Simplification of CFG

As we have seen, various languages can efficiently be represented by a context-free grammar. All the grammar are not always optimized that means the grammar may consist of some extra symbols(non-terminal). Having extra symbols, unnecessary increase the length of grammar. Simplification of grammar means reduction of grammar by removing useless symbols. The properties of reduced grammar are given below:

1. Each variable (i.e. non-terminal) and each terminal of G appears in the derivation of some word in L.
2. There should not be any production as X → Y where X and Y are non-terminal.
3. If ε is not in the language L then there need not to be the production X → ε.

Removal of Useless Symbols

A symbol can be useless if it does not appear on the right-hand side of the production rule and does not take part in the derivation of any string. That symbol is known as a useless symbol. Similarly, a variable can be useless if it does not take part in the derivation of any string. That variable is known as a useless variable.

For Example:

T → aaB | abA | aaT  

A → aA  

B → ab | b  

C → ad  

In the above example, the variable 'C' will never occur in the derivation of any string, so the production C → ad is useless. So we will eliminate it, and the other productions are written in such a way that variable C can never reach from the starting variable 'T'.

Production A → aA is also useless because there is no way to terminate it. If it never terminates, then it can never produce a string. Hence this production can never take part in any derivation.

To remove this useless production A → aA, we will first find all the variables which will never lead to a terminal string such as variable 'A'. Then we will remove all the productions in which the variable 'B' occurs.

Elimination of ε Production

The productions of type S → ε are called ε productions. These type of productions can only be removed from those grammars that do not generate ε.

Step 1: First find out all nullable non-terminal variable which derives ε.

Step 2: For each production A → a, construct all production A → x, where x is obtained from a by removing one or more non-terminal from step 1.

Step 3: Now combine the result of step 2 with the original production and remove ε productions.

Example:

Remove the production from the following CFG by preserving the meaning of it.

S → XYX  

X → 0X | ε  

Y → 1Y | ε  

Solution:

Now, while removing ε production, we are deleting the rule X → ε and Y → ε. To preserve the meaning of CFG we are actually placing ε at the right-hand side whenever X and Y have appeared.

Let us take

S → XYX  

If the first X at right-hand side is ε. Then

S → YX  

Similarly if the last X in R.H.S. = ε. Then

S → XY  

If Y = ε then

S → XX  

If Y and X are ε then,

S → X  

If both X are replaced by ε

S → Y  

Now,

S → XY | YX | XX | X | Y  

Now let us consider

X → 0X  

If we place ε at right-hand side for X then,

X → 0  

X → 0X | 0  

Similarly Y → 1Y | 1

Collectively we can rewrite the CFG with removed ε production as

S → XY | YX | XX | X | Y  

X → 0X | 0  

Y → 1Y | 1  

Removing Unit Productions

The unit productions are the productions in which one non-terminal gives another non-terminal. Use the following steps to remove unit production:

Step 1: To remove X → Y, add production X → a to the grammar rule whenever Y → a occurs in the grammar.

Step 2: Now delete X → Y from the grammar.

Step 3: Repeat step 1 and step 2 until all unit productions are removed.

For example:

S → 0A | 1B | C  

A → 0S | 00  

B → 1 | A  

C → 01  

Solution:

S → C is a unit production. But while removing S → C we have to consider what C gives. So, we can add a rule to S.

S → 0A | 1B | 01  

Similarly, B → A is also a unit production so we can modify it as

B → 1 | 0S | 00  

Thus finally we can write CFG without unit production as

S → 0A | 1B | 01  

A → 0S | 00  

B → 1 | 0S | 00  

C → 01