CMPSCI 601: Theory of Computation

Offered through the PEEAS distance learning program

Homework Assignment #2

David Mix Barrington

Assignment posted Fri 20 June 2003

Due Tue 1 July 2003, faxed before 2:00 p.m. EDT or express-mailed to arrive Tue 1 July 2003

Make sure you have read the statement on academic honesty on the homework assignment index page.

Question 1 (10): Define EQ_NFA to be the set of pairs (N₁,N₂) such that N₁ and N₂ are NFA's such that L(N₁)=L(N₂). Argue that EQ_NFA is decidable. Estimate the running time of your algorithm on input of size n.
Question 2 (40): This question concerns primitive recursion. You may assume any of the results about the programming language Bloop proved on homeworks #2 and #3 from the Spring 2003 course.
- (a,20) Let f(n) be any primitive recursive function. Prove that every function in FDTIME(f(n)) (that is, every function computable by a TM in time O(f(n))) is primitive recursive.
- (b,20) Let f be any primitive recursive function from N^k to N. Prove that there is a primitive recursive function g(n) such that f is in FDTIME(g(n)).
Question 3 (30): Each of the following languages is either recursive, r.e.-complete, or not r.e. at all. Determine which it is and justify your answer. In each case the Turing machines should be considered to be one-tape machines, that indicate their output only by their state when they halt. (Clarification made 30 June 2003.)
- (a,10) The set of one-tape Turing machines M such that M never writes a 1 on the tape when started on any valid input.
- (b,10) The set of one-tape Turing machines M such that M never changes any character on the tape when started on any valid input.
- (c,10) The set of one-tape Turing machines M such that there exists a string w such that M eventually writes a 1 when started on input w.
Question 4 (20): Prove the following variant of the Rice-Myhill-Shapiro Theorem from Lecture 8 (See also [P] Theorem 3.2 or Sipser Exercise 5.22.):
Suppose that A is a set of numbers such that if L(M_i) = L(M_j), then i and j are either both in A or both not in A. Also suppose that there is a recursive function f such that f is 1-1 and onto and for any string w, w is in A iff f(w) is not in A. Prove that A is neither r.e. nor co-r.e., that is, that neither A nor A-bar is recursively enumerable. (Correction made 26 June 2003.)

Last modified 30 June 2003