CSC 341 Grinnell College Spring, 2012
 
Automata, Formal Languages, and Computational Complexity
 

Assignments

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


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. 12 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 2h - 1 nodes.

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

    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 ssR, where sR 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}.


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