NAME ____________________________________________ CS 420/520 Theory of Computation, Spring 2019 at Indiana State University, taught by Jeff Kinne Exam 1 practice - regular languages Points - each part is graded as 1 point, half credit is possible. Note, the practice exam is a total of 13 points. The actual exam may be slightly shorter. 1) For each of these, if the following language is regular, give an RE, DFA, or NFA for the language. If it is not regular, prove it is not regular using the pumping lemma. 1a. L = {strings of 0's and 1's that contain 10010 as a substring} In the language: 10010, 01010010101 Not in the language: 1001, 0010, 101010101, 10011 1b. L = {strings of 0's and 1's with at least 2 0's and at least 2 1's} In the language: 0011, 1010, 100111100 Not in the language: 011, 110, 0, 101, 111111, 000010000 1c. L = {strings of 0's and 1's that start with 1010, end with 0101, and are at least 8 characters long} 1d. L = {strings of 0's and 1's of length a multiple of 3 and where each third bit is the parity of the previous 2 bits} Note: parity of 00 is 0, 01 is 1, 10 is 1, 11 is 0 In the language: 000, 110, 110011 Not in the langage: 0, 1, 01, 010, 111, 000100, 00010 2) For each of the following, give a Python3 regular expression for the given language. 2a. Floating point numbers In the language: 3.2344234, .234233423, 2.23423423 Not in the language: abc, pi, e, -, 2b. Sentence of the form - "REFRAIN. VERSE. REFRAIN." Note - REFRAIN and VERSE can be any combination of the following - letters, space, -, ' 3) Prove by induction or contradiction. 3a. There are infinitely many primes. 3b. 1 + 2 + 3 + ... + n = n(n+1)/2 3c. n! > 2^n (2 to the power n) for all n >= 4 3d. For real numbers a and b, if a is rational and ab is irrational then b is irrational 4) Write a table for the transition function of the following NFA or DFA. https://upload.wikimedia.org/wikipedia/commons/thumb/0/0e/NFAexample.svg/250px-NFAexample.svg.png 5) Describe the language accepted by the following NFA or DFA. https://i.stack.imgur.com/TwVxh.png 6) Prove the following. The class of non-regular languages is closed under complement. In other words, if a language L is not regular then the complement of L is not regular either.