CMPSCI 601: Theory of Computation

David Mix Barrington

Fall, 2004

Homework Assignment #2

Posted Tue 19 Oct 2004

Due Fri29 Oct 2004, 2:00 p.m. by fax to 413-545-1249

Covers Lectures 6-10

These problems all concern Turing machines and computability.

• Question 1: Informally describe Turing machines to compute the functions listed in Proposition 7.5 (slide 17 of Lecture 7, Spring 2004).

• Question 2: A Turing machine with two-dimensional tape is like an ordinary TM except that its "tape" is a two-dimensional array of cells -- each cell has an address (i,j) where i and j are non-negative integers. It has a single head which reads the contents of the cell where it is, decides what to do based on that cell and its state, and does it -- writing on the tape cell and moving left, right, up, or down. Prove that a Turing machine with two-dimensional tape can be simulated by an ordinary TM.

• Question 3: We say that a partial function f is computed by a nondeterministic TM N if for every input string x, (1) if f(x) is defined, at least one computation path of N on x gives output f(x), and (2) no two computation paths of N on x halt and output different strings, and (3) if f(x) is not defined, no computation path of f on x halts. Prove that every function computed by a nondeterministic TM is partial recursive. Explain why this definition leads to the same classes of total recursive functions and recursive sets as does the ordinary definition in terms of deterministic TM's.

• Question 4: The language BTHP (blank tape halting problem) is defined as {n: The TM M_n halts on blank input}. Prove that BTHP is r.e.-complete.

• Question 5: A tiling problem is a finite set of tile types, a set of rules as to which tiles may be adjacent horizontally and vertically, and a designated northwest and southeast tile. The problem is solvable if there exists positive integers m and n such that there is an m by n rectangle of tiles, with the northwest tile on the northwest corner and the southeast tile on the southeast corner, where every adjacent pair of tiles satisfies the rules. Prove that the set of solvable tiling problems is r.e.-complete.

Last modified 19 October 2004