代做Theories of Computation: Summative Assignment 1调试数据库编程

Theories of Computation: Summative Assignment 1

due on Mon 12 Feb, 15:00

Exercise 1. Consider the following εNFA:

(A) Give the NFA that is achieved by removing the above εNFA’s ε-transitions (using the algorithm taught),

[2 points]

(B) Give the DFA that is achieved by determinising your answer to (A),

[2 points]

(C) Give the minimal DFA that is achieved by minimising your answer to (B).

[2 points]

Exercise 2. Consider the language L of those words matched by a ∗ (ba) ∗ that contain twice as many a’s as b’s (e.g. aba, aababa are accepted, but not aaba or baa). Show that this language is not regular.

[4 points]

Exercise 3. For any regular expression E and n ∈ N, we write En as an abbreviation for the regular expression

In particular, considering the alphabet {x}, we have the regular expressions A = x3 and B = x7 , where A only accepts the word xxx and B only accepts the word xxxxxxx. It turns out that for every natural number n > 11, we can express xn by only concatenating the expressions A and B (e.g. x12 = AAAA and x38 = BABBBB).

(A) Show that this is the case for x13 and x14 ,

[2 points]

(B) Prove that this is the case for all n > 11 by using (course-of-values) induction.

[3 points]