CMPSCI 601: Theory of Computation

David Mix Barrington

Fall, 2004

Homework Assignment #4

Posted Wed 1 Dec 2004

Due Fri 9 Dec, 2:00 p.m. by fax to 413-545-1249

Covers Lectures 19-22

These problems all concern complexity measures and complexity classes.

• Question 1: Let A(x) be an NP predicate. Let M be a poly-time NDTM that has an oracle tape. It can read and write on the oracle tape, and the content of the oracle tape affects its computation in one way: It has a "query state" from which it can only continue if A(w) holds for the string w on the oracle tape. Prove that L(M) is in NP. You may find Fagin's Theorem useful, or you may want to describe an ordinary poly-time NDTM whose language is L(M).

• Question 2: Suppose that in the situation of Question 1, A was a co-NP rather than an NP predicate. Argue that there is a second order boolean query defining L(M), with two second-order quantifiers, the first existential and the second universal. Describe an alternating Turing machine game for L(M) where the players change control only once, from White to Black.

• Question 3: Consider the tiling problem of Question 5 of HW#2. Define the bounded tiling problem to be the language of pairs (T,1^s) where T is a tiling problem (a set of tile types and adjacency rules), s is a positive integer, and there is a solution to T that is an s by s square. Prove that the bounded tiling problem is NP-coomplete.

• Question 4: Consider the optimization problem COLORS where the input is an undirected graph G, the feasible solutions are colorings of the vertices of G such that adjacent vertices get different colors, the cost is the number of colors used, and the goal is to minimize this cost. Determine which of the four classes (slide 7, lecture 21) this problem is in, and prove your answer.

• Question 5: Consider the following game, a sort of generalization of tic-tac-toe (naughts-and-crosses). The players X and O alternate placing permanent markers of their symbol on an n by n grid of squares. A player wins if she creates a set of five of her own symbol in a row, consecutively, either horizontally, vertically, or diagonally. A position in the game (a partial assignment of X's and O's to the grid squares, is called open if neither player has five in a row, X-win or O-win if one player has one set of five in a row, and invalid if there is more than one set of five in a row.
  • (a) Show that a position may be classified as open, X-win, O-win, or invalid using O(log n) space. Is this property defined by a first-order formula?
  • (b) Describe an ATM game for the language of open positions where X has a winning strategy if she gets the next move. What alternating complexity class does your game put the language in? What conventional complexity class is it thus in?

  Last modified 1 December 2004