Unit Production Removal in Context Free Grammar

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Context-free grammars sometimes need to be simplified. Simplification means removing unit productions, removing null productions and removing useless productions. In the previous chapter, we explained how to remove useless symbols from CFG. In this chapter, we will see the steps with examples for unit production removal from CFG.

Unit Production and its Removal

Simplifying a CFG involves eliminating redundant or unnecessary elements without altering the language generated by the grammar. Unit productions, a specific type of production rule, can be removed to achieve this simplification.

In context-free grammars, a unit production is a production rule of the form "A → B", where both 'A' and 'B' represent non-terminal symbols.

In simpler terms, a unit production replaces a non-terminal with another single non-terminal.

Unit productions introduce a level of indirection in the derivation process of a CFG. By removing them, we can achieve a more concise and efficient grammar without affecting the language it generates. This simplification proves beneficial for various parsing algorithms and grammatical analysis techniques.

Steps to Remove Unit Productions

The process of eliminating unit productions from a CFG involves systematic replacement and iteration. Let's break down the steps −

• Identifying Unit Productions − Start by analyzing the production rules of the given CFG. Identify all production rules that is in the format like "A → B", where A and B are non-terminals.
• Replacing Unit Productions − For each identified unit production "A → B", perform the following:
  • Find all productions of B − Check the grammar for all production rules where 'B' appears on the left-hand side (LHS). Let's represent these productions as "B → X", where 'X' can be a terminal symbol, a string of terminal and/or non-terminal symbols, or even the empty string (ε).
  • Introduce new productions for A − For every production rule "B → X" found, add a new production rule "A → X" to the grammar. This essentially bypasses the intermediate step of going through 'B'.
• Removing the Original Unit Production − Once we have added the new production rules, eliminate the original unit production "A → B" from the grammar.
• Iteration − Repeat Steps 1 to 3 until no more unit productions remain in the grammar. This iterative process ensures the removal of all indirect unit productions as well.

Example of Unit Production Removal in CFG

Now we will see the idea through a real example. Let's consider a CFG with the following productions −

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

$$\mathrm{X \: \rightarrow \: a}$$

$$\mathrm{Y \: \rightarrow \: Z\:|\:b}$$

$$\mathrm{Z \: \rightarrow \: M}$$

$$\mathrm{M \: \rightarrow \: N}$$

$$\mathrm{N \: \rightarrow \: a}$$

Identifying Unit Productions

We can identify three unit productions in this grammar −

$$\mathrm{Y \: \rightarrow \: Z}$$

$$\mathrm{Z \: \rightarrow \: M}$$

$$\mathrm{M \: \rightarrow \: N}$$

Removing M → N

We see that 'N' has a production "N → a".

Therefore, we add a new production "M → a" to the grammar.

We remove the unit production "M → N".

The grammar now becomes −

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

$$\mathrm{X \: \rightarrow \: a}$$

$$\mathrm{Y \: \rightarrow \: Z \: | \: b}$$

$$\mathrm{Z \: \rightarrow \: M}$$

$$\mathrm{M \: \rightarrow \: a}$$

$$\mathrm{N \: \rightarrow \: a}$$

Removing Z → M

'M' now has the production "M → a".

We add a new production: "Z → a".

We remove the unit production "Z → M".

The grammar now becomes −

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

$$\mathrm{X \: \rightarrow \: a}$$

$$\mathrm{Y \: \rightarrow \: Z \: | \: b}$$

$$\mathrm{Z \: \rightarrow \: a}$$

$$\mathrm{M \: \rightarrow \: a}$$

$$\mathrm{N \: \rightarrow \: a}$$

Removing Y → Z

'Z' now has the production "Z → A".

We add a new production "Y → A".

We remove "Y → Z".

The grammar now becomes −

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

$$\mathrm{X \: \rightarrow \: a}$$

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

$$\mathrm{Z \: \rightarrow \: a}$$

$$\mathrm{M \: \rightarrow \: a}$$

$$\mathrm{N \: \rightarrow \: a}$$

Final Grammar

At this point, we have removed all the unit productions. However, we notice that the non-terminals Z, M, and N are not reachable from the start symbol 'S'.

Therefore, we can remove these unreachable symbols and their corresponding productions, resulting in the final simplified grammar −

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

$$\mathrm{X \: \rightarrow \: a}$$

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

Conclusion

Removing unit productions is a fundamental step in simplifying context-free grammars. It is a part of CFG simplification with other steps like null production removal, etc. This process eliminates unnecessary steps in derivations, making the grammar more efficient for parsing and analysis without altering the language it generates.