Homework Assignment #5

Due on paper in class or to CMPSCI main office by 4:00 p.m., Monday 4 May 2009

There are ten questions for 100 total points plus 10 extra credit. All but two are from the textbook, Introduction to the Theory of Computation by Michael Sipser (second edition). The number in parentheses following each problem is its individual point value.

Problem E-1 corrected on 29 April 2009.

• Problem E-1 (20) Recall the languages ACFG and ECFG, proved to be decidable in Theorems 4.7 and 4.8.

• (a,10) Adapt the proof of 7.16 to show that ACFG is in P.
• (b,0) I asked you to explain why the argument of Theorem 4.8 (though I said 7.8) cannot be used to show that ECFG is in P. But in fact I misremembered which argument is used in 4.8, and the argument there can be used to do this.
• (c,10) Prove that ECFG is in fact in P, adapting the argument of Theorem 4.8.

• Problem E-2 (25) Suppose that a binary relation R is defined on binary strings of each length, and that there is an algorithm to test whether R(x,y) is true given binary strings x and y of length n, and that this algorithm takes time polynomial in n.

• (a,10) Describe a nondeterministic procedure, with time polynomial in n, that can accept if and only if R is not an equivalence relation on strings of length n.
• (b,15) Suppose for this problem only that P = NP. Show that there exists a deterministic algorithm, with time polynomial in n, that decides whether R is an equivalence relation on strings of length n that has an equivalence class of size 1.

• Exercise 7.4 (5)

• Exercise 7.5 (5)

• Problem 7.12 (15)

• Problem 7.17 (5)

• Problem 7.19 (5)

• Problem 7.23, part (a) only (10)

• Problem 7.24 (10)

• Problem 7.30 (10 extra credit)