CMPSCI 501: Theory of Computation

David Mix Barrington

Spring, 2016

Homework Assignment #1

Posted Thursday 21 January 2016

Due on paper in class, Wednesday 3 February 2016

There are thirteen questions for 100 total points plus 10 extra credit. All but four are from the textbook, Introduction to the Theory of Computation by Michael Sipser (third edition, but the second edition numbers are the same for all but the last problem). Some problems are adapted as indicated. 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.

Correction in purple made 22 January 2016.

• Problem A-1 (15): Let G be an undirected graph. Define the binary relation P on nodes of G, so the P(x, y) is true if and only if there is a path from x to y.
  • (a, 5) Argue informally that P is an equivalence relation.
  • (b, 10) The equivalence classes of P are called connected components. Prove that if G is a graph with n nodes, e edges, and no cycles (no non-trivial simple paths from a node to itself, where a simple path is one that does not reuse an edge, and a trivial path is one with no edges), it has exactly n - e connected components. (Hint: Let n be arbitrary and use induction on e. You can't have a situation where e ≥ n, but your proof doesn't need to worry about that.)

• Problem A-2 (15): Prove the three following statements for all nonnegative integers n by ordinary induction on n. In both cases let Σ = {a, b, c}.
  • (a, 5) The number of palindromes of length 2n + 1 over Σ is exactly 3ⁿ⁺¹.
  • (b, 5) The number of strings of length n + 1 that never have the same letter twice in a row is 3(2)ⁿ.
  • (c, 5) The number of strings of length n where the letters occur in alphabetical order (a's before b's before c's) is exactly (n+1)(n+2)/2.

• Problem A-3 (15): Let G be a directed graph with at least one node, and no loops, such that for any distinct nodes x and y, there is an edge either from x to y or from y to x but not both. Prove that there must be a Hamilton path in G (a simple path that visits all the nodes). Hint: Use induction on all positive integers n to show that this is true for all such graphs with n vertices.

• Problem A-4 (5): As in Problem A-2, let Σ = {a, b, c} and let X be the set of all strings that never have the same letter twice in a row. Find the "index of X" as defined in Problem 1.52, that is, the number of Myhill-Nerode equivalence classes for X. Justify your answer.

• Exercise 1.10 (5). Don't miss the word "star". You may use any valid method.

• Exercise 1.20 (5). No proofs needed.

• Exercise 1.28 part c only, but also produce a DFA for the language. You may use any valid method (5)

• Problem 1.32 (10).

• Problem 1.48 (10).

• Problem 1.60 (5).

• Problem 1.61 (10).

• Problem 1.70 (10XC). (This is not in the second edition so I will repeat the question here.) We define the avoids operation for languages A and B to be "A avoids B = {w: w ∈ A and w doesn't contain any string in B as a substring}". Prove that the class of regular languages is closed under the avoids operation. (That is, if A and B are regular languages, so is "A avoids B".)

Last modified 22 January 2016