This document provides an introduction to computing the FIRST and FOLLOW sets for context-free grammars. It begins with definitions and rules for calculating FIRST(X) for each grammar symbol X. It then explains how to compute the FOLLOW set by applying rules until no new elements can be added. An example grammar is used to demonstrate calculating the FIRST and FOLLOW sets. Parsing tables are also presented for additional example grammars.

1. WELCOME TO A
JOURNEY TO
CS419
Dr. Hussien Sharaf
Computer Science Department
dr.sharaf@from-masr.com

2. FIRST

1.
2.
3.
Let X be a grammar symbol (a terminal or nonterminal or
), then the FIRST(X) can be computed by the following rules:
If X is terminal, then FIRST(X) is {X}.
If X → is a production, then add to FIRST(X).
If X is nontermianl and X → Y1 Y2 …… Yk is a production, then
place a in FIRST(X) if for some i, a is in FIRST(Yi), and is in
all of FIRST(Y1), ……., FIRST(Yi-1); that is, Y1 …...Yi-1 → . If is
in FIRST(Yj) for all j = 1, 2, ……., k, then add to FIRST(X).
For example ,everything in FIRST(Y1) is surely in FIRST(X). If
Y1 doesn’t derive , then we add nothing more to
FIRST(X), but if Y1 → , then we add FIRST(Y2) and so on.
Ali
Hussien
Dr. Hussien M. Sharaf
First(Ali)
Ali
2

3. FIRST SET EXAMPLE
Consider the following grammar:
E → TQ
Q → +TQ |
T → FR
R → *FR |
F → (E) | id
Then,
FIRST(E) = FIRST(T) = FIRST(F) = {(, id}.
FIRST(Q) = {+, }
FIRST(R) = {*, }

Dr. Hussien M. Sharaf
3

4. FOLLOW
To compute FOLLOW(A) for all nonterminals A, then
apply the following rules until nothing can be added to
any FOLLOW set:
1. Place $ in FOLLOW(S), where S is the start symbol and
$ is the input right endmarker.
i.e. $ ∈ FOLLOW(S)
2. If there is a production A → αBβ, then everything in
FIRST(β) except for is placed in FOLLOW(B).

i.e. (FIRST(β) ~
) ⊆ FOLLOW(B)
Ali Plays
Dr. Hussien M. Sharaf
Follow of Ali
Plays
4

5. FOLLOW
If there is a production A → αB, or [a production A
→ αBβ where FIRST(β) contains (i.e β → )], then
everything in FOLLOW(A) is in FOLLOW(B).
i.e. FOLLOW(A) ⊆ FOLLOW(B)
3.
Ali Plays
Dr. Hussien M. Sharaf
Follow of Ali
Plays
5

6. FOLLOW SET EXAMPLE
Consider again the previous grammar:
E → TQ
Q → +TQ |
First (‘)’) ⊆
T → FR
Follow(E) because
F → (E)
R → *FR |
E is the start
F → (E) | id
symbol
Then,
First(Q) ⊆ Follow(T)
Because 1. E → TQ
FOLLOW(E) = FOLLOW(Q) = { ) , $ }.
FOLLOW(T) = FOLLOW(R) ={ +, ), $ }
Follow(E) ⊆
Follow(T)
FOLLOW(F) = {*, +, ), $}

Because 2. Q →
Dr. Hussien M. Sharaf
First(Q) ⊆
Follow(T)
Because E → TQ
6

7. FOLLOW SET EXAMPLE
Consider again the previous grammar:
E → TQ
Q → +TQ |
T → FR
R → *FR |
F → (E) | id
FOLLOW(E) = FOLLOW(Q) = { ) , $ }.
FOLLOW(T) = FOLLOW(R) ={ +, ), $ }
FOLLOW(F) = { * , + , ) , $ }

First(R) ⊆ Follow(F)
Because T→ FR
Dr. Hussien M. Sharaf
First(R) ⊆ Follow(F)
Because 1. T→ FR
Follow(T) ⊆ Follow(F)
Because 2. R →
7

8. EXAMPLE 1
[ Alfred V. Aho, Compilers Principles, Techniques, and
Tools, 1986, Addison-Wesley, page 188]
Construct the parsing table for the following grammar:
1. E
TQ
2. Q
+TQ |
3. T → FR
4. R
5. F
E
Q
Dr. Hussien M. Sharaf
T
R
F
First Set
{(, id}
{+, }
{(, id}
{*, }
{(, id}
*FR |
(E) | id
Follow set
{$, )}
{$, )}
{+, ), $ }
{+, ), $}
{*, +, ), $}
8

9. EXAMPLE 1 (CONT.)
The parsing table for the previous grammar is:
1. E
TQ
2. Q
+TQ |
3. T → FR
id
E E
+
(
E
Q
id
Dr. Hussien M. Sharaf
$
TQ
Q
Q
R
R
T → FR
R
F
*FR |
(E) | id
)
+TQ
T → FR
R
F
*
TQ
Q
T
4. R
5. F
R
*FR
F
(E)
9

10. FIRST AND FOLLOW EXAMPLE
S →A
 A → BC | DBC
 B → bB ΄|
 B ΄→ bB΄|
 C →c|
 D →a|d
FIRST(S) = { a, b, c, d,
FIRST(A) = { a, b, c, d,
FIRST(B) = { b, }
FIRST(B ΄) = { b, }
FIRST(C) = { c, }
FIRST(D) = { a, d }

Dr. Hussien M. Sharaf
}
}
FOLLOW(S) = { $ }
FOLLOW(A) = { $ }
FOLLOW(B) = { c, $ }
FOLLOW(B ΄) ={ c, $ }
FOLLOW(C) = { $ }
FOLLOW(D) = { b, c, $ }
10

11. EXAMPLE 2 PAGE 156
The parsing table for the following grammar is:
1. S
If_Stat | other
2. If_Stat
if(Exp) S Else_part
if
S
If_Stat
S
3. Else_part
If_Stat S
other
4. Exp
else
else S |
true | false
true
false
$
other
If_Stat
if(exp) S
Else_part
Else_part
Exp
Dr. Hussien M. Sharaf
Else_part
else S
Else_part
exp
true
exp
false
11

12. EXAMPLE 3
The parsing table for the following grammar is:
1. E
TQ
6. R
*FR
2.
3.
4.
5.
+TQ
-TQ
7. R
8. R
9. F
10.F
/FR
Q
Q
Q
T
FR
+
-
*
/
E
Q
(
)
1
2
3
F
Dr. Hussien M. Sharaf
8
6
7
4
5
8
9
$
1
5
8
id
4
T
R
(E)
id
8
10
12

13. THANK YOU