Assignment 4: Properties of Regular Languages

Slide Set 4 Assignment: Properties of Regular Languages

This assignment is designed to evaluate and consolidate your mastery of the closure properties of regular languages, decision algorithms (elementary questions), and the Pumping Lemma, synthesizing the rigorous theoretical frameworks of Michael Sipser (Chapter 1.4) with the constructive algorithmic proofs of Peter Linz (Chapter 4).

Section A: True or False Questions

For each question, select True or False and provide a rigorous mathematical justification.

Question 1

If $L_1$ is a regular language over $\Sigma$ and $L_2$ is any language over $\Sigma$ (not necessarily regular), then the right quotient $L_1 / L_2$ is guaranteed to be regular.

Question 2

If $L_1$ is regular and $L_1 \cup L_2$ is regular, then $L_2$ must be regular.

Question 3

The Pumping Lemma for regular languages is a necessary but not sufficient condition for regularity. Thus, if a language satisfies the conditions of the Pumping Lemma, it is not guaranteed to be regular.

Question 4

Given standard DFA representations for two regular languages $L_1$ and $L_2$, we can decide whether $L_1 = L_2$ by constructing a DFA for the symmetric difference $(L_1 \cap \overline{L_2}) \cup (\overline{L_1} \cap L_2)$ and testing if its recognized language is empty.

Section B: Multiple Choice Questions (MCQs)

Select all options that apply. You must justify your choices mathematically.

Question 5

Let $L_1$ be a regular language and $L_2$ be a non-regular language over $\Sigma$. Which of the following statements are always true?

• Options:
  • A) $L_1 \cap L_2$ is guaranteed to be non-regular.
  • B) If $L_1$ is a finite language, then $L_2 - L_1$ must be non-regular.
  • C) If $L_1$ is a finite language, then $L_1 \cap L_2$ must be regular.
  • D) The concatenation $L_1 \cdot L_2$ is guaranteed to be non-regular.

Question 6

Let $h: \Sigma^* \to \Gamma^*$ be a homomorphism. Which of the following statements regarding regularity are correct? (Select all that apply)

• Options:
  • A) If $L \subseteq \Sigma^*$ is regular, then $h(L)$ is guaranteed to be regular.
  • B) If $h(L) \subseteq \Gamma^*$ is regular, then $L$ is guaranteed to be regular.
  • C) If $L' \subseteq \Gamma^*$ is regular, then the inverse homomorphism $h^{-1}(L')$ is guaranteed to be regular.
  • D) If $h^{-1}(L') \subseteq \Sigma^*$ is regular, then $L'$ is guaranteed to be regular.

Section C: Analytical, Numerical, & Proof Questions

Provide complete, step-by-step mathematical derivations, formal state diagrams, and proofs.

Question 7 (Inductive Proof of Intersection Closure)

Let $M_1 = (Q_1, \Sigma, \delta_1, q_1, F_1)$ and $M_2 = (Q_2, \Sigma, \delta_2, q_2, F_2)$ be two DFAs recognizing $L_1$ and $L_2$ respectively.

1. Define the Cartesian product DFA $M = (Q, \Sigma, \delta, q_0, F)$ that recognizes $L_1 \cap L_2$.
2. Prove rigorously by induction on the length of the string $w \in \Sigma^*$ that: $$\delta^*(q_0, w) = (q_i, p_j) \iff \delta_1^*(q_1, w) = q_i \text{ and } \delta_2^*(q_2, w) = p_j$$
3. Show how the choice of $F$ guarantees acceptance of $L_1 \cap L_2$.

Question 8 (Right Quotient State-Transition Analysis)

Let $\Sigma = \{a, b\}$. We have:

1. Construct the minimal DFA $M_1$ recognizing $L_1$.
2. Apply the state-filtering algorithm for right quotient (Theorem 4.4) to find the states $F'$ of $M_{L_1/L_2}$.
3. Provide the regular expression for $L_1 / L_2$.

Question 9 (The Pumping Lemma Games)

Rigorously prove that the following languages are not regular:

1. $L_a = \{ a^n b^k c^{n+k} \mid n, k \geq 0 \}$ using the standard Pumping Lemma.
2. $L_b = \{ w \in \{a,b\}^* \mid n_a(w) \neq n_b(w) \}$ using the specialized prime-factorial pumping technique.

Question 10 (Deciding DFA Equivalence via Symmetric Difference)

Consider two DFAs over $\Sigma = \{a, b\}$:

• $M_1$: States $Q_1 = \{A, B\}$, start state $A$, accept state $F_1 = \{B\}$.
  • $\delta_1(A, a) = B$, $\delta_1(A, b) = A$
  • $\delta_1(B, a) = B$, $\delta_1(B, b) = A$
• $M_2$: States $Q_2 = \{C, D, E\}$, start state $C$, accept state $F_2 = \{D, E\}$.
  • $\delta_2(C, a) = D$, $\delta_2(C, b) = C$
  • $\delta_2(D, a) = E$, $\delta_2(D, b) = C$
  • $\delta_2(E, a) = E$, $\delta_2(E, b) = C$

1. Construct the Cartesian product DFA $M_{diff}$ that recognizes the symmetric difference language $(L(M_1) \cap \overline{L(M_2)}) \cup (\overline{L(M_1)} \cap L(M_2))$.
2. Identify all reachable states and transitions in $M_{diff}$.
3. Apply the $ED_{DFA}$ emptiness testing algorithm to prove whether $L(M_1) = L(M_2)$.

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