# Proving Ambiguity in a Grammar

## Evaluating Ambiguity in a Grammar

In the study of formal languages and grammar, one common question that arises is determining the ambiguity of a given grammar. Ambiguity in grammar refers to a situation where a string can have more than one distinct parse tree, leading to potential confusion in interpreting the language's syntax. To determine whether a grammar is ambiguous, one must carefully analyze its rules and potential interpretations.

## Given Grammar

Consider the following grammar:

**<A> → <B> | <C>**

**<B> → <A> <B> | a**

**<C> → a | b | c**

## Question:

Prove that the following grammar is ambiguous:

**<A> → <B> | <C>**

**<B> → <A> <B> | a**

**<C> → a | b | c**

Is the given grammar ambiguous?

a) Yes, it's ambiguous

b) No, it's unambiguous

c) Ambiguity depends on context

d) Not enough information to determine ambiguity

### Final answer:

Without the actual grammar rules, it is impossible to determine whether the grammar is ambiguous. Ambiguity in grammar means that a string can have more than one distinct parse tree, but this evaluation cannot be done without specific information. Therefore, the correct answer is a).

### Explanation:

The student's question asks to prove whether a given grammar is ambiguous. To prove a grammar is ambiguous, one must demonstrate that there exists at least one string which can be derived in two different ways, leading to two distinct parse trees. Unfortunately, the grammar itself has not been provided in the question, and thus we do not have enough information to determine its ambiguity.

Ambiguity in grammar implies that a certain string can be parsed in more than one way. If the grammar provided produces multiple distinct parse trees for the same string, then the answer would be (a) Yes, it's ambiguous. If the grammar generates a unique parse tree for every valid string, then it would be unambiguous. However, without seeing the actual grammar rules, it is not possible to analyze and make a determination on the ambiguity of the grammar.

The grammar in question is ambiguous. The rule <A> → <B> | <C> allows for multiple interpretations. For example, if we have the input 'abc', we can derive 'a' followed by 'bc' or 'ab' followed by 'c'. Both derivations are valid, proving the ambiguity of the grammar. Therefore, the correct answer is a) Yes, it's ambiguous.