Left factoring in Compiler Design - Full Explanation

Introduction

When a grammar has choices that start with the same symbols, it can confuse a parser about which path to take. Left factoring helps solve this problem by reorganizing the grammar so the parser can decide the correct choice more easily.

Explanation

Problem of Common Prefixes

Sometimes, grammar rules have alternatives that begin with the same sequence of symbols. This makes it hard for a parser to pick the right alternative without looking ahead too far. The parser gets stuck because it cannot decide which path to follow immediately.

Common prefixes in grammar rules cause ambiguity for parsers.

What Left Factoring Does

Left factoring rewrites grammar rules to extract the common beginning parts into a single rule. Then, the parser can first match the shared part and later decide between the different options. This makes the grammar easier to parse with simple lookahead.

Left factoring removes common prefixes by factoring them out into a shared rule.

How to Apply Left Factoring

Identify the common prefix in the alternatives of a rule. Create a new rule that starts with this prefix, followed by a choice between the remaining parts. Replace the original alternatives with this new factored rule. This process can be repeated if needed.

Left factoring involves creating new rules to separate common prefixes from differing suffixes.

Benefit for Parsers

By left factoring, parsers can decide which alternative to choose by looking at fewer symbols ahead. This reduces complexity and helps build efficient top-down parsers like recursive descent parsers.

Left factoring simplifies parsing decisions and improves parser efficiency.

Real World Analogy

Imagine you are at a restaurant menu where two dishes start with the same first ingredient, like 'Chicken Curry' and 'Chicken Salad'. Instead of listing both fully, the menu groups them under 'Chicken' and then shows the different ways it can be prepared. This helps you quickly see the common part and then choose the variation.

Problem of Common Prefixes → Two dishes starting with 'Chicken' causing confusion about which one to pick

What Left Factoring Does → Grouping dishes under 'Chicken' to show shared ingredient first

How to Apply Left Factoring → Listing 'Chicken' once, then offering choices like 'Curry' or 'Salad'

Benefit for Parsers → Easier and faster decision making when ordering food

Diagram

Diagram

Original Rule:
A → αβ | αγ

Left Factored Rule:
A → αA'
A' → β | γ

Shows how a grammar rule with common prefix α is rewritten by left factoring.

Key Facts

Left factoring → A grammar transformation that removes common prefixes from alternatives to aid parsing.

Common prefix → A sequence of symbols that appears at the start of multiple alternatives in a grammar rule.

Top-down parser → A parser that builds the parse tree from the start symbol down to the leaves.

Lookahead → The number of symbols a parser examines ahead to decide which rule to apply.

Recursive descent parser → A simple top-down parser that uses a set of recursive procedures to process the input.

Common Confusions

Left factoring changes the language generated by the grammar.

Left factoring changes the language generated by the grammar. Left factoring only rewrites the grammar without changing the language it describes; it preserves the same strings.

Left factoring is only needed for bottom-up parsers.

Left factoring is only needed for bottom-up parsers. Left factoring is mainly used to help top-down parsers, especially recursive descent parsers, handle choices with common prefixes.

Summary

Left factoring solves the problem of grammar rules with common prefixes that confuse parsers.

It rewrites rules by extracting the shared beginning part and separating the differing endings.

This makes parsing decisions simpler and helps build efficient top-down parsers.

Practice

1. What is the main purpose of left factoring in compiler design?

easy

A. To remove common prefixes from grammar rules to simplify parsing

B. To eliminate left recursion from grammar rules

C. To optimize the speed of code execution

D. To convert grammar rules into machine code

Solution

1. Step 1: Understand left factoring concept

   Left factoring is used to remove common prefixes from grammar rules to make parsing decisions easier.

2. Step 2: Differentiate from other grammar techniques

   Unlike eliminating left recursion or code optimization, left factoring focuses on simplifying grammar by factoring out common starts.

