Reduction of Context-Free Grammars (CFG)

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In this chapter, we will explain how to make the context free grammars simpler. As we know, not all CFGs are created equal. Some may contain unnecessary or redundant parts that make them harder to work with. In this article, we will see how to simplify CFGs by removing these unnecessary elements. This process is called reduction or simplification of context-free grammars. Here, we will go through some examples for a better understanding.

Why Do We Need to Simplify of Context-Free Grammars?

We must understand why we need simplification of contextfree grammars. Simplifying CFGs is important for several reasons −

• It makes the grammar easier to understand and work with.
• It prepares the grammar for conversion into special forms, like Chomsky Normal Form.
• It can make parsing and other algorithms more efficient.

Types of Redundant Productions

There are three main types of productions (rules) in CFGs that we often want to remove:

• Useless productions
• ε (epsilon) productions
• Unit productions

Let us look at each of these in detail.

Removing Useless Productions

Useless Productions are rules in the grammar that can never be used to generate a string in the language. They come in two characteristics −

• Productions that can never be reached from the start symbol
• Productions that can never lead to a string of terminals

Example of Useless Productions

Let us look at an example grammar −

$$\mathrm{S\: \rightarrow\: abS\: | \: abA \: | \: abB}$$

$$\mathrm{A\: \rightarrow\: cd}$$

$$\mathrm{B\: \rightarrow\: aB}$$

$$\mathrm{C\: \rightarrow\: dc}$$

In this grammar, the production C → dc is useless because C can never be reached from the start symbol S.

The production B → aB is useless because it can never terminate (lead to a string of only terminals).

How to Remove Useless Productions?

To remove useless productions, we follow these steps −

• Find all variables that can derive terminal strings.
• Remove productions with variables that can't derive terminal strings.
• Find all variables reachable from the start symbol.
• Remove productions with variables not reachable from the start symbol.

After applying these steps, our grammar becomes −

$$\mathrm{S\: \rightarrow\: abS \: | \: abA}$$

$$\mathrm{A\: \rightarrow\: cd}$$

Removing ε Productions

The ε productions (also called lambda or null productions) are rules that allow a variable to be replaced by nothing. They look like A → ε.

Example of ε Productions

Consider this grammar −

$$\mathrm{S \: \rightarrow\: ABCd}$$

$$\mathrm{A \: \rightarrow\: BC}$$

$$\mathrm{B \: \rightarrow\: bB \: | \: \epsilon}$$

$$\mathrm{C \: \rightarrow\: cC \: | \: \epsilon}$$

Here, both B and C have ε productions.

How to Remove ε Productions?

To remove ε productions:

• Find all "nullable" variables (variables that can derive ε).
• For each production, create new productions by optionally removing nullable variables.
• Remove the original ε productions.

After applying these steps, our grammar becomes −

$$\mathrm{S \: \rightarrow\: ABCd \: | \: ABd \: | \: ACd \: | \: BCd \: | \: Ad \: | \: Bd \: | \: Cd \: | \: d}$$

$$\mathrm{A \: \rightarrow\: BC \: | \: B \: | \: C}$$

$$\mathrm{B \: \rightarrow\: bB \: | \: b}$$

$$\mathrm{C \: \rightarrow\: cC \: | \: c}$$

Removing Unit Productions

Finally we have the question, what are Unit Productions? The unit productions are rules where a variable derives just another single variable, like A -> B.

Example of Unit Productions

Consider this grammar −

$$\mathrm{S \: \rightarrow\: Aa \: | \: B}$$

$$\mathrm{A \: \rightarrow\: b \: | \: B}$$

$$\mathrm{B \: \rightarrow\: A \: | \: a}$$

Here, S → B, A → B, and B → A are unit productions.

How to Remove Unit Productions

To remove the unit productions we need to follow the steps.

• Add all non-unit productions to the new grammar.
• For each variable A, find all variables B such that A* ⇒ B (A derives B in zero or more steps).
• For each such pair (A, B), add A → x for each non-unit production B → x in the original grammar.

After applying these steps, our grammar becomes −

$$\mathrm{S \: \rightarrow\: Aa \: | \: b \: | \: a}$$

$$\mathrm{A \: \rightarrow\: b \: | \: a}$$

From the overall process, when simplifying a CFG, it's important to follow these steps in order −

• Remove ε productions
• Remove unit productions
• Remove useless productions

Following this order ensures that we get the correct result.

Conclusion

In this chapter, we explained the process of simplifying context-free grammars. We covered the three types of redundant productions: useless productions, ε productions, and unit productions.

Simplifying CFGs is an important step in working with formal languages. It makes our grammars cleaner, more efficient, and ready for further transformations.