Automata and Formal Language (chapter 6) ppt

Then there exists an equivalent grammar G^ = V^, T^, S, P^ that does not contain any useless variables or productions... • Remove those variables and the productions involving them.. • E

Simplification of Context-Free

Grammars

Quan Thanh Tho qttho@cse.hcmut.edu.vn

Simplification of Context-Free

Grammars

• Some useful substitution rules.

• Removing useless productions.

• Removing λ -productions.

• Removing unit-productions.

Some Useful Substitution Rules

G = (V, T, S, P)

A → x1Bx2 ∈ P

B → y1 | y2 | | yn ∈ P

L(G) = L(G^) G^ = (V, T, S, P^)

A → x1y1x2 | x1y2x2 | | x1ynx2 ∈ P^

A → yi | yiZ (i =1, m) ∈ P^

Z → xi | xiZ (i =1, n) ∈ P^

Removing Useless Productions

S → aSb | λ | A

A → aA

S → A is redundant as A cannot be transformed into a terminal string.

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

2 Repeat until no more variables are added to V1:

For every A ∈ T for which P has a production of the form

A → x1x2 xn (xi ∈ T* ∪ V1) add A to V1.

3 Take P as all the productions in P with symbols in (V ∪ T)*

Theorem 6.1 (cont’d)

Proof:

• Draw the variable dependency graph for G1 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^)

Example 6.5

S → aS1b S → aS1b | ab

S1 → aS1b | λ S1 → aS1b | ab

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

Theorem 6.2 (cont’d)

Proof:

• Find the set VN of all nullable variables of G:

1 For all productions A → λ , put A into VN.

2 Repeat until no more variables are added to VN:

For all productions

B → A1A2 An (Ai ∈ VN) add B to VN.

Theorem 6.2 (cont’d)

Proof:

• For each production in P of the form:

A → x1x2 xm (m ≥ 1, xi ∈ 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 xi are nullable, then A → λ is not put into P^

Removing Unit-Productions

Any production of a context-free grammar of the form:

A → B

is called a unit-production

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

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 → y1 | y2 | | yn ∈ P^ add A → y1 | y2 | | yn to P^.

Theorem 6.4

Let L be a context-free language that does not contain λ

Then there exists a CFG that generates L and does not have any useless productions, λ -productions, or unit-productions

Theorem 6.4 (cont’d)

Let L be a context-free language that does not contain λ

Then there exists a CFG that generates L and does not have any useless productions, λ -productions, or unit-productions

Proof:

1 Remove λ-productions

2 Remove unit-productions

3 Remove useless-productions

Two Important Normal Forms

• Chomsky normal form.

• Greibach normal form.

Chomsky Normal Form

A context-free grammar G = (V, T, S, P) is in Chomsky normal form iff all productions are of the form:

A → BC

or

A → a

where A, B, C ∈ V and a ∈ T.

Theorem 6.5

Any context-free grammar G = (V, T, S, P) such that λ ∉ L(G)has an equivalent grammar G^ = (V^, T, S, P^) in Chomsky normal form

Theorem 6.5 (cont’d)

Proof:

First, construct an equivalent grammar G1 = (V1, T, S, P1)

• V1 = V ∪ {Ba | a ∈ T} P1 = P ∪ {Ba → a | a ∈ T}

• Remove all terminals from productions of length > 1:

1 Put all productions A → a into P1.

2 Repeat until no more productions are added to P1:

For each production

A → x1x2 xn (n ≥ 2, xi ∈ T ∪ V) add A → C1C2 Cn to P1

• 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 → C1C2 Cn (n > 2) add A → C1D1, D1 → C2D2, , Dn-2 → Cn-1Dn to P^.

Example 6.8

S → ABa

A → aaB

B → aC

Greibach Normal Form

A context-free grammar G = (V, T, S, P) is in Greibach normal form iff all productions are of the form:

A → ax

where a ∈ T and x ∈ V*.

Theorem 6.6

Any context-free grammar G = (V, T, S, P) such that λ ∉ L(G)has an equivalent grammar G^ = (V^, T, S, P^) in Greibach normal form

Example 6.9

A2 → A1A2 | b

A1 → A2A2 | a