Parse trees and derivations in Compiler Design - Mini Project: Build & Apply

Understanding Parse Trees and Derivations

📖 Scenario: You are learning how programming languages are understood by computers. One key idea is how sentences in a language are broken down into parts using parse trees. These trees show how a sentence is built step-by-step from basic rules called grammar productions. This helps computers check if sentences are correct and understand their meaning.

🎯 Goal: Build a simple parse tree for a small sentence using a given grammar. Then, write the steps (derivations) that show how the sentence is formed from the grammar rules.

📋 What You'll Learn

Create a dictionary representing grammar rules with exact keys and values

Add a variable for the sentence to be parsed

Write code to build a parse tree structure for the sentence using the grammar

List the derivation steps that show how the sentence is generated from the grammar

💡 Why This Matters

🌍 Real World

Parse trees and derivations are used by compilers and interpreters to understand and check programming languages and natural languages.

💼 Career

Understanding parse trees is essential for jobs in compiler design, language processing, and software development tools.

1

Set up the grammar rules dictionary

Create a dictionary called grammar with these exact entries: 'S': ['NP VP'], 'NP': ['Det N'], 'VP': ['V NP'], 'Det': ['the'], 'N': ['cat'], 'V': ['chased'].

Compiler Design

# Create the grammar dictionary with the given rules
# Your code here

Hint

Use a Python dictionary with keys as non-terminals and values as lists of productions.

2

Define the sentence to parse

Create a variable called sentence and set it to the string 'the cat chased the cat'.

Compiler Design

grammar = {
    'S': ['NP VP'],
    'NP': ['Det N'],
    'VP': ['V NP'],
    'Det': ['the'],
    'N': ['cat'],
    'V': ['chased']
}

# Define the sentence string
# Your code here

Hint

Assign the exact sentence string to the variable sentence.

3

Build the parse tree structure

Create a dictionary called parse_tree that represents the parse tree for the sentence. Use this exact nested structure: {'S': {'NP': {'Det': 'the', 'N': 'cat'}, 'VP': {'V': 'chased', 'NP': {'Det': 'the', 'N': 'cat'}}}}.

Compiler Design

grammar = {
    'S': ['NP VP'],
    'NP': ['Det N'],
    'VP': ['V NP'],
    'Det': ['the'],
    'N': ['cat'],
    'V': ['chased']
}

sentence = 'the cat chased the cat'

# Build the parse tree dictionary
# Your code here

Hint

Use nested dictionaries to show how each part of the sentence breaks down by grammar rules.

4

Write the derivation steps

Create a list called derivations with these exact strings in order: 'S', 'NP VP', 'Det N VP', 'the N VP', 'the cat VP', 'the cat V NP', 'the cat chased NP', 'the cat chased Det N', 'the cat chased the N', 'the cat chased the cat'.

Compiler Design

grammar = {
    'S': ['NP VP'],
    'NP': ['Det N'],
    'VP': ['V NP'],
    'Det': ['the'],
    'N': ['cat'],
    'V': ['chased']
}

sentence = 'the cat chased the cat'

parse_tree = {
    'S': {
        'NP': {'Det': 'the', 'N': 'cat'},
        'VP': {
            'V': 'chased',
            'NP': {'Det': 'the', 'N': 'cat'}
        }
    }
}

# Write the derivation steps list
# Your code here

Hint

List each step showing how the sentence is formed by replacing non-terminals with their productions.

Practice

1. What does a parse tree represent in compiler design?

easy

A. The list of tokens from lexical analysis

B. The sequence of machine instructions generated

C. The memory layout of variables

D. The hierarchical structure of a sentence based on grammar rules

Solution

1. Step 1: Understand the role of parse trees

   Parse trees visually show how a sentence is broken down into parts according to grammar rules.

2. Step 2: Differentiate from other compiler components

   Unlike tokens or machine instructions, parse trees focus on syntax structure, not code or memory.

