CMPSCI 401: Theory of Computation
David Mix Barrington
Homework Assignment #5
Posted Thursday 23 April 2009
Due on paper in class or to CMPSCI main office by 4:00 p.m., Monday 4 May
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.
Students are responsible for understanding and following
the academic honesty
policies indicated on this page.
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.
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)
Last modified 29 April 2009