Assignment 6: Pushdown Automata and Deterministic Parsing

This assignment evaluates and deepens your understanding of Pushdown Automata (PDAs), nondeterministic versus deterministic computation, equivalence with Context-Free Grammars (CFGs), Deterministic Context-Free Languages (DCFLs), and deterministic parsing principles, synthesizing Michael Sipser (Chapters 2.2 & 2.4) and Peter Linz (Chapter 7).

Section A: True or False Questions

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

Question 1

In a Pushdown Automaton (PDA), acceptance by final state and acceptance by empty stack are equivalent in power for nondeterministic PDAs (NPDAs). However, for Deterministic PDAs (DPDAs) without an explicit endmarker, acceptance by empty stack recognizes a strictly smaller class of languages than acceptance by final state (specifically, prefix-free DCFLs).

Question 2

The class of Deterministic Context-Free Languages (DCFLs) is closed under union, intersection, and complementation, matching the closure properties of regular languages.

Question 3

The regular languages form a proper subset of Deterministic Context-Free Languages (\(Regular \subset DCFL\)), which in turn form a proper subset of Context-Free Languages (\(DCFL \subset CFL\)).

Question 4

In the standard top-down CFG-to-PDA conversion algorithm (simulating a leftmost derivation on the stack), when a variable \(A\) is on top of the stack, the PDA nondeterministically selects a production rule \(A \to w\) and pushes \(w\) onto the stack in reverse order.

Section B: Multiple Choice Questions (MCQs)

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

Question 5

Let \(M = (Q, \Sigma, \Gamma, \delta, q_0, z_0, F)\) be a Pushdown Automaton. Which of the following mathematical conditions are strictly required for \(M\) to be classified as a Deterministic Pushdown Automaton (DPDA)? (Select all that apply)

• Options:
  • A) For every \(q \in Q\), \(a \in \Sigma \cup {\varepsilon}\), and \(X \in \Gamma\), the transition set contains at most one element (\(|\delta(q, a, X)| \le 1\)).
  • B) For every \(q \in Q\) and \(X \in \Gamma\), if \(\delta(q, \varepsilon, X) \neq \emptyset\), then \(\delta(q, a, X) = \emptyset\) for all \(a \in \Sigma\).
  • C) The stack must be emptied completely whenever the machine enters an accepting state \(q \in F\).
  • D) The machine must not contain any \(\varepsilon\)-transitions under any circumstances.

Question 6

Which of the following Context-Free Languages are Deterministic Context-Free Languages (DCFLs)? (Select all that apply)

• Options:
  • A) \(L_1 = {a^n b^n \mid n \geq 0}\)
  • B) \(L_2 = {w c w^R \mid w \in {a, b}^*}\)
  • C) \(L_3 = {w w^R \mid w \in {a, b}^*}\)
  • D) \(L_4 = {a^i b^j c^k \mid i, j, k \geq 0 \text{ and } (i = j \text{ or } i = k)}\)

Section C: Analytical, Numerical, & Proof Questions

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

Question 7 (NPDA Design and Configuration Trace)

Design a Nondeterministic Pushdown Automaton \(M = (Q, \Sigma, \Gamma, \delta, q_0, z, F)\) that accepts the language: \[L = {a^n b^m \mid 1 \leq n \leq m \leq 2n}\]

1. Explain the stack strategy used to enforce the counting relationship \(n \leq m \leq 2n\).
2. Provide the formal 7-tuple specification and list all transition rules.
3. Trace the sequence of Instantaneous Descriptions (IDs) / configurations for the input string \(w = aabbb\).

Question 8 (CFG to NPDA Conversion)

Consider the Context-Free Grammar \(G = (V, T, S, P)\) with \(V = {S, A}\), \(T = {a, b}\), start symbol \(S\), and productions: \[S \to a S b \mid A\] \[A \to a A \mid \varepsilon\]

1. Convert \(G\) into an equivalent single-state NPDA \(M\) using the top-down (leftmost derivation) algorithm.
2. List the full transition function \(\delta\) of \(M\).
3. Trace the computation (stack contents and unread input) of \(M\) accepting the string \(w = aab\).

Question 9 (DPDA and Complementation)

Let \(L = {a^n b^n \mid n \geq 1}\).

1. Construct a formal DPDA \(M\) that accepts \(L\) by final state and reads the entire input string on all inputs.
2. Modify \(M\) to construct a DPDA \(M^c\) that accepts the complement language \(\bar{L} = \Sigma^* - L\).
3. Explain why unread input or infinite \(\varepsilon\)-loops break simple state-swapping in DPDAs, and how your construction avoids these pitfalls.

Question 10 (Formal Proof: Non-Determinism of \(ww^R\))

Prove that the language \(L_{pal} = {w w^R \mid w \in {0, 1}^*}\) is not a Deterministic Context-Free Language (DCFL).