5
q7 q3 q6
q3
step3.
Enacting π1, π2 and so on until πn= πn+1
π1= {{q3},{q0,q1,q5,q6},{q2,q4},{q7}}
π2= {{q3},{q0,q6},{q1,q5},{q2,q4},{q7}}
π3= {{q3},{q0,q6},{q1,q5},{q2,q4},{q7}}
From the above equation, we have,
π3=π2
Hence the minimized automaton is as follows:-
States A B
[q0,q6] [q0,q6] [q1,q5]
[q1,q5] [q2,q4] [q0,q6]
[q2,q4] [q1,q5] [q3]
[q3] [q0,q6] [q3]
[q7] [q3] [q0,q6]
The figure is shown below:-
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