Class Test-2
TAFL (NCS-402)
Solution
Section-A
Q1. Prove or disprove the following for regular expressions r, s and t:
(r + s)* = r* + s*
Replace r by {a} and s by {b}. The left side becomes all strings of a's and b's (mixed), while the right side
consists only of strings of a's (alone) and strings of b's (alone). A string like ab is in the language of the left side
but not the right.
Q2. Using Pumping Lemma, prove that L = {0n; n is a perfect square} is not regular.
Q3. Design a Moore machine which counts the occurrence of substring aab in the
input string over the alphabet Σ = {a,b}.
The transition table for Moore machine is as follows:
Present State
q0
q1
q2
q3
Next state on
a
q1
q2
q2
q1
Output
b
q0
q0
q3
q0
0
0
0
1
Q4. Differentiate among Regular Expression, Regular Language and Regular Grammar.
Regular Expression: The regular expressions are useful for representing certain sets of strings in an algebraic
fashion. Actually these describe the languages accepted by finite state automata.
We give a formal recursive definition of regular expressions over Σ as follows:
1. Any terminal symbol (i.e. an element of Σ), ε and Φ are regular expressions. When we view a in Σ as a regular
expression, we denote it by a.
2. The union of two regular expressions R1 and R2 written as R1 + R2, is also a regular expression.
3. The concatenation of two regular expressions R1 and R2, written as R1R2, is also a regular expression.
4. The iteration (or closure) of a regular expression R written as R*, is also a regular expression.
5. If R is a regular expression, then (R) is also a regular expression.
6. The regular expressions over Σ are precisely those obtained recursively by the application of the rules 1-5 once
or several times.
Regular Grammar and Language:
A production of the form Aa or A aB, where A, B Є N and a Є Σ, is called a type 3 production. A grammar
is called a type 3 or regular grammar if all its productions are type 3 productions. A production S ε is allowed
in type 3 grammar but in this case S does not appear on the right-hand side of any production. A language
generated by a regular grammar is called a type 3 language or a regular language.
Q5. Give a regular expression for language L= {anbm : n ≥ 4,m ≤ 3}
Regular expression for language L= {anbm : n ≥ 4,m ≤ 3 } is
aaaaa*( Λ + b + bb + bbb)
Section-B
Q1. Using Arden’s Theorem, find the regular expression for the finite automata,
M = (Q , Σ, δ, q0, F) where Q={A,B,C}, Σ={a,b}, q0 = A, F = {B, C} and δ is defined as
State
Input = a
Input = b
A
A
B
B
B
C
C
C
A
Write Arden’s Equations after checking for availability of ε moves:
Since there is no ε move, we can directly write down the equations as follows:
A=Aa+Cb+ε
…………..(1)
B=Ab+Ba
…………..(2)
C= Bb+Ca
……………(3)
Applying Arden’s on equation 2 & 3, we get
B= Aba*
………….(4)
and,
C= Bba*
………..(5)
Now using eq. 4 & 5,
C= Aba*ba* ………..(6)
Now using eq. 1 & 6,
A= Aa+ Aba*ba*b+ε ….(7)
Apply Arden’s Theorem on eq. 7,
A= (a+ba*ba*b)* ………(8)
Using eq. 8,
B= (a+ba*ba*b)* ba*
And,
C= (a+ba*ba*b)* ba*ba*
So, Regular Expression can be obtained by adding the values of B & C,
= (a+ba*ba*b)* ba* + (a+ba*ba*b)* ba*ba*
Section-C
Q.1Construct Deterministic finite automata
expression:
10 + (0 + 11)0*1
equivalent
to
following
regular