3. Final Answer:

   The hierarchical structure of a sentence based on grammar rules -> Option D

4. Quick Check:

   Parse tree = hierarchical structure [OK]

Hint: Parse tree = sentence structure, not code or memory [OK]

Common Mistakes:

• Confusing parse tree with tokens list
• Thinking parse tree shows machine code
• Mixing parse tree with memory layout

2. Which of the following is the correct way to represent a derivation step in grammar?

easy

A. S → aSb

B. S = a + Sb

C. S := aSb

D. S <- aSb

Solution

1. Step 1: Identify standard grammar notation

   In grammar, derivations use the arrow symbol → to show how a symbol expands.

2. Step 2: Check each option's correctness

   S → aSb uses → correctly; others use assignment or incorrect symbols.

3. Final Answer:

   S → aSb -> Option A

4. Quick Check:

   Derivation uses → symbol [OK]

Hint: Look for arrow → in grammar rules [OK]

Common Mistakes:

• Using assignment symbols instead of arrow
• Confusing programming assignment with grammar rules
• Using incorrect arrow symbols

3. Given the grammar rules:
S → aSb | ab
What is the leftmost derivation of the string aabb?

medium

A. S → aSb → aaSbb → aabb

B. S → ab → aabb

C. S → aSb → ab

D. S → aSb → aab

Solution

1. Step 1: Apply leftmost derivation rules

   Start with S, replace leftmost S using S → aSb to get aSb.

2. Step 2: Apply rule again to inner S

   Replace inner S with aSb again to get aaSbb, then replace S with ab to get aabb.

3. Final Answer:

   S → aSb → aaSbb → aabb -> Option A

4. Quick Check:

   Leftmost derivation = S → aSb → aaSbb → aabb [OK]

Hint: Replace leftmost non-terminal first [OK]

Common Mistakes:

• Replacing rightmost non-terminal first
• Skipping intermediate derivation steps
• Confusing production rules

4. Identify the error in this derivation sequence for grammar S → aSb | ab to produce aabb:
S → aSb → ab → aabb

medium

A. Used wrong production rule for S

B. String aabb cannot be derived from this grammar

C. Incorrectly replaced inner S with ab too early

D. No error, derivation is correct

Solution

1. Step 1: Analyze the derivation steps

   The derivation replaces S with aSb, then immediately replaces S with ab but in the wrong position.

2. Step 2: Check correctness of replacement order

   The inner S should be replaced after expanding the outer S fully; here it skips a step causing incorrect string formation.

3. Final Answer:

   Incorrectly replaced inner S with ab too early -> Option C

4. Quick Check:

   Early replacement error = Incorrectly replaced inner S with ab too early [OK]

Hint: Check order of non-terminal replacements carefully [OK]

Common Mistakes:

• Replacing non-terminals out of order
• Assuming all strings are derivable
• Ignoring grammar rules sequence

5. Consider the grammar:
E → E + T | T
T → T * F | F
F → (E) | id
Which parse tree correctly represents the expression id + id * id?

hard

A. Parse tree where '*' is root, left child '+' with two 'id' leaves, right child 'id'

B. Parse tree where '+' is root, left child 'id', right child '*' with two 'id' leaves

C. Parse tree with '+' as root and both children as 'id'

D. Parse tree with '*' as root and both children as 'id'

Solution

1. Step 1: Understand operator precedence in grammar

   Multiplication (*) has higher precedence than addition (+), so '*' groups tighter.

2. Step 2: Build parse tree accordingly

   The root is '+', left child is 'id', right child is '*', which has two 'id' children.

3. Final Answer:

   Parse tree where '+' is root, left child 'id', right child '*' with two 'id' leaves -> Option B

4. Quick Check:

   Operator precedence respected = Parse tree where '+' is root, left child 'id', right child '*' with two 'id' leaves [OK]

Hint: Remember * binds tighter than + in parse trees [OK]

Common Mistakes:

• Ignoring operator precedence
• Swapping '+' and '*' as root
• Misplacing leaves in parse tree