LL(1) parser

Let’s take a real-world example of interpreting a simple arithmetic expression in a calculator using an LL(1) parser. Imagine you have a small calculator program that can evaluate expressions involving addition and multiplication, such as:

3 + 5 * 2

The calculator uses an LL(1) parser to process and evaluate the input expression by building a parse tree. Here's how this works step by step.

Grammar Definition

For this simple calculator, we can define the grammar (syntax rules) to represent expressions involving numbers, addition, and multiplication. The grammar follows typical arithmetic precedence, where multiplication has higher precedence than addition:

E  → T E'          (E for Expression)
E' → + T E' | ε     (E' for Expression Prime)
T  → F T'          (T for Term)
T' → * F T' | ε     (T' for Term Prime)
F  → ( E ) | num    (F for Factor)

• E: Represents an entire expression, such as 3 + 5 * 2.
• E': Handles the continuation of the expression after a term, for example, dealing with the + in 3 + 5 * 2.
• T: Represents a term, such as 5 * 2.
• T': Handles the continuation of a term after a factor, for example, dealing with the * in 5 * 2.
• F: Represents a factor, which can be a number (e.g., 3, 5, or 2) or a parenthesized expression (e.g., (3 + 5)).

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Real-World Example: Parsing 3 + 5 * 2

Let’s see how an LL(1) parser would process the expression 3 + 5 * 2.

1. Lookahead Input:
   Input: 3 + 5 * 2 $
   Stack: E $ (Start with the start symbol E)

2. Expand E (Expression):

   • The parser sees E and applies the production E → T E'.
   • Stack: T E' $
   • Input: 3 + 5 * 2 $

3. Expand T (Term):

   • The parser sees T and applies T → F T'.
   • Stack: F T' E' $
   • Input: 3 + 5 * 2 $

4. Expand F (Factor):

   • The parser sees F and applies F → num (since 3 is a number).
   • Stack: num T' E' $
   • Input: 3 + 5 * 2 $

5. Match num:

   • The parser matches the num with 3.
   • Stack: T' E' $
   • Input: + 5 * 2 $

6. Expand T' (Term Prime):

   • The parser sees T' and applies the epsilon production T' → ε (because the lookahead is +, and T' doesn’t expand for +).
   • Stack: E' $
   • Input: + 5 * 2 $

7. Expand E' (Expression Prime):

   • The parser sees E' and applies the production E' → + T E' (since the lookahead is +).
   • Stack: + T E' $
   • Input: + 5 * 2 $

8. Match +:

   • The parser matches the +.
   • Stack: T E' $
   • Input: 5 * 2 $

9. Expand T (Term):

   • The parser sees T and applies T → F T' (since the lookahead is 5).
   • Stack: F T' E' $
   • Input: 5 * 2 $

10. Expand F (Factor):

    • The parser sees F and applies F → num (since 5 is a number).
    • Stack: num T' E' $
    • Input: 5 * 2 $

11. Match num:

    • The parser matches the num with 5.
    • Stack: T' E' $
    • Input: * 2 $

12. Expand T' (Term Prime):

    • The parser sees T' and applies T' → * F T' (since the lookahead is *).
    • Stack: * F T' E' $
    • Input: * 2 $

13. **Match ***:

    • The parser matches the *.
    • Stack: F T' E' $
    • Input: 2 $

14. Expand F (Factor):

    • The parser sees F and applies F → num (since 2 is a number).
    • Stack: num T' E' $
    • Input: 2 $

15. Match num:

    • The parser matches the num with 2.
    • Stack: T' E' $
    • Input: $

16. Expand T' (Term Prime):

    • The parser sees T' and applies the epsilon production T' → ε (because the lookahead is $, the end of the input).
    • Stack: E' $
    • Input: $

17. Expand E' (Expression Prime):

    • The parser sees E' and applies the epsilon production E' → ε.
    • Stack: $
    • Input: $

At this point, the stack and input both have $, indicating that the parsing is successful, meaning the input expression 3 + 5 * 2 is syntactically valid.

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Why LL(1) Parsers?

