CMPSCI 601: Theory of Computation

David Mix Barrington

Spring, 2003

Homework Assignment #3

Due Wednesday 12 Mar 2003 in class

(Wed 19 Mar for off-campus students)

(Extra credit question added 2 March)

(Point values of questions fixed to match those of subparts, 10 March)

• Question 1 (40):

  Three questions concerning Bloop, defined in HW#2 and the resulting questions and answers:

  • (a,20) Prove that a function from N^k to N is computable by a Bloop program iff it is primitive recursive.
  • (b,5) Prove that every Bloop-computable function is total recursive. Assume that any function definition must end with a return statement.
  • (c,15) Prove that there is a total recursive function that is not Bloop computable. Hint: Look on the web or elsewhere for a definition of Ackermans's function. Another candidate is the function called "BLUEDIAG" in Hofstadter, where BLUEDIAG(n) is f_n(n)+1, where f_n is the n'th Bloop computable function in some canonical listing. Here you have to argue that a Turing machine can find and implement this function given n as input. (Spelling of "Ackermann" corrected 6 March.)

• Question 2 (15): Argue that the language EMPTY-CFL, the set of grammars G such that L(G) is empty, can be decided in polynomial time. (For example, in a time that is polynomial in the number of symbols in G, or in the number of rules, or in the length of the string needed to describe G.)

• Question 3 (15): When we proved the Rice-Myhill-Shapiro Theorem in Lecture #7, we assumed that the language EMPTY was disjoint from A. (That is, we assumed that all the Turing machines that never halt were outside of A.) Prove the theorem (that A is not recursive) assuming that EMPTY is a subset of A. Why are these the only two possibilities?

• Question 4 (30+10):
  • (a,10) Show that for any n, the boolean formula

    (x₁ or x₂) and (x₃ or x₄) and ... and (x_2n-1 or x_2n)

    has a disjunctive normal form with 2ⁿ different terms.

  • (b,10) It can be shown (see (d) below) that no DNF expression without at least this many terms can represent the expression in part (a). Assuming this fact, explain why there is no polynomial time algorithm that inputs a boolean expression and outputs an equivalent DNF expression. (Here "polynomial time" means polynomial in the length of the input expression.)
  • (c,10) Describe a polynomial-time algorithm that inputs a DNF expression and determines whether it is satisfiable.
  • (d, 10 extra credit) Prove that fact mentioned in 4(b) above -- that no DNF expression with fewer than 2ⁿ terms can represent the boolean expression in 4(a). Hint: You can show that no term with fewer than n literals in it can appear in a correct DNF expression, because it will be true for some setting of the variables where the expression of 4(a) is false. Then you can show that each of the terms in your DNF expression is necessary -- that there are 2ⁿ different settings that each require a separate term to be included in the DNF. (Typos, referring to Question 3 rather than 4, corrected 5 March 2003.)

Last modified 10 March 2003