Automata and Formal Language (chapter 6) ppt

Simplification of Context-Free
Grammars
Quan Thanh Tho

2
Simplification of Context-Free
Grammars
•
Some useful substitution rules.
• Removing useless productions.
•
Removing λ-productions.
•
Removing unit-productions.
3
Some Useful Substitution Rules
G = (V, T, S, P)
A → x
1
Bx
2
∈ P
B → y
1
| y
2
| | y
n
∈ P
L(G) = L(G^)
G^ = (V, T, S, P^)

A → x
1
y
1
x
2
| x
1
y
2
x
2
| | x
1
y
n
x
2
∈ P^
4
Example
G = ({A, B}, {a, b}, A, P)
A → a | aaA | abBc
B → abbA

| b
G^ = (V, T, S, P^)
A → a

| aaA

| ababbAc | abbc
5
Example 6.1
G = (V, T, S, P)
A → Ax
1
| Ax
2
| | Ax
n
∈ P
A → y
1
| y
2
| | y
m
∈ P
L(G) = L(G^)
G^ = (V∪{Z}, T, S, P^)
A → y
i
| y
i
Z (i =1, m) ∈ P^
Z → x
i
| x
i

Z (i =1, n) ∈ P^
6
Example 6.2
G = ({A, B}, {a, b}, A, P)
A → Aa | aBc | λ
B → Bb

| ba
A → aBc

| aBcZ

| Z | λ A → aBc

| aBcZ

| Z | λ
Z → a | aZ Z → a | aZ
B → Bb | ba B → ba | baY
Y → b | bY
7
Removing Useless Productions
S → aSb | λ | A
A → aA
S → A is redundant as A cannot be transformed into a
terminal string.
8
Removing Useless Productions
G = (V, T, S, P)
A ∈ V is useful iff there is w ∈ L(G) such that:

S ⇒* xAy

⇒* w
A production is useless it it involves any useless variable.
9
Example 6.3
G = ({S, A, B}, {a, b}, S, P)
S → A
A → aA

| λ
B → bA
10
Example 6.4
G = ({S, A, B, C}, {a, b}, S, P)
S → aS | A | C S → aS | A S → aS | A
A → a A → a A → a
B → aa B → aa
C → aCb
11
Example 6.5
G = ({S, A, B, C}, {a, b}, S, P)
S → aS | A | C S → aS | A S → aS | A
A → a A → a A → a
B → aa B → aa
C → aCb dependency graph
S A B
12
Theorem 6.1
Let G = (V, T, S, P) be a context-free grammar.

Then there exists an equivalent grammar G^ = (V^, T^, S,
P^)
that does not contain any useless variables or productions.
13
Theorem 6.1 (cont’d)
Proof:
•
Construct (V
1
, T, S, P
1
) such that V
1
contains only variables A
for which A ⇒* w ∈ T*.
1. Set V
1
to ∅.
2. Repeat until no more variables are added to V
1
:
For every A ∈ T for which P has a production of the form
A → x
1
x
2
x
n
(x
i

∈ T*∪V
1
)
add A to V
1
.
3.

Take P
1
as all the productions in P with symbols in (V
1
∪ T)*.
14
Theorem 6.1 (cont’d)
Proof:
•
Draw the variable dependency graph for G
1
and find all
variables that cannot be reached from S.
•
Remove those variables and the productions involving them.
•
Eliminate any terminal that does not occur in a useful
production.
⇒ G^ = (V^, T^, S, P^)

15
Removing λ-Productions

•
Any production of a context-free grammar of the form:
A → λ
is called a λ-production.
•
Any variable A for which the derivation:
A ⇒* λ
is possible is called nullable.
16
Example 6.5
S → aS
1
b S → aS
1
b | ab
S
1
→ aS
1
b | λ S
1
→ aS
1
b | ab
17
Theorem 6.2
Let G = (V, T, S, P) be a CFG such that λ ∉ L(G).
Then there exists an equivalent grammar G^ having no
λ-productions.
18

Theorem 6.2 (cont’d)
Proof:
•
Find the set V
N
of all nullable variables of G:
1. For all productions A → λ, put A into V
N
.
2. Repeat until no more variables are added to V
N
:
For all productions
B → A
1
A
2
A
n
(A
i
∈ V
N
)
add B to V
N
.
19
Theorem 6.2 (cont’d)
Proof:

•
For each production in P of the form:
A → x
1
x
2
x
m
(m ≥ 1, x
i
∈ V∪T)
put into P^ that production as well as all those generated by
replacing null variables with λ in all possible combination.
Exception: if all x
i
are nullable, then A → λ is not put into P^.
20
Example 6.5 (cont’d)
S → ABaC S → ABaC | BaC | AaC | ABa | aC | Aa |Ba | a
A → BC A → B | C | BC
B → b | λ B → b
C → D | λ C → D
D → d D → d
V
N
= {A, B, C}
21
Removing Unit-Productions
Any production of a context-free grammar of the form:
A → B

is called a unit-production.
22
Theorem 6.3
Let G = (V, T, S, P) be a CFG without λ-productions.
Then there exists an equivalent grammar G^ = (V, T, S, P^)
that does not have any unit-productions.
23
Theorem 6.3 (cont’d)
Proof:
1. Put into P^ all non-unit-productions of P.
2. Repeat until no more productions are added to P^:
For every A and B ∈ V such that A ⇒* B and B → y
1
| y
2
| | y
n
∈ P^
add A → y
1
| y
2
| | y
n
to P^.
24
Example 6.6
S → Aa | B
B → A | bb
A → a | bc | B

25
Example 6.7
S → Aa | B S → Aa
A → a | bc | B A → a | bc
B → A | bb B → bb
S ⇒* A S → a | bc | bb
S ⇒* B A → bb
A ⇒* B B → a | bc
B ⇒* A