Simplification of Context-Free Grammars

Simplification of Context-Free Grammars • Some useful substitution rules... Then there exists an equivalent grammar G^ = V^, T^, S, P^ that does not contain any useless variables or prod

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 -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^

Removing Useless Productions

S  aSb |  | A

A  aA

S  A is redundant as A cannot be transformed into a

terminal string.

Removing Useless Productions

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 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)*.

Theorem 6.1 (cont’d)

Proof:

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.

Example 6.5

S1  aS1b |  S1  aS1b | ab

Theorem 6.2

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

Then there exists an equivalent grammar G^ having no

-productions

Theorem 6.2 (cont’d)

Proof:

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

1 For all productions A   , put A into VN.

For all productions

B  A1A2 An (Ai  VN)

Theorem 6.2 (cont’d)

Proof:

• For each production in P of the form:

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

Exception: if all xi are nullable, then A   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 -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^.

Example 6.6

S  Aa | B

B  A | bb

A  a | bc | B

Theorem 6.4

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

Theorem 6.4 (cont’d)

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

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   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 G 1 = (V 1 , T, S, P 1 ).

• V 1 = V  {B a | a  T} P 1 = P  {B a  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 P 1 :

For each production

A  x 1 x2 xn (n  2, x i  TV)

add A  C 1 C 2 C n to P 1

where Ci = xi if xi  V or C i = Ba if xi = a

Theorem 6.5 (cont’d)

Proof:

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

• V^ = V 1

• 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

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   L(G)has an equivalent grammar G^ = (V^, T, S, P^) in Greibach normal form

Example 6.9

A2  A1A2 | b

A1  A2A2 | a