Left recursion elimination in Compiler Design - Step-by-Step Execution

Concept Flow - Left recursion elimination

Start with grammar

↓

Detect left recursion

↓

Rewrite rules

↓

Create new non-terminal

↓

Replace original rules

↓

Check no left recursion

↓

Done

The process starts by identifying left recursion in grammar rules, then rewriting them to remove it by introducing new rules without left recursion.

Execution Sample

Compiler Design

A -> A alpha | beta
Rewrite to:
A -> beta A'
A' -> alpha A' | ε

This code shows how a left-recursive rule is rewritten to remove left recursion by introducing a new non-terminal A'.

Analysis Table

Step	Action	Rule Before	Rule After	Notes
1	Identify left recursion	A -> A alpha | beta	Same	A is left-recursive because it calls itself first
2	Create new non-terminal A'	N/A	A' -> alpha A' | ε	New rule to handle recursion
3	Rewrite original rule	A -> A alpha | beta	A -> beta A'	Original rule rewritten to start with beta
4	Check for left recursion	A -> beta A'	No left recursion	Left recursion eliminated
5	Process complete	N/A	N/A	Grammar ready for parsing

💡 Left recursion removed by rewriting rules with new non-terminal

State Tracker

Variable	Start	After Step 2	After Step 3	Final
Grammar Rules	A -> A alpha | beta	A' -> alpha A' | ε added	A -> beta A' rewritten	No left recursion

Key Insights - 3 Insights

Why do we need to create a new non-terminal like A'?

Because directly rewriting the original rule without a new non-terminal would still cause left recursion. The new non-terminal helps to factor out the recursion safely, as shown in step 2 and 3 of the execution table.

What does the ε (epsilon) represent in the new rule?

ε represents the empty string, allowing the recursion to stop. This is important to avoid infinite loops and is shown in the new rule A' -> alpha A' | ε in step 2.

How do we know left recursion is fully removed?

After rewriting, the original non-terminal no longer calls itself first. Step 4 confirms no left recursion remains by checking the rewritten rule.

Visual Quiz - 3 Questions

Look at the execution table, at which step is the new non-terminal A' introduced?

A Step 2

B Step 1

C Step 3

D Step 4

💡 Hint

Check the 'Action' and 'Rule After' columns in the execution table for when A' first appears.

According to the variable tracker, what is the state of the grammar after step 3?

A Original left-recursive rule unchanged

B Original rule rewritten to start with beta and new non-terminal added

C New non-terminal added but original rule unchanged

D Left recursion still present

💡 Hint

Look at the 'After Step 3' column in the variable tracker for the grammar rules.

If we remove the ε option from A' -> alpha A' | ε, what would happen?

A Grammar would become ambiguous

B Left recursion would still be eliminated

C Grammar would allow infinite recursion

D Parsing would be faster

💡 Hint

Refer to the key moment explaining the role of ε in stopping recursion.

Concept Snapshot

Left recursion elimination removes rules where a non-terminal calls itself first.
Rewrite A -> A alpha | beta as:
A -> beta A'
A' -> alpha A' | ε
This removes infinite recursion and makes grammar suitable for top-down parsing.

Full Transcript

Left recursion elimination is a process in compiler design to rewrite grammar rules that call themselves first, which causes problems in parsing. The process starts by identifying such rules, then creating a new non-terminal to factor out the recursion. The original rule is rewritten to start with a non-recursive option, followed by the new non-terminal. The new non-terminal handles the recursive part and includes an empty string option to stop recursion. This ensures the grammar can be parsed by top-down parsers without infinite loops. The execution table shows each step of this rewriting, and the variable tracker shows how the grammar changes. Key moments clarify why the new non-terminal and epsilon are necessary. The quiz tests understanding of these steps and their effects.

Practice

1. What is the main reason to eliminate left recursion in a grammar?

easy

A. To simplify bottom-up parsing

B. To make the grammar ambiguous

C. To increase the number of production rules

D. To prevent infinite loops in top-down parsers

Solution

1. Step 1: Understand left recursion impact

   Left recursion causes a parser to repeatedly call the same rule without consuming input, leading to infinite loops in top-down parsers.

2. Step 2: Identify the goal of elimination

   Eliminating left recursion removes this infinite loop problem, making the grammar usable for top-down parsing methods like recursive descent.

