What is Left Recursion and How to Eliminate It?

What is Left Recursion?

In computer science, left recursion is a concept that is used in formal language theory and compiler design. In basic words, it happens when a grammatical rule refers to itself directly or indirectly from the left side. Simply defined, it's like seeing into a mirror and seeing yourself holding another mirror, which mirrors you holding another mirror, creating an unending cycle.

Left recursion can cause parsing problems in the context of formal grammar, which specifies the syntax of computer languages. When parsing a string of code, the parser may become trapped in an infinite loop attempting to determine which grammatical rule to apply, resulting in inefficient and incorrect processing.

Let's break it down a bit. In a grammar rule like A → Aα | β, where A is a non-terminal symbol, and α and β are sequences of symbols, the left recursion is evident because A appears on the left side of the arrow and in the production rules. The problem here is that when parsing, the parser sees A and thinks it needs to expand A again, leading to an infinite loop.

Why to Eliminate Left Recursion?

Left recursion is a tough concept in formal grammar, so let's simplify it down into simpler words. Assume you're teaching the rules of a game to a friend. Now, if those rules cause your friend to wander in circles while attempting to follow the directions, that's a little like left recursion in grammar.

In the domain of programming languages and compilers, we wish to prevent left recursion. Why? It all comes down to understanding the rules that form the structure of a language. To demonstrate, consider the following hypothetical scenario.

Consider a grammar rule like this:

In this case, 'A' is a non-terminal symbol, whereas 'alpha' and 'beta' represent sequences of other symbols. The problem is that 'A' occurs at the start of the rule and also in the expansion on the right side. This results in a loop, similar to a never-ending merry-go-round.

When we try to utilize this rule to parse or interpret a statement in our language, we may become trapped in an infinite loop, similar to how your buddy may become trapped in an unending cycle of obeying game rules. To avoid this, we want to eliminate left recursion.

By rewriting the rule to look like:

we break the loop. "Start with 'beta' and if there are more, go to 'A' prime", the new rule states. If there is a 'A' prime, follow it with a 'alpha' and repeat or stop if there is nothing left." In this manner, we may make progress without going round and round in circles.

In technical terms, removing left recursion allows parsers (programs that analyze and comprehend code) to work more effectively. It makes determining the structure of a program written in a certain language easier. Consider it as simplifying the game rules so that your friend may comprehend and play without becoming trapped in an unending cycle. It's all about making language norms plain and simple to understand to ensure smooth communication between people and robots.

Types of Left Recursion in Compiler Design

Left recursion is a concept in compiler design that may be an ally or an opponent. It describes a case in which a non-terminal in a language generates a string that starts with itself. This may appear to be an innocuous self-reference, but left recursion can disrupt the parsing process, resulting in ambiguity and confusion.

Direct left recursion and indirect left recursion are the two main kinds of left recursion.

1. Direct Left Recursion:

This is the most straightforward form, in which a non-terminal refers to itself directly in its production rules. Consider the following grammar excerpt for better comprehension:

Here, 'A' produces a string that begins with 'A' in this case. The difficulty with straight left recursion is that it can trap a parser in an indefinite loop by extending the same non-terminal over and over again.

2. Indirect Left Recursion:

This method is a little more complex. It happens when a series of non-terminals refer to each other in a way that eventually leads back to the first non-terminal. Here's an illustration:

In this example, 'A' refers to itself indirectly via 'B', resulting in a loop that might nevertheless confuse the parser.

Left recursion is a problem for recursive descent parsers, which can become caught in an infinite loop when attempting to read such grammars. During the parsing step, methods such as left factoring and left recursion removal are used to overcome the issue.

Example:

Consider the following grammar with direct left recursion:

To eliminate direct left recursion, we can rewrite it as:

Here, Expr' and Term' are used to break the left recursion and enable unambiguous parsing.

How to Eliminate Left Recursion?

Left recursion is a typical difficulty faced by developers in parsing grammars that might hinder parser building. It happens when a non-terminal symbol forms a string that begins with itself, either directly or indirectly. This recursion can lead to infinite loops and ambiguity, thus it must be avoided. In this section of this article, we'll look at a basic algorithm and how it's implemented in different programming languages. Left recursion is a typical problem in parsing grammars that can hinder parser building. It happens when a non-terminal symbol forms a string that begins with itself, either directly or indirectly. This recursion can lead to infinite loops and ambiguity, thus it must be avoided. In this section, we'll look at a basic algorithm and how it's implemented in different programming languages.

Algorithm

1. Identify Left Recursion: Examine your grammar rules for any left-recursive creations. Left recursion often appears as a non-terminal symbol that refers to itself at the start of one of its outputs.
2. Transform Left-Recursive Rules: Create new non-terminals for each left-recursive rule to eliminate left recursion. As a rule A -> Aα | β, transform it into A -> βA', where A' represents the remaining alternatives and is defined as A' -> αA' | 'ε' (epsilon).
3. Update Original Rule: Replace the existing left-recursive rule with the new one. The left recursion is broken yet the original semantics are preserved.
4. Handle Indirect Left Recursion: If your grammar has indirect left recursion, ensure that the transformation is applied repeatedly until all left recursion is removed.

Implementation

Python

Java

Eliminating left recursion is an important step in creating clear and efficient parsers, ensuring that your language is correctly interpreted and processed by parsing tools.