1 simplification of context-free grammars some useful substitution rules. removing useless...

1

Simplification of Context-Free Grammars

• Some useful substitution rules.

• Removing useless productions.

• Removing -productions.

• Removing unit-productions.

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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^

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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

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Some Useful Substitution Rules

G = (V, T, S, P)

A Ax1 | Ax2 | ... | Axn P

A y1 | y2 | ... | ym P

L(G) = L(G^)

G^ = (V{Z}, T, S, P^)

A yi | yiZ (i =1, m) P^

Z xi | xiZ (i =1, n) P^

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Example

G = ({A, B}, {a, b}, A, P)

A Aa | aBc |

B Bb | ba

A aBc | aBcZ | Z | A aBc | aBcZ | Z |

Z a | aZ Z a | aZ

B Bb | ba B ba | baY

Y b | bY

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Removing Useless Productions

S aSb | | A

A aA

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

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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.

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Example

G = ({S, A, B}, {a, b}, S, P)

S A

A aA |

B bA

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Example

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

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Example

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

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Theorem

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.

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Theorem

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.

Proof: ?

13

Theorem

Proof:

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

for which A * w T*.

1. Set V1 to .

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 P1 as all the productions in P with symbols in (V1 T)*.

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Theorem

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^)

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Removing -Productions

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

A

is called a -production.

• Any variable A for which the derivation:

A *

is possible is called nullable.

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Example

S aS1b S aS1b | ab

S1 aS1b | S1 aS1b | ab

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Theorem

Let G = (V, T, S, P) be a CFG such that L(G).

Then there exists an equivalent grammar G^ having no

-productions.

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Theorem

Proof:

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

1. For all productions A , 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.

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Theorem

Proof:

• For each production in P of the form:

A x1x2... xm (m 1, xi VT)

put into P^ that production as well as all those generated by

replacing null variables with in all possible combination.

Exception: if all xi are nullable, then A is not put into P^.

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Example

S ABaC S ABaC | BaC | AaC | ABa | aC | Aa |Ba | a

A BC A B | C | BC

B b | B b

C D | C D

D d D d

VN = {A, B, C}

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Removing Unit-Productions

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

A B

is called a unit-production.

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Theorem

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

Then there exists an equivalent grammar G^ = (V, T, S, P^)

that does not have any unit-productions.

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Theorem

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^.

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Example

S Aa | B

B A | bb

A a | bc | B

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Theorem

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

Then there exists a CFG that generates L and does not have

any useless productions, -productions, or unit-productions.

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Theorem

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

Then there exists a CFG that generates L and does not have

any useless productions, -productions, or unit-productions.

Proof:

1. Remove -productions.

2. Remove unit-productions.

3. Remove useless-productions

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Two Important Normal Forms

• Chomsky normal form.

• Greibach normal form.

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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.

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Theorem

Any context-free grammar G = (V, T, S, P) such that L(G)

has an equivalent grammar G^ = (V^, T, S, P^) in Chomsky

normal form.

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TheoremProof:

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

• V1 = V {Ba | a T} P1 = {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 TV)

add A C1C2... Cn to P1

where Ci = xi if xi V or Ci = Ba if xi = a

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Theorem

Proof:

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 C1C2... Cn (n 2)

add A C1D1, D1 C2D2, ... , Dn-2 Cn-1Dn to P^.

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Example

S ABa

A aaB

B aC

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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*.

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Theorem

Any context-free grammar G = (V, T, S, P) such that L(G)

has an equivalent grammar G^ = (V^, T, S, P^) in Greibach

normal form.

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TheoremProof:

• Rewrite the grammar in Chomsky normal form.

• Relabel variables A1, A2, ..., An.

• Rewrite the grammar so that all productions have one of the following forms:

Ai Ajxj (j > i)

Zi Ajxj (j n, Zi introduced to eliminate left recursion)

Ai axi (a T and xi V*)

• Start from An axn to derive Greibach productions.

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Example

A2 A1A2 | b

A1 A2A2 | a

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Homework

• Exercises: 3, 4, 5, 6, 7, 8, 17, 22 of Section 6.1 - Linz’s book.

• Exercises: 2, 3, 4, 6, 9, 10, 11 of Section 6.2 - Linz’s book.

• Presentations: Section 6.3 and Section 7.4.

1 simplification of context-free grammars some useful substitution rules. removing useless...

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