In this example, the LL(1) parser successfully processes the input by looking ahead one symbol at a time and predicting which rule to apply based on the current input token and the parser’s state. This technique is fast and efficient for simple grammars like arithmetic expressions.

Real-World Scenario:

This parsing technique is widely used in programming languages to parse arithmetic expressions, function calls, or other syntactically constrained constructs. For example, in your calculator app:

• You input 3 + 5 * 2.
• The parser uses an LL(1) strategy to break down the expression into its components (3, +, 5, *, 2).
• Based on the grammar rules, it ensures that the operations are executed in the correct order (multiplication before addition), and the expression is evaluated correctly to return the result:
  3 + (5 * 2) = 3 + 10 = 13

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QUIZ Questions

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Here are 10 multiple-choice questions (MCQs) on LL(1) Parsers suitable for GATE CSE preparation:

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1. What does the "1" in LL(1) stand for?

A. One leftmost derivation step
B. One lookahead symbol
C. One production rule
D. One terminal symbol

Answer: B. One lookahead symbol

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2. Which of the following is NOT a characteristic of an LL(1) parser?

A. It constructs a leftmost derivation of the input
B. It uses a bottom-up parsing strategy
C. It requires a single lookahead token to make parsing decisions
D. It generates a predictive parsing table

Answer: B. It uses a bottom-up parsing strategy

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3. Which of the following statements is true for an LL(1) grammar?

A. The grammar must be ambiguous
B. The grammar cannot have left recursion
C. The grammar allows arbitrary lookahead
D. The grammar must have right recursion only

Answer: B. The grammar cannot have left recursion

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4. What is the main reason left recursion must be eliminated in LL(1) parsers?

A. It reduces the size of the parsing table
B. Left recursion causes infinite recursion in top-down parsing
C. It improves the efficiency of parsing
D. Left recursion causes conflicts in the follow set

Answer: B. Left recursion causes infinite recursion in top-down parsing

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5. For a grammar to be LL(1), which condition must hold for every non-terminal A with productions $A → α_1 | α_2 | ... | α_n$?

A. $FIRST(α_i) \cap FIRST(α_j) = \emptyset$ for all $i \neq j$
B. $FOLLOW(A) \cap FIRST(α_i) = \emptyset$
C. $FIRST(α_i) \cap FOLLOW(A) = \emptyset$
D. $FIRST(A) \cap FOLLOW(A) = \emptyset$

Answer: A. $FIRST(α_i) \cap FIRST(α_j) = \emptyset$ for all $i \neq j$

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6. What is the purpose of the predictive parsing table in an LL(1) parser?

A. To store all possible productions for each non-terminal
B. To guide the parser in selecting the correct production based on the lookahead symbol
C. To resolve ambiguities in the grammar
D. To store the parsing stack during the parsing process

Answer: B. To guide the parser in selecting the correct production based on the lookahead symbol

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7. Which of the following statements is true regarding the FIRST set in an LL(1) parser?

A. FIRST set contains only terminal symbols
B. FIRST set is used to determine the follow set of non-terminals
C. FIRST set is used to handle epsilon (ε) productions
D. FIRST set contains the start symbol of the grammar

Answer: A. FIRST set contains only terminal symbols

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8. Given the following grammar, which of the following FIRST sets is correct for non-terminal A?

A → B C | ε
B → a | ε
C → b

A. FIRST(A) = {a, b, ε}
B. FIRST(A) = {a, ε}
C. FIRST(A) = {a, ε, b}
D. FIRST(A) = {a, b}

Answer: C. FIRST(A) = {a, ε, b}

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9. Which of the following grammars is NOT suitable for LL(1) parsing?

A. A → a A'
A' → b | ε
B. A → A a | b
C. S → A b
A → c | ε
D. A → B C
B → b
C → c | ε

Answer: B. A → A a | b

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10. What should be the entry in the LL(1) parsing table for non-terminal A and terminal 'b' if $FIRST(A)$ contains 'a', and $FOLLOW(A)$ contains 'b'?

A. Error
B. A → ε
C. A → a A
D. A → b

Answer: B. A → ε