CSE 200
Computability and Complexity
Calibration Homework
Due April 9
April 2, 2012
1. Let Regkbe the class of languages accepted by a deterministic finite automaton with at most k
states. Prove that for every k > 0, Regkis a strict subset of Regk+1.
2. Let fbe a non-decreasing, positive integer-valued function over the positive integers. Prove that
if f(2n)โO(f(n)), then there is a kso that f(n)โO(nk). Is the coverse always true? Prove it or
give a counter-example.
3. Let pbe a prime number. Prove that (pโ1)! + 1 is divisible by p.
4. Let Gbe an undirected graph. An independent set Iis a set of vertices so that for any two nodes
uโIand vโI, there is no edge between uand v. Consider the distribution on graphs where each
pair of nodes has an edge between them with probability 1/2, and these choices are independent.
Do an experiment to determine the expected size of the maximum independent set in such a graph,
as a function of n, the number of nodes. Give a conjecture as to the order of this function. Describe
your experiment and give data in a clear format.
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