Introduction to Compiler Design
Midterm Examination Solution
Name: _____________
Reg-No: __Solution_____
1. (20 pts) [CFG ] Consider the grammar
and the string:
5/9 2007
G:
score: _____________
SÆSS+|
SS# |
a
a a + a #
(a) Give a rightmost derivation for the string. (5pts)
Ans: S ÆSS# ÆSa# Æ SS+a# Æ Sa+a# Æaa+a#.
(b) Give a parse tree for the string.
Ans:
S(
S(
(5 pts)
S(a), S(a), + ), S(a), # ).
Note: we use S(t1,t2,,,,tn) to represent a tree with root S and subtrees t1,t2,…,tn.
(c) Is the grammar ambiguous or unambiguous ?
why? (5pts)
Ans: No, since it can be shown that for every input, there is at most one parse tree for it.
(d) Describe the language generated by this grammar? ( 5 pts)
Ans:
It is the set of the postfix forms of all expressions which are composable by two binary
operators (+ and #) and one constant a.
2. [5 pts] Consider the grammar : S Æ S a | a
Which class of grammar does this grammar belong to?
(a) LL(1)
(b)
LL(2)
(c) LR(0)
(d) LR(1)
(e)
LALR(1)
Ans: ____cde______
3. (30 pts) [LL(1) parsing] Given the following grammar G:
S
ÆE$
E Æ T E'
T
Æ F T'
T' Æ * F T'
E' Æ + T E'
|
ε
F
Æ (E)
(a) Find all nonterminals in G that are nullable. (4pts)
1
|
|d
ε
Ans:
______________T',E'__________
(b) Fill in the following table with FIRST and FOLLOW sets for the grammar: (10 pts)
N
FIRST(N)
FOLLOW(N)
S
(,d
{}
E
(,d
),$
E'
+
),$
T
(,d
+,),$
T'
*
+,),$
F
(, d
*,+,),$
(c) Complete the following LL(1) table for the grammar:
N\Σ
(10 pts)
d
+
*
(
)
$
S
E$
err
err
E$
err
err
E
TE'
err
err
TE'
err
err
E'
err
+TE'
err
err
ε
ε
T
FT'
err
err
FT'
err
err
T'
err
ε
*FT'
err
ε
ε
F
FÆ d
error
error
FÆ(E)
error
error
(d) Is this grammar LL(1) ? why? (4 pts)
Ans: __Yes, since there is at most one rule in every cell of the LL(1) table.
___________________________________________________________________
4. (10pts) Let G = (N,Σ, S,P) be a CFG where N is the set of non-terminal symbols, S is the set of
terminal symbols and P the set of production rules. Define the relation First ⊆ N x Σ as the set of
all pairs (A,a) such that a can appear in the first position of some sentential forms derived from A, i.e.,
First = { (A, a) ∈ N X Σ | A Æ* a β for some β ∈ (N U Σ)*
}. The relation First can be defined
inductively as follows:
(a) Basis:
(A, a) ∈ First
for all A Æ B1B2…Bn a … ∈ P, where n ≥ 0 and B1,…,Bn are all
nullable.
(b) Closure:
If (C,c)∈First and B1,..,Bn are nullable and A ÆB1B2…Bn C… ∈ P, then (A,c) ∈First .
Now let Last = { (A, a) ∈ N X Σ
|
∃β∈(N U Σ)*
2
s.t.
A Æ* β a }.
List an inductive definition for the relation Last :
(a) Basis:
If there exists A Æ … a B1,…,Bn∈ P where n ≥ 0 and each Bk ( 1≤ k ≤ n ) is nullable,
then (A,a) ∈ Last.
(b) Closure:
If there is a rule A Æ … C B1,…,Bn∈ P where n ≥ 0 and each Bk ( 1≤ k ≤ n ) is nullable,
and (C,d) ∈ Last, then (A,d) ∈ Last.
5. [10 pts] Write regular expressions for the following languages:
(a) All strings over {a,b} containing at least 2 b's.
Ans: (a + b)* b (a + b)* b (a + b)*
(b) Decimal number 0 ~ 255. Note that all leading zero's except 0 are not accepted.. (4pts)
Ans:
(0 + 1 +…+ 9)
// 0 ~ 9
+
(1 +…+ 9) (0 + 1 +…+ 9)
// 10 ~99
+
1(0+1 +…+ 9) (0 + 1 +…+ 9) // 100 ~199
+
2(0+1 +…+ 4) (0 + 1 +…+ 9) // 200~249
+
25(0+1 +…+ 5)
// 250 ~255
(c) Binary numbers that are greater than 101. Note. leading zero's are permitted.
Ans: 0* (100 + 101) (1+0)+ + 0* (110 + 111) ( 1 + 0)*
6. [20 pts ; minimization of FA] For the finite automata F given below,
(a) Complete the following chart to find all pairs of states that cannot be merged without
affecting the accepted language. [12pts]
3
1
2
x1
3
x0
x0
4
x1
x1
x0
5
-
x1
x0
x1
6
x1
x1
x0
-
x1
7
x2
x1
x0
x1
x2
x1
8
x1
-
x0
x1
x1
x1
x1
1
2
3
4
5
6
7
8
2 ≈8, 4 ≈6 , 1 ≈5
(b) Draw the minimized finite automata according the result of (a).
0
[15]
0
[[3]]
1
1
[28]
1
1
[46]
0
0
1
[7]
0
7. (15 pts) [From Regular Expression to DFA] Consider the regular expression α given below:
(a+b)* ba(a+b)
(a) Write an NFA which recognizes the language represented by α and contains no more than 4
states. Remember to label each state. [5 pts]
(b) Convert the NFA you got at (a) to an equivalent DFA by using the subset construction. Note:
you must identify each state of your DFA with the set of old NFA states it corresponds to and
make sure all unused states are removed. [10pts]
4
(a) : Ans :
a,b
0
b
a
1
a,b
2
(b) : Ans :
3
a
a
b
0
b
b
01
b
a
02
b
a
a
013
8. (25 pts)[LR parsing] Consider the grammar
0.
S Æ E$
1.
E Æ E E +
2. E Æ E E !
(a) Build the LR(0) NFA for this grammar [5pts]
5
3. E Æ a
03
E
SÆy E$
1
ε
ε
E
EÆy EE+
$
SÆEy$
EÆ EyE+
E
SÆE$y
+
EÆ EEy+
2
E
EÆy EE!
ε
3
a
EÆy a
EÆ EyE!
E
EÆ a y
ε
!
EÆ EEy!
EÆ EE+y
EÆ EE! y
ε
to 1,2 and 3
(b) Build the LR(0) DFA for this grammar [10pts]
SÆy E$
(1)EÆy EE+
(2)EÆy EE!
(3)EÆy a
a
E
SÆEy$
+123
(4) EÆ EyE+
(5) EÆ EyE!
a
SÆE$y
EÆ EE+y
E
a
EÆ a y
$
+
!
12345+
(6) EÆ EEy+
(7) EÆ EEy!
E
a
(c) Is this an LR(0) grammar ? Give evidence. [5pts]
Yes, because there is no s/r or r/r conflict in the LR(0) parsing table.
6
EÆ EE! y
(d) Is this an SLR(1) grammar ? Give evidence. [5pts]
Yes, since it is already an LR(0) grammar and every LR(0) grammar is a SLR(1) grammar.
7