Eliminating Ambiguity in Context-Free Grammars (CFGs) for Compiler Design

Learn how to resolve ambiguity in context-free grammars (CFGs), a critical step in compiler design. This guide explains how ambiguous grammars can lead to incorrect code generation and presents techniques for eliminating ambiguity, ensuring a single, unambiguous interpretation of programming language syntax.

Eliminating Ambiguity in Context-Free Grammars (CFGs)

What is Ambiguity in a CFG?

In compiler design, context-free grammars (CFGs) define the syntax of a programming language. A grammar is ambiguous if a single input string can be derived in more than one way. This means that there's more than one possible parse tree (a tree-like representation of the grammatical structure) for the same input string. This is a problem because a compiler needs a single, unambiguous interpretation of the code to translate it correctly. Ambiguous grammars can lead to errors in code generation.

Basic CFG Terminology

• V: Set of non-terminal symbols (variables representing grammatical constructs).
• T: Set of terminal symbols (actual characters in the language).
• P: Production rules (rules defining how non-terminals can be replaced by other symbols).
• S: Start symbol (the top-level non-terminal).

Example of Ambiguity

Consider the grammar:

E → E + E | E * E | id

The string “id + id * id” has multiple parse trees because the grammar doesn't explicitly define operator precedence (order of operations). This ambiguity leads to different interpretations.

Resolving Ambiguity: Precedence and Associativity

Ambiguity can sometimes be eliminated by:

1. Applying Precedence Rules:

Define the order of operations. For example, multiplication (*) has higher precedence than addition (+). Modifying the grammar to reflect these rules removes ambiguity.

E → E + T | T
T → T * F | F
F → id

2. Handling Associativity:

Specify how operators of the same precedence are grouped. Left-associativity (operators group from left to right) is achieved using left recursion in the grammar. Right-associativity (operators group from right to left) is done using right recursion. For example:

• Left-associative (+): E → E + T | T
• Right-associative (^): E → E ^ T | T

Example: Removing Ambiguity from Subtraction

The grammar:

X → X - X | id

is ambiguous. Using left associativity, we can create an unambiguous version:

X → X - Y | Y
Y → id