Quiz3 | Theory of Computation | Akash Anand | MTH 401A | IIT Kanpur
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The document is a quiz question asking the student to: 1) Describe the set Fc such that the language accepted by the pushdown automaton Pc with final states Fc is the complement of the language accepted by the original automaton P. 2) Prove that the construction works to recognize the complement language. 3) Note where the proof breaks down if P is not assumed to be deterministic.
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![MTH 401: Theory of Computation October 4, 2016
Department of Mathematics and Statistics Time: 10 minutes
Indian Institute of Technology - Kanpur Maximum Score: 10
Quiz 3
Name
Roll Number
Let P = (Q, Σ, Γ, q0, Z0, δ, F) be a deterministic pushdown machine that accepts by final
states. First describe the set Fc ⊆ Q so that for the pushdown machine
Pc = (Q, Σ, Γ, q0, Z0, δ, Fc),
we have L(Pc) = Σ∗
L(P), then prove that your construction indeed works. Where does
you proof break down if P is not assumed to be deterministic? [2+6+2]
Solution: Let Fc = Q F. Then,
w ∈ L(Pc) ⇐⇒ (q0, w, Z0) ∗
(q, , α) for some q ∈ Fc
⇐⇒ (q0, w, Z0) ∗
(q, , α) for some q ∈ Q F
⇐⇒ w ∈ L(P) ⇐⇒ w ∈ Σ∗
L(P).
Hence proved. The second last “ ⇐⇒ ” requires P to be deterministic.](https://image.slidesharecdn.com/quiz3-161116101756/85/Quiz3-Theory-of-Computation-Akash-Anand-MTH-401A-IIT-Kanpur-1-320.jpg)
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Quiz3 | Theory of Computation | Akash Anand | MTH 401A | IIT Kanpur
- 1. MTH 401: Theory of Computation October 4, 2016 Department of Mathematics and Statistics Time: 10 minutes Indian Institute of Technology - Kanpur Maximum Score: 10 Quiz 3 Name Roll Number Let P = (Q, Σ, Γ, q0, Z0, δ, F) be a deterministic pushdown machine that accepts by final states. First describe the set Fc ⊆ Q so that for the pushdown machine Pc = (Q, Σ, Γ, q0, Z0, δ, Fc), we have L(Pc) = Σ∗ L(P), then prove that your construction indeed works. Where does you proof break down if P is not assumed to be deterministic? [2+6+2] Solution: Let Fc = Q F. Then, w ∈ L(Pc) ⇐⇒ (q0, w, Z0) ∗ (q, , α) for some q ∈ Fc ⇐⇒ (q0, w, Z0) ∗ (q, , α) for some q ∈ Q F ⇐⇒ w ∈ L(P) ⇐⇒ w ∈ Σ∗ L(P). Hence proved. The second last “ ⇐⇒ ” requires P to be deterministic.