CMPSCI 240: Reasoning About Uncertainty

David Mix Barrington

Fall, 2009

Homework Assignment #7

Posted Tuesday 17 November 2009

Due on paper in class, Monday 30 November 2009

There are six questions for 50 total points plus 10 extra credit.  All are from the textbook, A Mathematical Foundation for Computer Science. Note that the book has both Exercises and Problems -- make sure you are doing a Problem and not the Exercise with the same number. 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 12.3.3 (not 12.2.3) corrected on 23 November 2009 -- see below.

• Problem 11.10.1 (10)
• Problem 12.1.2 (10)
• Problem 12.1.5 (10XC)
• Problem 12.2.2 (10)
• Problem 12.2.3 (10)
• Problem 12.3.3 (10)
Correction (23 Nov, fixed 24 Nov): As pointed out on the Q and A page, while there is always a steady state of the form described it may not be an attracting steady state. In fact there is an attracting steady state if and only if the undirected graph G has an odd cycle -- a path from some vertex v to itself that never revisits any other vertex and has an odd number of edges. So here is the revised question:
• (a,5) Given any connected undirected graph G, give the steady state distribution for the resulting Markov chain as the problem suggests, and prove that it is a steady state distribution.
• (b,5) Suppose that the connected graph G has the property that for all integers greater than some fixed number t, there is a path of length t from any vertex u to any vertex v. Then prove that the steady state of part (a) is attracting. You may quote the Steady State Theorem.
• (c,10XC) Show that if G has an odd cycle, it has the property in part (b). (Hint: Remember that a path can reuse an edge. Show that there are paths of all sufficiently long odd lengths and of all sufficiently long even lengths.)

Last modified 24 November 2009