CMPSCI 601: Theory of Computation

David Mix Barrington

Spring, 2003

Homework Assignment #5

Due by Wednesday 9 Apr 2003 9:05 a.m.

(Wed 16 Apr for off-campus students)

The first half of these problems are from LPL and should be submitted to the Grade Grinder with instructor name kazu@cs.umass.edu as in HW#4. The second half are to be handed in on paper at or before the class, or otherwise as arranged with Kazu.

Each LPL question counts five points.

• Questions 1-10 (50 points): 13.5, 13.6, 13.9, 13.13, 13.17, 13.18, 13.27, 13.28, 13.29, 19.17.

• Question 11 (30 points): This question defines and deals with the Arithmetic Hierarchy, the very top class in Neil's "descriptive world" picture. The AH is a set of complexity classes Σ_i and Π_i for each positive integer i. We say that a set S is in Σ_k if there is some primitive-recursive predicate φ such that S is the set:

  {n: ∃x₁ ∀x₂...Q_kx_k φ(n,x₁,x₂,...,x_k)}.

  Similarly, S is in the class Π_k if there is some primitive recursive predicate φ such that S is the set:

  {n: ∀x₁ ∃x₂...Q_kx_k φ(n,x₁,x₂,...,x_k)}. Here Q_k is a quantifier, either ∃ or ∀ depending on the alternation of quantifiers in the prefix. Here are some problems concerning these classes:

  • (a,5): Prove that K is in the class Σ₁. Recall that COMP is a primitive recursive predicate.
  • (b,10): Prove that a set S is r.e. iff it is in Σ₁, and that S is co-r.e. iff S is in Π₁.
  • (c,15): For each of the following four sets, find the smallest class you can in the AH that contains them, and prove that they are in this class. (You do not have to prove that they are in no smaller class.)
    • MEMBER = {P(a,n): a ∈ W_n}. (Recall that P is the pairing function, and that W_n = L(M_n).)
    • TOTAL = {n: M_n computes a total function}
    • INFINITE = {n: W_n is an infinite set of numbers}
    • COFINITE = {n: W_n contains all but finitely many numbers} (Note co-FINITE is equal to INFINITE, not COFINITE!)
    • REGULAR = {n: W_n is a regular language}

• Question 12 (20 points): We have defined REACH to be the set {(G,s,t): G is a directed graph with a path from s to t}. Define a levelled graph to be a graph where the vertices are each given a level number, and every edge out of a vertex in level i goes to a vertex in level i+1. Now define LEVELLED-REACH to be the set {(G,s,t): G is a levelled directed graph with a path from s to t}.

  Prove that LEVELLED-REACH is NL-complete. You will need to explain both why this set is in NL and why every NL set L-reduces to it. (Hint: Show that REACH L-reduces to LEVELLED-REACH. Why is this sufficient?)

Last modified 1 April 2003