Tutorial 1 solution

COMPILER CONSTRUCTION
Week 2 Tutorial Solutions
Regular Expressions, NFA and DFA
1. Let ∑ = {a,b}. Write regular expressions for the languages over ∑ that contain:
a. All strings.
(a|b)*
b. The empty string.
∈
c. The string abb.
abb
d. The strings ba and aba.
ba|aba
e. All strings beginning with ab.
ab(a|b)*
f. All strings beginning with a a and ending with a b.
a(a|b)*b
g. All strings that contain exactly two a’s.
b*ab*ab*
h. All strings in which every a is followed by a b.
(b|ab)*
2. Construct nondeterministic finite automata (NFA) from the following regular
expressions:
a
a
a. (a|b)(a|b)
7
6
1
2 ∈
∈
∈

∈
•
/
\
0
5
|
|
∈
∈
∈
∈
/ \ / \
b
b
9
a b a b
8
3
4

10

∈

b. a(a|b)+
•
/
\
a

+
|
/ \
a b

∈
0

a

1

∈

3

a

4

∈
7

2
∈
5

b

∈

6

∈

8


∈

c. (ab|b)*a
•
/
\
*
a
|
|
/
\
•
b
/ \
a b
d. abb(a|b)?a
•
/
\
•
a
/

\
•
?
/
\
|
•
b / \
/ \
a b
a
b

∈

a

2

b

3

4

∈

∈
0

1

7

∈

∈

8

a

9

∈
b

5

6

∈

∈
0

a

1

b

2

b

3

∈
∈

a

5

6

b

7

8

∈

4

∈
∈
∈

10

a

11

9

Note: You may use Thompson’s construction algorithm.
3. Construct deterministic finite automata (DFA) from the NFA’s constructed in
question 2.
a.
a
B
D
a
b
a
A={0,1,3}
{2,5,6,8}
{4,5,6,8}
a
A
B={2,5,6,8}
{7,10}
{9,10}
b
C={4,5,6,8}
{7,10}

{9,10}
b
D={7,10}
b
C
E
E={9,10}
b.
a
a
b
a
A={0}
{1,2,3,5}
B
A
b
B={1,2,3,5}
{4,7,8,2,3,5}
{6,7,8,2,3,5}
b
C={4,7,8,2,3,5} {4,7,8,2,3,5}
{6,7,8,2,3,5}
D={6,7,8,2,3,5} {4,7,8,2,3,5}
{6,7,8,2,3,5}
c.
B
a
b
a

a
A={0,1,2,5,8}
{3,9}
{6,7,8,1,2,5}
b
A
a
B={3,9}
{4,7,8,1,2,5}
C={6,7,8,1,2,5}
{3,9}
{6,7,8,1,2,5}
b
b
D={4,7,8,1,2,5}
{3,9}
{6,7,8,1,2,5}
C
D
b

a
C
a
D
b


d.
A={0}

B={1}
C={2}
D={3,4,6,8,9,10}
E={5,10,11}
F={7,10}
G={11}

a
{1}
{5,10,11}
{11}
{11}
-

b
{2}
{3,4,6,8,9,10}
{7,10}
-

G
a
A

B

C

D

b

a

B
a

a
a

b

a
a
b

b

F

E

A
b

b

a

4. Minimise the following DFA:

Partition 1: {A,E,D} {B,C}
For a,
{B,B,B} {C,B}
For b,
{E,E,D} {D,D}

A

a

a

C
b

b
B

D

E
b

b

5. Write a phrase using wildcards to find the
following words in a document written in MSWord:
a. Any number.
<[0-9]@>
b. “get” and “got”.

<g[eo]t>
c. Any word that starts with a ‘s’ and ends with ‘t’. E.g. “sit”, “shoot”,…
<s*t> (which includes blanks “sa t”)
<s[! ]@t> (which does not include “st”)
d. Any word whose beginning is “re”. E.g. “repeat”, “recharge” but not “trend”
<(re)

a