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Table of contents
1.
Introduction
2.
Left factored grammar in compiler design
3.
Problems with Left Factored Grammar
4.
Left Factoring in Compiler Design
5.
Benefits of Left Factoring in Compiler Design
6.
Challenges in Left Factoring in Compiler Design
7.
Implementation of left factoring
8.
Comparison with other parsing techniques
9.
Frequently Asked Questions
9.1.
What is top-down parsing?
9.2.
What is left recursion in compiler design?
9.3.
What is the difference between left recursion and left factoring in compiler design?
9.4.
Why is left factoring a problem?
10.
Conclusion
Last Updated: Mar 27, 2024
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# Left Factoring in Compiler Design

Gaurav Gandhi
2 upvotes
Master Python: Predicting weather forecasts
Speaker
Ashwin Goyal
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## Introduction

Syntax analysis or parsing is the process of analyzing a string of symbols using grammar rules. It is done either in a bottom-up manner or a top-down manner. There are multiple problems that the Syntax analyzer faces while performing top-down parsing. One of which is left factoring.

This article will teach us about left factoring in compiler design and why it is important for us. We will also learn how to perform left factoring in our grammar.

So without any delay, let's dive deep into the topic.

## Left factored grammar in compiler design

Grammar in the following form will be considered to be having left factored-

``S â‡’ aX | aY | aZ``

S, X, Y, and are non-terminal symbols, and is terminal. So left, factored grammar is having multiple productions with the same prefix or starting with the same symbol. In the example, â‡’ aX, â‡’ aY, and â‡’ aZ are three different productions with the same nonâ€“terminal symbol on the left-hand side, and the productions have a common prefix a. Hence the above grammar is left-factored.

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## Problems with Left Factored Grammar

Left-factored grammar creates an ambiguous situation for top-down parsers. Top-down parsers cannot choose the production to derive the given string since multiple productions will have a common prefix. This creates an ambiguous situation for the top-down parser.

To tackle this situation, we need to convert the left-factored grammar into an equivalent grammar without any productions having a common prefix. This is done using left factoring in Compiler design.

## Left Factoring in Compiler Design

Left factoring is used to convert a left-factored grammar into an equivalent grammar to remove the uncertainty for the top-down parser. In left factoring, we separate the common prefixes from the production rule.

The following algorithm is used to perform left factoring in the grammar-

Suppose the grammar is in the form:

``A â‡’ Î±Î˛1 | Î±Î˛2 | Î±Î˛3 | â€¦â€¦ | Î±Î˛n | Îł ``

Where is a non-terminal and Î± is the common prefix.

We will separate those productions with a common prefix and then add a new production rule in which the new non-terminal we introduced will derive those productions with a common prefix.

``````A â‡’ Î±A`
A` â‡’ Î˛1 | Î˛2 | Î˛3 | â€¦â€¦ | Î˛n``````

The top-down parser can easily parse this grammar to derive a given string. So this is how left factoring in compiler design is performed on a given grammar.

## Benefits of Left Factoring in Compiler Design

Left factoring has multiple benefits associated with it. Some of the benefits of left factoring are-

• Error Detection: With the help of left factoring, error detection becomes easier since the prefix in the input string which caused the error while parsing can now be identified easily.

• The complexity of rules is decreased:  The complexity of complex and larger rules with common prefixes is decreased as we factor out the common prefix. This makes the grammar more readable, and the rules become simpler.

• Consistent Grammar: Left factoring helps make the grammar consistent since it follows predefined standards. It ensures that the grammar is well-defined and predictable.

## Challenges in Left Factoring in Compiler Design

Many challenges are faced while performing left factoring in the grammar. Some of the challenges are-

• Introduces ambiguity: Performing left factoring in grammar can sometimes give rise to ambiguity as it introduces new rules, making it possible for the parser to parse a string in multiple ways.

• Decrease in Parsing Efficiency: In left factoring, multiple new production rules are added by factoring out the common prefix. New non-terminals are also introduced by us. This increases the size of the grammar and hence makes it more complex.

• Identification of common prefixes: Sometimes, identifying common prefixes is difficult if the size of the grammar is already very large.

## Implementation of left factoring

Let's see the implementation of left factoring with the help of an example.

The given production rule is-

``````A â‡’ aB | aC | aD
B â‡’ b
C â‡’ c
D â‡’ d``````

A, B, C, and D are non-terminals, while a,b,c, and d are terminals.

We are going to separate those productions which have a common prefix. In this example, A â‡’ aB, A â‡’ aC, and A â‡’ aC are three such productions. We take the common prefix and create a new non-terminal A`.

A â‡’ aA` where  A` is the new non-terminal we introduced.

We then add a new production rule in which the new non-terminal we introduced will derive those productions with a common prefix.

``A` â‡’ B | C |  D``

Therefore the final grammar would be-

``````A â‡’ aA`
A` â‡’ B | C |  D
B â‡’ b
C â‡’ c
D â‡’ d``````

The top-down parser can easily parse this grammar to derive a given string. So this is how left factoring in compiler design is performed on a given grammar.

## Comparison with other parsing techniques

We have seen how and why left factoring is used in our grammar. Apart from left factoring, there are other parsing techniques as well. Some of them are-

• LL Parsing algorithm: It is one of the top-down parsing techniques. This algorithm maintains a parsing table for the given grammar, a stack, and a parser while parsing a string. It determines whether the given grammar/parsing table can form the specified string. If it cannot be produced, then we get an error. Left factoring in compiler design can improve efficiency in LL parsing algorithms.

• CYK algorithm: CYK (Chomsky Normal Form) is a basic algorithm for parsing context-free grammar (CFG). This algorithm checks if a string can be parsed using a given grammar. Left factoring reduces the size of the parsing table so that the parsing efficiency can be increased.

• LR Parsing algorithm: LR parsing is a bottom-up parsing technique for context-free grammar. The input string is parsed from left to right, producing a right-most derivation. It reduces the top-level grammar productions as it builds from the leaves. Left factoring minimizes the number of states in the LR parsing table to improve efficiency.

## Frequently Asked Questions

### What is top-down parsing?

Top-down parsing is one of the techniques in which the parser starts with the highest-level non-terminal symbol of the grammar and works downwards to derive the input string. It involves choosing a production rule for the current non-terminal symbol and recursively expanding it.

### What is left recursion in compiler design?

Left recursion in compiler design is an unwanted situation that sometimes arises in our grammar while parsing the syntax. If there is a production rule in which a non-terminal symbol derives itself as the first symbol on the right side, it is called left recursion.

### What is the difference between left recursion and left factoring in compiler design?

In left-factored grammar, productions have a common prefix, i.e., a grammar in the form â‡’ aX | aY | aZ. In contrast, in left recursive grammar, there is a production rule in which a non-terminal symbol derives itself as the first symbol on the right side. It will be in the form â‡’ Sa | Ab | cb, where is non-terminal.

### Why is left factoring a problem?

Left factoring is a problem because it creates ambiguity in understanding and translating programming code. It's like having sentences that start similarly but go in different directions, confusing the translation process.

## Conclusion

Congrats, Ninja!! You've learned what left factoring in compiler design is and why removing left factoring from our grammar is necessary. You have also learned how to remove it from your grammar so that it doesn't create any uncertainty for the top-down parser.

We have also added some FAQs related to the left factoring in compiler design.

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