Question

The number of states required to accept the string ending with 010 is:

• A. \(2 \)
• B. \(3 \)
• C. \(1 \)
• D. \(4 \)

Hint:

A DFA for detecting a fixed-length pattern requires at least as many states as the length of the pattern plus one. Each state tracks progress toward recognizing the sequence.

Answer 1

The Correct Option is D

Step 1: Understanding the Problem A deterministic finite automaton (DFA) is required to recognize strings ending with "010". Each bit transition should help track whether the last three bits match "010".
Step 2: Constructing the DFA To track the substring "010":
- State \( q_0 \): The start state, waiting for the first bit.
- State \( q_1 \): After reading '0'.
- State \( q_2 \): After reading '01'.
- State \( q_3 \): After reading '010' (Accept state). Thus, 4 states are needed to properly track the sequence "010".
Step 3: Evaluating the Options
- Option (A) \(2\): Insufficient states to track all bits.
- Option (B) \(3\): Can only track up to "01", missing full sequence.
- Option (C) \(1\): Cannot track multiple bits.
- Option (D) \(4\): Correct, as the DFA needs 4 states.