Introduction
As we know, various languages can efficiently be represented by contextfree Grammar (CFGs). The Grammar is not always optimized, consisting of some extra symbols(nonterminal). Having additional symbols unnecessarily increases the length of Grammar.
By simplifying ContextFree Grammars, we can remove all the unnecessary symbols while maintaining the converted grammar equivalent to the original Grammar. If two grammars produce the same language, they are equivalent.
Simplifying Grammars is required to convert them to Normal forms later.In this article we will cover Simplification of CFG(ContextFree Grammars).
Also see, Turing Machine in TOC.
What is the Simplification of CFG?
It is possible in a CFG that the derivation of strings does not require the use of all the production rules and symbols. Moreover, there might be some unit and null productions. The term "simplification of CFGs" refers to the removal of certain productions and symbols.
ContextFree Grammar can be made simpler by removing all the extraneous symbols while yet preserving a converted grammar that is equivalent to the original Grammar.
 Each variable (i.e., nonterminal) and terminal of G is used to derive some word in language L.
 There should not be any production like X â†’ Y when X and Y are nonterminal.

If Îµ is not in L, then the production X â†’ Îµ is unnecessary.
Letâ€™s discuss all the 3 types of reduction processes in detail.
 Useless productions
 Îµ productions
 Unit productions
Removal of Useless Symbols
If a symbol does not exist on the righthand side of the production rule and does not participate in the derivation of any string, that symbol is considered a useless symbol. Similarly, if a variable does not participate in the derivation of any string, it is useless.
Example:
For example, simplify the belowgiven Grammar G:
S â†’ aaB  abA  aaS
A â†’ aA
B â†’ ab  b
C â†’ ae
In the above example, first, find the useless productions,
 The variable 'C' will never appear in the derivation of any string, so the production C > ae is useless. Therefore, we will remove it, and the other productions are written so that variable C can never reach from the initial variable 'T.'
 Production A > aA is also useless because there is no way to end it. If it never ends, it can never produce a string. As a result, this production can never participate in any derivation.
The process to remove the above found useless productions:
 To eliminate useless productions like A > aA, we first find all variables that will never lead to a terminal string, such as 'A.' We remove all of the productions in which variable 'B' appears. As a result, the modified grammar:
S â†’ aaB  aaS
B â†’ ab  b
C â†’ ae
2. Then we try to find all the variables that can never be reached from the initial variable S, such as 'C.' We remove all of the productions in which variable 'C' appears.
The grammar G is now free of all the useless productions.
S â†’ aaB  aaS
B â†’ ab  b
Also read  Arden's theorem