Ambiguity in Context-Free Grammars

Introduction:
Context-free grammars are widely used in computer science and linguistics to describe the syntax of programming languages and natural languages, respectively. A grammar is considered ambiguous if it generates more than one parse tree for a given sentence. In other words, the meaning of the sentence becomes ambiguous due to multiple interpretations.

Explanation:
The correct answer to the question is option 'B' because ambiguity in context-free grammars is precisely defined as the existence of multiple parse trees for a sentence. Let's understand this in more detail:

Context-Free Grammars:
A context-free grammar consists of terminals, non-terminals, production rules, and a start symbol. The production rules define how the non-terminals can be rewritten using terminals and other non-terminals. The start symbol represents the initial non-terminal from which the derivation begins.

Ambiguity:
Ambiguity arises when a given sentence can be derived by different parse trees, leading to multiple interpretations. This can occur due to various reasons, such as:

1. Multiple Derivations:
- In an ambiguous grammar, a single sentence can have multiple valid derivations, each corresponding to a different parse tree.
- Each derivation represents a different interpretation or meaning of the sentence.

2. Conflicting Production Rules:
- Ambiguity can arise when there are conflicting production rules for a non-terminal.
- For example, if a non-terminal has two production rules with different derivations, it can lead to ambiguity.

3. Common Prefixes:
- If there are common prefixes among the right-hand sides of the production rules, ambiguity can occur.
- When parsing a sentence, the parser may have multiple choices at a particular step, leading to different parse trees.

4. Left Recursion:
- Ambiguity can also arise due to left recursion in the grammar.
- Left recursion occurs when a non-terminal can directly or indirectly derive itself.

Conclusion:
In conclusion, a context-free grammar is considered ambiguous if it produces more than one parse tree for a given sentence. Ambiguity can arise due to multiple derivations, conflicting production rules, common prefixes, or left recursion. It is important to identify and resolve ambiguity in grammars to ensure clear and unambiguous interpretations of sentences.