3. Final Answer:

   To prevent infinite loops in top-down parsers -> Option D

4. Quick Check:

   Left recursion causes infinite loops = Prevent infinite loops [OK]

Hint: Left recursion causes infinite loops; remove it for top-down parsing [OK]

Common Mistakes:

• Thinking left recursion improves parsing speed
• Confusing left recursion with ambiguity
• Believing bottom-up parsers require left recursion elimination

2. Which of the following is the correct way to start eliminating immediate left recursion from a rule A -> A α | β?

easy

A. Rewrite as A -> β A' and A' -> α A' | ε

B. Rewrite as A -> α A | β

C. Rewrite as A -> β | α A' and A' -> ε

D. Rewrite as A -> A' β and A' -> α | ε

Solution

1. Step 1: Identify immediate left recursion pattern

   The rule A -> A α | β has immediate left recursion because A appears on the left side of its own production.

2. Step 2: Apply standard elimination method

   Rewrite as A -> β A' and A' -> α A' | ε to remove left recursion by introducing a new non-terminal A'.

3. Final Answer:

   Rewrite as A -> β A' and A' -> α A' | ε -> Option A

4. Quick Check:

   Immediate left recursion elimination uses new non-terminal A' [OK]

Hint: Use new non-terminal and right recursion to remove immediate left recursion [OK]

Common Mistakes:

• Not introducing a new non-terminal
• Swapping α and β incorrectly
• Leaving left recursion in the rewritten rules

3. Given the grammar rule E -> E + T | T, after eliminating left recursion, what is the correct equivalent set of productions?

medium

A. E -> T E'E' -> + T E' | ε

B. E -> E' + TE' -> T | ε

C. E -> T + E'E' -> T | ε

D. E -> T | E' + TE' -> + T | ε

Solution

1. Step 1: Identify left recursion in E -> E + T | T

   The rule is immediately left recursive because E appears on the left side of its own production.

2. Step 2: Apply left recursion elimination formula

   Rewrite as E -> T E' and E' -> + T E' | ε where E' is a new non-terminal handling the recursion on the right side.

3. Final Answer:

   E -> T E' and E' -> + T E' | ε -> Option A

4. Quick Check:

   Left recursion removed by new non-terminal E' [OK]

Hint: Rewrite left recursion as right recursion with new non-terminal [OK]

Common Mistakes:

• Placing '+' before T incorrectly
• Not adding epsilon (ε) to new rule
• Mixing order of productions

4. Consider the grammar rule S -> S a | b. Which of the following corrected versions properly eliminates left recursion?

medium

A. S -> S' bS' -> a | ε

B. S -> b | a S'S' -> S | ε

C. S -> b S'S' -> a S' | ε

D. S -> b a S'S' -> S | ε

Solution

1. Step 1: Identify immediate left recursion in S -> S a | b

   The rule is left recursive because S appears on the left side of its own production.

2. Step 2: Rewrite to remove left recursion

   Rewrite as S -> b S' and S' -> a S' | ε introducing new non-terminal S' to handle recursion on the right side.

3. Final Answer:

   S -> b S' and S' -> a S' | ε -> Option C

4. Quick Check:

   Left recursion removed by new non-terminal S' [OK]

Hint: New non-terminal after base case removes left recursion [OK]

Common Mistakes:

• Placing recursion on the left side again
• Not including epsilon (ε) in new rule
• Swapping terminals and non-terminals incorrectly

5. Given the indirect left recursion in the grammar:
A -> B a | c
B -> A b | d
Which of the following is the correct first step to eliminate left recursion?

hard

A. Remove B from the grammar

B. Substitute B in A's productions to get immediate left recursion

C. Replace B with d in all rules

D. Directly rewrite A as A -> c A' and A' -> a A' | ε

Solution

1. Step 1: Identify indirect left recursion

   Here, A calls B and B calls A, causing indirect left recursion.

2. Step 2: Substitute B's productions into A

   Replace B in A -> B a | c with A b | d to get A -> A b a | d a | c, creating immediate left recursion.

3. Step 3: Eliminate immediate left recursion

   Now apply immediate left recursion elimination on A.

4. Final Answer:

   Substitute B in A's productions to get immediate left recursion -> Option B

5. Quick Check:

   Indirect recursion -> substitution -> immediate recursion [OK]

Hint: Convert indirect to immediate recursion by substitution first [OK]

Common Mistakes:

• Trying to eliminate without substitution
• Removing non-terminals incorrectly
• Ignoring indirect recursion