COSC 312 — Algorithm Analysis and Automata
Spring 2012

Directory

• Contact Information
• Catalog Description
• Prerequisites
• Texts
• Schedule
  
• Topics
• Homework and Tests
• Exams and Grading 
• For Students with Disabilities
• Handouts (in pdf form)
  
  
• MacLennan’s home page
• Department of Electrical Engineering & Computer Science
• UTK Home Page
  
  
• ETS Major Field Test in Computer Science content [pdf]
  

Contact Information

Teaching Assistants:
Zahra Mahoor
Office: Min Kao 204
Hours: MW 10:00–12:00 or make an appointment
Email:  zmahoor at utk.edu

Cale Nelson

Office: Min Kao 207
Email: cnelso23 at utk.edu

Classes: MWF 12:20–1:10, Min Kao 406

This page: http://web.eecs.utk.edu/~mclennan/Classes/312

Catalog Description

Prerequisites

Texts

There are two required textbooks for COSC 312:

• Michael Sipser: Introduction to the Theory of Computation, Course Technology. $146 from Amazon.com, but used copies are available from $90. [S]
  Make sure to see the Errata [pdf]; it would be a good idea to write in your copy of the book the significant corrections for the chapters we will use (chs. 0–4, 7).
  
  
• Clifford A. Shaffer, Data Structures and Algorithm Analysis Edition 3.2 (C++ Version). Free online. [Sh] 

Schedule

1. Finite Automata (S[ipser] 1.1–2)
2. Regular Expressions and Languages (S 1.3–4)
   
3. Context-free Grammars and Languages (S 2.1)
   
4. Pushdown Automata (S 2.2–3)
   
5. Turing Machines and Other Models of Computation (S 3.1–3)
   
6. Uncomputable Problems (S 4.1–2; see also Shaffer §17.3.2)
7. Complexity (S 7.1–4; see also Shaffer ch. 17)
8. Algorithm Analysis (Shaffer §2.4, chs. 3, 14)
   

Topics

• Basics Of Counting
• Algorithm Analysis
• Basic Automata Theory
• Computability

Basics Of Counting

Topics: Most of the “basics of counting” are covered in COSC 311, and so COSC 312 will cover the following:

• Solving recurrence relations [Sh 2.4, 14.2]
• The Master theorem [Sh 14.2.3]

Learning Outcomes

1. Analyze a problem to create relevant recurrence equations or to identify important counting questions.

Algorithm Analysis

Topics

• Asymptotic analysis of upper and average complexity bounds [S 7.1; Sh 3.4]
  
• Identifying differences among best, average, and worst case behaviors [Sh 3.2; S 7.1]
• Big Oh, little oh, omega, and theta notation [Sh 3.4]
• Standard complexity classes [Sh 3.1]
• Empirical measurements of performance[Sh 3.11]
  
• Time and space tradeoffs in algorithms [Sh 3.9]
  
• Using recurrence relations to analyze recursive algorithms [Sh 2.4, 14.2]
  

Learning Outcomes

1. Explain the use of Big Oh, omega, and theta notation to describe the amount of work done by an algorithm.
2. Use Big Oh, omega, and theta notation to give asymptotic upper, lower, and tight bounds on time and space complexity of algorithms.
3. Determine the time and space complexity of simple algorithms.
4. Deduce recurrence relations that describe the time complexity of recursively defined algorithms.

Basic Automata Theory

Topics: Because there may be some uncertainty about how deeply topics in automata and computabilty theory should be covered, we have included the relevant sections from the Sipser text. Also some of the topics, denoted in italics, are optional and may be covered at the instructor’s discretion:

• Finite-state machines (Sipser 1.1–2)
• Regular expressions and real-world applications (lexical analysis, scripting languages) (Sipser 1.3)
• Relationship between finite-state machines and regular expressions (Sipser 1.3)
• The pumping lemma for regular languages (Sipser 1.4)
• Context free grammars and real-world applications (parsing) (Sipser 2.1)
• Greibach and Chomsky Normal Forms (Sipser 2.1)
• Pushdown automata (Sipser 2.2)
• Relationship between PDAs and context free grammars (Sipser 2.2)
• The pumping lemma for context free languages (Sipser 2.3)

Learning Outcomes

1. Design a finite state machine to accept a specified language
2. Design a regular expression to formally specify a language
3. Describe how finite state machines and regular expressions are related
4. Design a pushdown automata to accept a specified language
5. Design a context free grammar to formally specify a language
6. Describe how pushdown automata and context free grammars are related
7. Describe the difference between a finite state machine and a pushdown automata and describe why a pushdown automata can accept more languages than a finite state machine.

Computability

Topics: Because there may be some uncertainty about how deeply topics in automata and computabilty theory should be covered, we have included the relevant sections from the Sipser text.