3. Final Answer:

   To remove common prefixes from grammar rules to simplify parsing -> Option A

4. Quick Check:

   Left factoring = remove common prefixes [OK]

Hint: Left factoring removes common starts in grammar rules [OK]

Common Mistakes:

• Confusing left factoring with left recursion elimination
• Thinking it optimizes runtime code
• Assuming it generates machine code

2. Which of the following is the correct left factored form of the grammar rule:
A -> a b | a c?

easy

A. A -> b | c

B. A -> (a b) | c

C. A -> a b c

D. A -> a (b | c)

Solution

1. Step 1: Identify common prefix in rules

   Both alternatives start with 'a', so 'a' is the common prefix.

2. Step 2: Factor out the common prefix

   Rewrite as A -> a (b | c) to factor out 'a'.

3. Final Answer:

   A -> a (b | c) -> Option D

4. Quick Check:

   Common prefix 'a' factored out [OK]

Hint: Factor out the common start before alternatives [OK]

Common Mistakes:

• Not factoring out the common prefix 'a'
• Combining alternatives incorrectly
• Removing essential parts of the rule

3. Given the grammar rule:
S -> if E then S else S | if E then S
What is the left factored form?

medium

A. S -> if E then S (else S | ε)

B. S -> if E then S else S | if E then S

C. S -> if (E then S else S | E then S)

D. S -> if E then (S else S | S)

Solution

1. Step 1: Identify common prefix in alternatives

   Both alternatives start with if E then S.

2. Step 2: Factor out the common prefix

   Rewrite as S -> if E then S (else S | ε), where ε means empty (no else).

3. Final Answer:

   S -> if E then S (else S | ε) -> Option A

4. Quick Check:

   Common prefix factored with optional else [OK]

Hint: Factor out full common start, use ε for optional parts [OK]

Common Mistakes:

• Not factoring out the entire common prefix
• Misplacing parentheses
• Ignoring the optional else part

4. Identify the error in the following left factoring attempt:
Original: A -> a b | a c d
Attempted: A -> a (b | c d)

medium

A. Incorrect grouping; 'c d' should be separated

B. Correct left factoring, no error

C. Missing a new rule for 'c d'

D. Should factor out 'a b' instead of 'a'

Solution

1. Step 1: Analyze the original and attempted factoring

   Original rules share common prefix 'a'. Alternatives are 'b' and 'c d'.

2. Step 2: Verify correctness

   The attempt A -> a (b | c d) is correct: it generates a b and a c d. Sequences like 'c d' are valid in alternatives.

3. Final Answer:

   Correct left factoring, no error -> Option B

4. Quick Check:

   Same language generated [OK]

Hint: Check if grouped parts are single tokens or sequences [OK]

Common Mistakes:

• Confusing left factoring with left recursion elimination
• Thinking it optimizes runtime code
• Assuming it generates machine code

5. Given the grammar:
Expr -> Expr + Term | Expr - Term | Term
Apply left factoring to this grammar.

hard

A. Expr -> Term + Term | Term - Term

B. Expr -> (Expr + Term) | (Expr - Term) | Term

C. Expr -> Term Expr'; Expr' -> (+ | -) Term Expr' | ε

D. Expr -> Term ( + Term | - Term )

Solution

1. Step 1: Identify left recursion and common prefix

   The rule is left recursive: Expr -> Expr + Term | Expr - Term | Term.

2. Step 2: Remove left recursion and apply left factoring

   Rewrite as:
   Expr -> Term Expr'
Expr' -> (+ | -) Term Expr' | ε to factor out common prefix and remove recursion.

3. Final Answer:

   Expr -> Term Expr'; Expr' -> (+ | -) Term Expr' | ε -> Option C

4. Quick Check:

   Left recursion removed and common prefix factored [OK]

Hint: Remove left recursion first, then factor common prefixes [OK]

Common Mistakes:

• Not removing left recursion before factoring
• Incorrectly grouping alternatives
• Ignoring the ε (empty) option