CMPSCI 250: Introduction to Computation

David Mix Barrington

Fall, 2010

Homework Assignment #5

Posted Sunday 14 November 2010

Due on paper in (discussion) class, Tuesday 23 November 2010

There are thirteen questions for 100 total points plus 10 extra credit. All are from the textbook, Mathematical Foundation for Computer Science, from Volume II. 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.

Corrections and notes in green added 17 November 2010.

• Problem 9.9.1 (10) The reference to "Problem T.6.3" should be to Problem 9.8.3. Actually this problem needs a number of changes, starting with the fact that (0,0) is currently unreachable. We will keep the graph the same but change the following:

  • As previously observed, there are typos in the description of the graph in the book. The north edges are from (i,j) to (i,j+1) and the south edges are from (i,j) to (i,j-1).

  • There is an ASCII picture of the graph here.

  • The start is now (1,1), and the goal is (3,3).

  • As on Manhattan, east-west blocks are longer than north-south blocks. Every edge going east or west has length 3, and every edge going north or south has length 1.

  • For the A^* search, the function h is now 3|i-3| + |j-3|, the "pedestrian distance" from (i,j) to (3,3), along streets but ignoring the one-way restrictions. (The vertical lines denote absolute value.)

• Problem 9.10.1 (10 extra credit) Note that you don't know the "techniques from Chapter 6" as they are now in CMPSCI 240. If you don't have a better idea, answer (a) with an overestimate based on the fact that each of the eight dark squares might have nothing, a white man, a black man, a white king, or a black king. We will give some extra credit for any reasonable estimate -- the exact number of positions that could occur in a game would require a computer search, I think.

• Excursion 9.11, Writing Exercise 1 (10) Note that a "winning strategy" for one player consists of a set of instructions that will allow her to win against any sequence of moves by her opponent. She only needs responses for situations that can come up in a game where she is playing her strategy.

• Problem 5.1.1 (5)

• Problem 5.1.2 (5)

• Problem 5.2.2 (10)

• Problem 5.2.4 (10) As John pointed out to me, this problem is hardly worth ten points after yesterday's discussion. But you get them anyway for a correct regular expression.

• Problem 5.4.1 (10)

• Problem 5.4.2 (5) This problem is wrong -- for five points, give an example of regular expressions S and T, over a one-letter alphabet, such that (ST)^* and S^*T^* are different languages. I think I intended to ask you to prove that (ST)ⁿ and SⁿTⁿ are the same over a one-letter alphabet, which is true. But for now, just find and justify a counterexample to the claim in the book.

• Problem 5.5.1 (10)

• Problem 14.1.1 (5) (not 14.1.11)

• Problem 14.1.6 (10)

• Problem 14.2.4 (10)

Last modified 17 November 2010