1.
2.
3.
Qualifying Examination of Computational Theory
Convert the NFA shown in Fig.1 into DFA. (10%)
Minimize the DFA shown in Fig.2. (10%)
Prove that the following language is NOT regular by using Pumping Lemma:
(10%)
L = {w|#(a) ≠ #(b)in w, where w ∈ Σ ∗ , Σ = {𝑎, 𝑏}}.
4.
Prove that the following language is NOT context free by using the Pumping
Theory of context free language. (10%)
L = {w|#(a) > #(b) > #(c), w ∈ {𝑎, 𝑏, 𝑐}∗ }
5.
Given Σ = {a, b}, prove that the following language is context sensitive. (10%)
6.
L = {𝑎𝑛 𝑏 2𝑛 𝑎3𝑛 }
Prove that the following problem is unsolvable by using problem reduction. The
candidate well-known unsolvable problem should be the Halting or non-Halting
problem. (15%)
𝐆𝐢𝐯𝐞 𝐚 𝐬𝐭𝐫𝐢𝐧𝐠 𝐰 𝐭𝐨 𝐚 𝐓𝐌 𝐌, 𝐰𝐢𝐥𝐥 𝐭𝐡𝐞 𝐜𝐨𝐦𝐩𝐮𝐭𝐚𝐭𝐢𝐨𝐧 𝐩𝐫𝐨𝐝𝐮𝐜𝐞 𝐚 𝐜𝐞𝐫𝐭𝐚𝐢𝐧 𝐜𝐨𝐧𝐟𝐢𝐠𝐮𝐫𝐚𝐭𝐢𝐨𝐧
(𝐪, 𝒂𝟏 … 𝒂𝒊 … ) ?
7.
Prove that the following problem is unsolvable: (15%)
Given a Linear Bounded Automata A and a string w, will A halt on w within
polynomial time steps?
(Hint: The candidate problem: will a Context Sensitive Grammar generate all string?)
8. Verify that the following problems are P or NP? Please explain the reasons. (10)
A. Equivalence of two DFA.
B. Given two graphs G1 and G2, is G1 a subgraph of G2?
C. Given an NFA M, will M accept any string?
9. Prove that the following problem is NP-complete: (10%)
G=(V, E) is a weighted graph. Can we create a subgraph G1=(V1,E1) in G such
that the total edge weight of E1 = the total edge weight of E-E1?
Fig1. NFA of problem 1
Fig.2 DFA of problem 2