CMPSCI 601: Theory of Computation

David Mix Barrington

Spring, 2003

Homework Assignment #2

Due Monday 3 Mar 2003 in class

(Mon 10 Mar for off-campus students)

(deadlines extended on 18 Feb due to snow)

• Question 1 (10):
  • (a,5) Let N be any nondeterministic Turing machine. Recall that L(N) is the set of strings on which N has at least one accepting path. Prove that L(N) is recursively enumerable.
  • (b,5) Let N be any nondeterministic Turing machine that is guaranteed to eventually halt on any input and on any computation path. When it halts, it either accepts or rejects, and L(N) is the set of strings for which it has at least one accepting path. Prove that L(N) is recursive.

• Question 2 (30):We have said that there are one-to-one, onto functions (bijections) from the set N of numbers to the set S^* of binary strings, and from the set (N cross N) of pairs of numbers to N. In this question we consider specific bijections among these sets. Conveniently, there are Javascript implementations of these for you to play with, thanks to 2001 CMPSCI 601 student Chris Chandler. Here are definitions of these functions:

  • The function rho from strings to numbers takes a string w to ((int) 1w) - 1, where ((int) w) is the number coded by w in the usual way.

  • If x and y are two strings, we form a string p(x,y) as follows. Let h be the homomorphism taking 0 to 00 and 1 to 01. Then p(x,y) is h(x) followed by the empty string (if y is empty), 0 (if y is 0), or a 1 followed by rho^-1(rho(y) - 2)) otherwise.

  • The pairing function P from (N cross N) to N takes numbers i and j to rho(p(rho^-1(i), rho^-1(j))). That is, it uses the bijection rho to convert the string pairing function p into a pairing function on numbers. Note that a typo in this definition was corrected on Sat 15 Feb 2003.

  • The inverse of p is the pair of functions l and r, so that p(l(w),r(w)) = w. Similarly the inverse of P is the pair of functions L and R, so that P(L(n),R(n)) = n.

  Now for your questions:

  • (a,5) Describe in words how to calculate rho^-1 on a number.

  • (b,10) Argue that the regular language

    (00+01)^*(epsilon + 0 + 1S^*)

    equals the language S^*, and that every string has exactly one way to write it as a string in (00+01)^* followed by a string in (epsilon + 0 + 1S^*). (Here epsilon is the empty string, and I am using "+" for the union operator.)

  • (c,15) Argue that both alleged bijections (rho and P) are really bijections. That is, argue that any valid inputs to Chris' script will give outputs, that functions alleged to be inverses of each other really are, and that no two different inputs will give the same output.

• Question 3 (40): Here are the two programming languages Bloop and Floop, more or less as defined in Godel, Escher, Bach by Douglas Hofstadter. Each language has variables of type "number", called input(0), input(1), ..., cell(0), cell(1), ..., and output. Recall again that "number" means "element of N", that is, "non-negative integer".

  The basic program is a function from some fixed number of inputs to one output, and only a constant number of variables may be used in a program. (The "1" in "input(1)" cannot be replaced by a variable, for example.) The basic statement is an assignment, with a variable on the left-hand side and an expression in variables, function calls, and number constants, using plus and times, on the right-hand side. The function ends when a value is assigned to "output" (as in a C/Java "return" statement).

  The difference between the two languages is in their control structures. In Bloop you have a bounded loop. If P is a block of code and x is a variable not assigned to in P, then "loop x times P" is a block of code. In Floop you have "free loops" as well as bounded loops: if P is a block of code containing a "break" statement, then "loop P" is a block of code equivalent to C/Java "while (true) P". (Also see the clarifications of the syntax on the questions and answers page.)

  • (a,10) Write the following Bloop functions:
    1. equals (x,y) [return 1 if x=y, 0 otherwise]
    2. positive (x) [return 1 if x > 0, 0 otherwise]
    3. twoToThe(x) [return 2^x, recall 2⁰=1]
    4. prime(x) [return 1 if x is prime, 0 otherwise]
    5. sumOfDivisors(x) [return sum of all y such that y < x and y divides x]
    6. perfect(x) [return 1 if x = sumOfDivisors(x), 0 otherwise]
  • (b,5) Show how to simulate an "if" statement in Bloop. If P and Q are blocks, "if x then P else Q" should execute P if positive(x) = 1 and execute Q otherwise. Note that you are then encouraged to use the "if" construct for other Bloop programs, such as those for part (a).
  • (c,5) Show how to execute a general "while" loop in Floop. If P is a block (which may modify x), implement "while (positive(x)) P".
  • (d,10) Write a Floop program that halts if and only if an odd perfect number exists, that is, an odd number x such that perfect(x) = 1. (No one currently knows whether one exists, though it's considered unlikely.)
  • (e,10) Argue informally that any Floop program can be simulated on a Turing machine. ("Informally" doesn't mean just saying "because of the Church-Turing thesis" -- I want some description of how the simulation will work.) It may help to note that function calls can be implemented as in C or Java, with a stack of activation records.

• Question 4 (20):
  • (a,10) Show that an infinite set of numbers is recursive iff it can be enumerated in strictly increasing order. (See Proposition 3.5 in [P], page 61.) (Note this question was rephrased on 24 Feb 2003.)
  • (b,10) Recall that M_n is the n'th Turing machine in the canonical ordering. Prove that the set {n: L(M_n) is not empty} is r.e.-complete. That is, prove that it is r.e., and that some r.e.-complete language is reducible to it.

Last modified 24 February 2003