sliek môn máy tự động chương 6 simplication of context-free

Simplification of Context-Free GrammarsSome useful substitution rules.. Chapter 6: Simplification of Context-Free Grammars... Removing Useless ProductionsG = V, T, S, P A production is u

Chapter 6: Simplification of Context-Free

Grammars

November 13, 2009

Simplification of Context-Free Grammars

Some useful substitution rules

Removing useless productions

Removing λ-productions

Removing unit-productions

Chapter 6: Simplification of Context-Free Grammars

Some Useful Substitution Rules

A → a | aaA | abbAc | abbc

Chapter 6: Simplification of Context-Free Grammars

Removing Useless Productions

S → aSb | λ | A

A → aA

S → A is redundant as Acannot be transformed into aterminal string

Removing Useless Productions

G = (V, T, S, P)

A production is useless if it involves anyuseless variable

Chapter 6: Simplification of Context-Free Grammars

Theorem 6.1 - Linz ’s book

Theorem

Let G = (V, T, S, P) be a context-free grammar Then there

contain any useless variables or productions

Chapter 6: Simplification of Context-Free Grammars

Proof of theorem 6.1 - Linz ’s book

Construct (V1, T, S, P1) such that V1 contains only variables

1 Set V1to ∅.

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 P1 as all the productions in P with symbols in (V1∪ T ) ∗

Proof of theorem 6.1 - Linz ’s book

variables that cannot be reached from S

Remove those variables and the productions involving them.Eliminate any terminal that does not occur in a usefulproduction

⇒ G∗ = (V∗, T∗, S, P∗)

Chapter 6: Simplification of Context-Free Grammars

Proof of theorem 6.1 - Linz ’s book

Chapter 6: Simplification of Context-Free Grammars

Theorem 6.2 - Linz ’s book

Theorem

Let G = (V, T, S, P) be a CFG such that λ ∈ L(G) Then there

Proof of theorem 6.2 - Linz ’s book

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

Chapter 6: Simplification of Context-Free Grammars

Proof of theorem 6.2 - Linz ’s book

For each production in P of the form:

A → x1x2 xm (m ≥ 1, xi ∈ V ∪ T )

replacing null variables with λin all possible combination

Exception: if all xi are nullable, then A → λ is not put into

P∗

Removing Unit-Productions

Definition

Theorem 6.3 - Linz ’s book

Theorem

Let G = (V, T, S, P) be a CFG without λ-productions Then there

have any unit-productions

Chapter 6: Simplification of Context-Free Grammars

Proof of theorem 6.3 - Linz ’s book

For every A and B ∈ V such that

A ⇒∗B and B → y1| y2| | yn ∈ P ∗

add A → y 1 | y 2 | | y n to P∗.

Example 6.7

S → Aa | B

A → a | bc | B

B → A | bb

Theorem 6.4 - Linz ’s book

Theorem

Let L be a context-free language that does not contain λ Thenthere exists a CFG that generates L and does not have any uselessproductions, λ-productions, or unit-productions

Proof of theorem 6.4 - Linz ’s book

Two Important Normal Forms

Chomsky normal form

Greibachnormal form

Chomsky Normal Form

Theorem 6.5 - Linz ’s book

Theorem

normal form

Proof of theorem 6.5 - Linz ’s book

Remove all terminals from productions of length > 1:

1 Put all productions A → a into P 1

2 Repeat until no more productions are added to P 1 :

For each production

A → x 1 x 2 x n (n ≥ 2, x i ∈ T ∪ V ) add A → C 1 C 2 C n to P 1

where C i = x i if x i ∈ V or C i = B a if x i = a

Chapter 6: Simplification of Context-Free Grammars

Proof of theorem 6.5 - Linz ’s book

Construct G∗ = (V∗, T, S, P∗) from G1 = (V1, T, S, P1)

V∗= V1

Reduce the length of the right sides of the productions:

1 Put all productions A → a and A → BC into P∗.

2 Repeat until no more productions are added to P∗:

For each production A → C 1 C 2 C n (n ≥ 2) add A → C 1 D 1 , D 1 → C 2 D 2 , , D n−2 → C n−1 D n to P∗.

Greibach Normal Form

Theorem 6.6 - Linz ’s book

Theorem

normal form

Chapter 6: Simplification of Context-Free Grammars

Proof of theorem 6.6 - Linz ’s book

Rewrite the grammar in Chomsky normal form

Exercises: 3, 4, 5, 6, 7, 8, 17, 22 of Section 6.1 - Linzs book

Exercises: 2, 3, 4, 6, 9, 10, 11 of Section 6.2 - Linzs book