• Turing machines (Sipser 3.1)
• Church-Turing Thesis (Sipser 3.3)
• Uncomputable functions (informal definition: Sipser 4.2; Sh 17.3.2)
• The halting problem (Sipser 4.1–2)
• Implications of uncomputability

Learning Outcomes

1. Describe a Turing machine and explain the significance of a Turing machine.
2. Explain why a Turing machine is more powerful than a pushdown automata.
3. Create simple programs using a Turing machine.
4. Explain what it means for a function to be uncomputable.
5. Describe the Church-Turing Thesis and explain its significance.
6. Describe the halting problem and explain its significance.
7. Provide a proof that the halting problem is undecidable.

Homework and Tests

• We will assign weekly homework, which will count a total of 15% of your Homework + Test average. In order to allow me to discuss homework in class and to be fair to all students, this is the late homework policy: First, the presumption is that you will normally turn in homework on time. The individual homeworks are designed to be easily doable in the (approximate) week allocated to each. Also, if you get behind in the homework it will be difficult to catch up, since the material is cumulative. Turning homework in on time means at the beginning of class on the day it is due. Homework turned in after class is considered late. (Of course I will consider extenuating circumstances.) I will take off a half letter grade for each day late, including weekends. The date will be determined by when the TA or I receives the work, not by when you deposit it. Of course, I will consider extenuating circumstances.
• In addition, there will be three tests (each covering two chapters), each of which will count 25% of your Homework + Test average.
• Finally, there will be a quiz over algorithm analysis, counting 10% of your Homework + Test average.

Homework Assignments

1. Due Jan 27: S (p. 83) Ex. 1.4 (c, f), 1.5 (c, f), 1.6 (e, i), 1.32.
   
2. Due Feb 3: S Ex. 1.7 (c), 1.8 (a), 1.9 (a), 1.10 (a), 1.14 (b)*, 1.16 (b); extra credit: 1.15. 
   *Hint for 1.14b: Swap the accept and non-accept states of the NFA in 1.16a and show it doesn’t work (i.e., accept the complement of the original machine).
3. Due Feb 10: S Ex. 1.18 (do for 1.6 e, i), 1.19 (b), 1.20 (b, h), 1.21 (b), 1.29 (b).
   
4. Due Feb 20: S Ex. 2.1 (c, d), 2.4 (b, e), 2.9, 2.14, 2.16; extra credit: 2.6 (b). 
5. Due Feb. 27: S Ex. 2.5 (for 2.4e), 2.7 (for 2.6d), 2.10, 2.11, 2.26.
6. Due Mar. 5: S Ex. 3.1 (c, d), 3.2 (c, e), 3.6, 3.8 (c), 3.15b (Hint: Use a NTM).
7. Due Mar. 16: S Ex. 4.2, 4.4, 4.12*, 4.15**.
   *Hint for 4.12: Re-express the subset condition as an empty language.
   **Hint for 4.15: Re-express the condition as the intersection of two regular languages.  
8. Due Apr 4: S Ex. 7.1 (a, b, e, f), 7.6, 7.8.
   
9. Due Apr 11: S Ex. 7.4, 7.9, 7.10*.
   *Hint: See Ex. 4.3 for ALL_DFA.
   
10. Due Apr 18: S Ex. 7.5, 7.7, 7.11; extra credit: 7.21.
    

Tentative Test Schedule

• Feb 17: Test 1 (S1) 
• Mar 16: Test 2 (S2–3) 
• Apr 23: Test 3 (S4, 7)
• Apr 27: Quiz (Sh 3, 14)
  

Final Exam and Grading

The Final Exam is 10:15 a.m. – 12:15 p.m. on Thursday, May 3. The cumulative Final Exam will be two hours worth of questions similar in difficulty to those on the Tests.

For Students with Disabilities

Students who have a disability that require accommodation(s) should make an appointment with the Office of Disability Services (974-6087) to discuss their specific needs as well as schedule an appointment with me during my office hours.

Handouts

• Errata for Sipser [pdf]
• Problem solving steps from Polya’s How to Solve It [pdf]
• Summary of proof techniques from Solow [pdf]
• Context-free grammar for Algol 60 [pdf, rtf]
  
• Models of Computation, Turing Machines, and the Limits of Turing Computation slides [pdf]. For an example of a physical TM, see A Turing Machine in the Classic Style.
  
• HaltingProblem.cpp (C++ program illustrating the undecidability of the Halting Problem)
  
• RiceTheorem.cpp (C++ program illustrating Rice’s Theorem)
  
• Hierarchy of complexity classes [wikipedia] (principal class discussed in this course) 
• Map of complexity classes 
• Complexity Zoo (468 named and defined complexity classes!)