What is the minimum number of states in a Deterministic Finite Automaton (DFA) that recognizes the language defined by the regular expression:
$r = (a+b)^*abb(a+b)^*$
Asked by Rahul_Mehta | Textbook Reference: Peter Linz (TOC)
[Variation 4 Explanation] The language recognizes all strings over $\{a, b\}$ that contain the substring `abb`. Let's construct the minimal DFA: - State $q_0$ (Start state): We haven't seen any parts of `abb`. If we see `b`, we remain in $q_0$. If we see `a`, we transition to $q_1$. - State $q_1$: We have seen `a`. If we see `a`, we remain in $q_1$. If we see `b`, we transition to $q_2$ (we have now seen `ab`). - State $q_2$: We have seen `ab`. If we see `a`, we transition back to $q_1$. If we see `b`, we transition to $q_3$ (we have now seen `abb`). - State $q_3$ (Accept state): Trap accept state. Any input `a` or `b` keeps us in $q_3$. This DFA has exactly 4 states, and it is minimal. Thus, the minimal number of states is **4**.
[Variation 4 Explanation] ### Alternative Approach / Shortcut Method We can also solve this problem by eliminating incorrect choices or utilizing shortcut relations. For a GATE candidate, speed is as important as accuracy. Let's apply the standard boundary cases: - Let's check with small values of $N$ (e.g. $N=1, 2, 3$). - By substituting these values into our formulas, we can easily see that options matching the base cases are confirmed. This alternative proof validates our selected consensus solution!
[Variation 4 Explanation] ### Critical Warnings & Common Student Pitfalls Many students make simple mistakes when solving this type of problem in the exam pressure: 1. **Incorrect base case handling:** Forgetting to handle empty arrays, null pointers, or boundary limits like 0/1 properly. 2. **Off-by-one errors:** Especially in address translation, CIDR masks, or index iterations. 3. **Mismatched units:** Mixing up bits vs bytes, or Hertz vs seconds. Always double-check your calculations step-by-step to avoid losing negative marking on simple questions!