CS 500: Assignments

Homework:

• Homework #1 (do not turn in: strictly for warm-up): Exercises 3.5, 3.8, 3.11. 3.15, and 3.16.
  Solutions are available as a PS file.
• Homework #2 (due Monday, Feb. 23): Exercises 3.17 and 3.19.
  Solutions are available as a PS file.
• Homework #3 (due Monday, March 10): Exercises 4.9, 4.13, 5.1, and 5.5.
  Solutions are available as a PS file.
• Homework #4 (due Wednesday, April 2): Exercise 5.16
  Solutions are available as a PS file.
• Homework #5 (due Monday, April 21): Exercises 6.20, 6.22, 6.23, and 6.26.
  Solutions are available as a PS file.
• Homework #6 (due Wednesday, April 30): Exercises 7.1 (p. 232), 7.4 (p. 252), 7.17, and 7.19.
  Solutions are available as a PS file.
• Homework #7 (due Friday, May 9): Exercises 7.21, 7.22, 7.23, and 7.28.
  Hint on 7.23: create long strings for each edge (or each vertex: there is a very similar solution) of the VC problem; by long, I mean that they should include every element of the instance more than once, with the repetition used to formulate "or" constraints (an edge is covered by one endpoint or the other); note that, in constrast, having multiple strings achieves an "and" constraint. Then add one more string to force the structure of the solution (to avoid repeats in the solution).
  Solutions are available as a PS file.

Tests:

• Test #1 (Chapters 4 and 5), due on Wednesday, April 9. Each of the five problems is worth 25pts; thus one of the problems is strictly bonus -- you can use it to improve your score.
  • Consider all Turing machines over some alphabet Sigma; order Sigma in some fashion, Sigma = (c₁,...,c_k), where k is the size of the alphabet. Now restrict these Turing machines in the following way: in any transition
    delta(q,c_i) = (q',c_j,L/R)
    we must have j at least as large as i. In other words, these restricted TMs can only replace a character by itself or by one that is farther down the order on Sigma. Are these restricted Turing machines still universal?
  • Give primitive recursive definitions for
    • the least common multiple of x and y;
    • the greatest common divisor of x and y;
    • the number of divisors of x and the number of prime divisors of x .
    You can use any of the functions defined in class, in homework, or in text exercises.
  • Classify each of the following sets and its complement as recursive, nonrecursive r.e., or non-r.e.
    • The set of all monotonically increasing functions (i.e., functions such that f(x) is larger than f(y) whenever x is larger than y).
    • The set of all x such that, for every y such that phi_x(y) converges, there exists a z such that phi_x(z) converges and phi_x(z) = 2 phi_x(y)}.
    • The set of all infinite r.e. languages (read: sets).
  • Prove that every infinite r.e. set has an injective enumerating function.
  • Prove or disprove: R.e. sets are closed under pointwise sum, i.e., if S and T are two r.e. sets, then the set U = { x | there exists y in S and z in T with x=y+z } is also r.e.
  Solutions are available in Postscript form. A short discussion of the results is also available.
  
• Test #2 (Chapters 6 and 7; due May 15): available as a Postscript file.
  Further notes after grading homework sets 6 and 7:
  • For NP-completeness proofs, clearly separate the parts -- separate paragraphs, for instance, each labeled with the role it fulfills. Write proofs, not just descriptions of the reductions -- refer to Tables 7.1 and 7.2 for a quick check of what each of your proofs must include.
  • Remember that the most fatal mistake you can make in a reduction is to use the solution in order to define the reduction -- that immediately makes your reduction take exponential time, since we know of no other way to obtain the solution...
  • If you use sources on the web or in a paper or anything else that is not directly from class material (text and lectures), you MUST acknowledge it, giving the exact reference; and this acknowledgment must be made for each idea, for each paragraph, for each citation, not just once at the end. Anything else is plagiarism, which is worse than plain cheating, and will be treated accordingly...
  Solutions are available in Postscript form. A short discussion of the results is also available.