Assignments

The accompanying table indicates assignments and due dates for Computer Science 341 and are subject to the following notes.

• Unless otherwise indicated, textbook references are to Michael Sipser, Introduction to the Theory of Computation, Second Edition, Thomson/Course Technology, 2006, ISBN: 13: 976-0-534-95097-2 and 10: 0-534-95097-3.

• Supplemental problems are stated later on this page.

• Unless otherwise stated, collaboration IS allowed on problems from the text, but collaboration IS NOT allowed on assignments from the supplemental problems.

Due Date	Collaboration	Chapter	Problems	Notes
Wed., Feb. 1	yes	0	0.4, 0.7, 0.9, 0.11	The formal definition of a graph is a set of vertices and a set of edges. Thus, for Exercise 0.9, you need to translate the content of the given figure into the definition of appropriate sets. (Of course, explanation is needed.)
	no	0	Supp. Prob. 1, 2	Be sure to state any induction hypothesis carefully!
Fri., Feb. 10	yes	1	1.4be, 1.5ceg, 1.6aijl, 1.8ab
	no	1	Supp. Prob. 3a
Wed., Feb. 15	no	1	1.21 ab, 1.46ac, 1.53
Mon, Feb. 20	Hour Test 1			Covers Chapters 0, 1
Wed., Mar. 7	yes	2	2.4b, 2.9, 2.14, 2.26, 2.31
	no	2	Supp. Prob. 4
Fri. Mar 9	no	—	Lecture Summary	Several paragraphs describing Feb. 29 lecture on Recursive Function Theory
Fri, Apr. 6	no — take home test	3	3.6, 3.8b, 3.13	(give a brief justification for 3.13)
		3	Supp. Prob. 5
		4	4.10 or 4.12	your choice
		3	Supp. Prob. 6
Wed., Apr. 18	yes (strongly encouraged)	5	5.3, do two of 5.12-5.15
Wed., Apr. 18	no	5	5.4, 5.9, 5.17
Wed., Apr. 25	yes (strongly encouraged)	5	5.19, do three of 5.21-5.24, 5.29, 5.34	additional problems may be done for extra credit
Fri., Apr. 27	no	Take-home Test Distributed	Covers Chapters 1-5, 7
Mon., April 30	Class Presentations on NP Complete Problems
Wed., May 2	Class Presentations on NP Complete Problems
Fri., May 4	Class Presentations on NP Complete Problems
Mon., May 7	no	Take-home Test Due	Covers Chapters 1-5, 7
Due Date	Collaboration	Chapter	Problems	Notes

Supplemental Problems

1. Prove: 1² + 2² + ... + n² = n(n+1)(2n+1)/6 for all positive integers n.

2. Definition: The height of a non-empty tree is defined as the number of nodes on the longest path from a leaf of a tree to its root; the height of an empty tree is defined to be 0.

   Nodes in a Binary Tree: Suppose a binary tree T has height h. Prove that T contains at least h nodes and at most 2^h - 1 nodes.

3. Comments in Java: Java allows two types of comments:

   • Any string [on one or more lines] that starts with /*, ends with */ and does not contain */ except at its end.
   • Any part of a line that begins with // and concludes at the end of the line.

   For the purposes of this problem, suppose all characters within a Java program are from the set {/, *, a, b, N}, where N represents the end-of-line character. (In a real Java program, of course, the characters a, b would be extended to all letters, digits, punctuation, etc., but that seems too extensive for this exercise.)

   1. Write an NFA with no more than 10 states that accepts exactly the comments in Java (within this limited alphabet).
   2. Using the construction(s) in the book, translate your NFA from part a into a DFA.

4. Cond Statements in Scheme: Suppose the symbols condition and statement have been defined for the Scheme language through a context-free grammar.

   1. Give a context-free grammar that generates exactly the Cond statements of Scheme.
   2. Write a state diagram of a PDA that accepts exactly the Cond statements of Scheme (using condition and statement as simple input symbols).

5. Turing Machine for 2 a's: Consider an input alphabet Σ = {a, b}, and let L = {strings w over Σ | w contains two consecutive a's}. Design a Turing machine that accepts the language L. Write out your machine in full, both using a complete transition table and using a state diagram.

   Consider the input alphabet for this problem to be {a, b}.

6. Turing Machine for Palindromes: Design a Turing machine that reads a string s and returns the string ss^R, where s^R is the reverse of the string s. For example, given the string "abbaa", the Turing machine should halt after "abbaaaabba" is printed on the tape. As in Supplemental Problem 5, write out your machine in full, both using a complete transition table and using a state diagram, and consider the input alphabet to be {a, b}.

This document is available on the World Wide Web as

http://www.walker.cs.grinnell.edu/courses/341.sp12/assignments.shtml

created 5 January 1999 last revised 22 February 2012
For more information, please contact Henry M. Walker at walker@cs.grinnell.edu.