Question

Consider the following definition of a lexical token id for an identifier in a programming language, using extended regular expressions: \[ \text{letter} \;\;\rightarrow\;\; [A\!-\!Za\!-\!z] \] \[ \text{digit} \;\;\rightarrow\;\; [0\!-\!9] \] \[ \text{id} \;\;\rightarrow\;\; \text{letter (letter | digit)}^* \] Which one of the following Non-deterministic Finite-state Automata with $\epsilon$-transitions accepts the set of valid identifiers? (A double-circle denotes a final state). 

• A. A
• B. B
• C. C
• D. D

Hint:

Identifiers in most programming languages must start with a letter, followed by letters or digits. This is why the automaton must force a letter first, and only then allow repetition of both letters and digits.
 

Answer 1

The Correct Option is C

To determine which Non-deterministic Finite-state Automata (NFA) with ε-transitions accepts the set of valid identifiers based on the given regular expression:

\[ \text{id} \;\;\rightarrow\;\; \text{letter (letter | digit)}^* \]

we need to understand and construct the NFA that adheres to the following rules:

• An identifier must start with a \(\text{letter}\), which is \([A\!-\!Za\!-\!z]\).
• After the first letter, it can have any combination of letters or digits \(\text{(letter | digit)}^*\), meaning zero or more occurrences of letters or digits are allowed.

Let's analyze the provided NFAs to determine which one implements this logic correctly.

1. Option A does not correctly handle the subsequent characters as it doesn't properly link ε-transitions to allow multiple characters as needed.
2. Option B has separate paths for letters and digits after the initial letter, but it might get stuck in loops and doesn't allow seamless transitions.
3. Option C correctly starts with a letter, and through ε-transitions, it allows any number of letters or digits to follow, reflecting the regular expression correctly.
4. Option D is restrictive as it mishandles the post-initial-character logic and misses the star operator's flexibility in repetition.

Analyzing each option carefully, we conclude that option C correctly represents the NFA for the given regular expression. Here’s why:

• It begins with a \(\text{letter}\).
• Through ε-transitions, it seamlessly allows any combination of letters and digits after the initial letter.

Therefore, the correct answer is option C, which accurately represents the given regular expression for an identifier.