Syllabus

We assume that you have taken 6.042, Mathematics for Computer Science. 6.045 is, at heart, a mathematics course, and we assume that you are reasonably facile with mathematical concepts. In particular, we assume that you are comfortable with formal mathematical proofs, and can write them up properly.

The book for this class is Introduction to the Theory of Computation by Michael Sipser.

We hope that students who find the textbook unenlightening can consult these books for a different explanation:

• Martin, John. Introduction to Languages and the Theory of Computing.
  
  
• Kozen, Dexter. Automata and Complexity.
  
  
• Johnson, David S., and Michael Garey. Computers and Intractability : A Guide to the Theory of NP-Completeness.
  
  
• Ullman, Jeffrey, John Hopcroft, and Rajeev Motwani. Introduction of Automata, Languages, and Computation.
  
  
• Minsky, Marvin. Computation of Infinite Machines.A

On completion of 6.045, students will be able to explain the basic methods and conclusions of the Theory of Computation. They will be able to apply these methods to problems from different fields and be guided by the results in searching for computational solutions to the problems.

In particular, students will be able to:

1. Explain the theoretical limits on computational solutions of undecidable and inherently complex problems.
   
   
2. Describe concrete examples of computationally undecidable or inherently infeasible problems from different fields.
   
   
3. Devise and analyze the complexity of procedures to determine properties of computationally bounded automata.
   
   
4. Understand formal definitions of machine models.
   
   
5. Prove the undecidability or complexity of a variety of problems.

Students will be able to:

1. Synthesize finite automata with specific properties.
   
   
2. Convert among multiple representations of finite automata.
   
   
3. Use pigeon-holing arguments and closure properties to prove particular problems cannot be solved by finite automata.
   
   
4. Calculate assymptotic estimates of the computational complexity of simple procedures from automata, language and graph theory.
   
   
5. Prove undecidability using diagonalization and reducibility methods.
   
   
6. Use the relationship between recognizability and decidability to determine decidability properties of problems.
   
   
7. Describe concrete examples of undecidable problems from different fields.
   
   
8. Define, and explain the significance of, the "P = NP?" question and NP-completeness.
   
   
9. Describe concrete examples of NP-complete problems from different fields.
   
   
10. Prove lower bounds on time and space complexity using diagonalization and polynomial time reducibility methods.
    
    
11. Define deterministic and nondeterministic computation time and space, and explain the relationships among them.
    
    
12. Describe concrete examples of decidable problems that are known to be unsolvable in polynomial time.

Objectives: Related Outcomes

• Undecidability and complexity: 5, 6, 7, 8, 9, 10, 11, 12
  
• Example undecidable/complex problems: 6, 9, 11, 12
  
• Automata: 1, 2, 3
  
• Formal models: 1, 2, 3
  
• Prove undecidability/complexity: 3, 4, 5, 6, 7, 8, 10, 11
  
  

Outcomes: Reflected Objectives

• Synthesize DFA's: 3, 4
  
• Regular expressions, etc.: 3, 4
  
• Nonregularity: 3, 4, 5
  
• Assymptotic complexity: 5
  
• Undecidable examples: 1, 2
  
• Undecidability: 1, 5
  
• Recognizability: 1, 5
  
• P and NP: 1, 5
  
• NP examples: 1, 2
  
• Undecidability and complexity: 1, 5
  
• Time-space: 1, 2, 5
  
• Examples: 1, 2

Lectures:
Two sessions / week
1.5 hours / session

Recitations:
One session / week
1 hour / session

There will be (approximately) weekly homework assignments, three in-class quizzes, and a final exam. The final grade will be computed using the following weights:

• Homework: 30%
• Quizzes: 30%
• Final Exam: 30%
• Participation in Class and Recitation Sections: 10%

Homework will be due approximately every week, at the beginning of lecture. Extensions are possible with advance warning. The greater the warning, the more likely the odds of an extension. No extensions will be granted the day a homework is due, and no homework will be accepted after solutions are posted.

With regards to homeworks: full credit will be given for correct answers and proofs, of course. We will also grant partial credit for partial solutions and solutions with minor flaws. We will also give a small amount of partial credit for answers which read in full, "I don't know." Likewise, proofs with gaps will receive partial credit, and the partial credit granted will increase if the gaps are explicitly noted. We will give no credit for wildly incorrect answers which are obviously only there in the hopes of getting partial credit. Please only write down answers in which you are confident. Wild guesses only waste our time. Making yourself believe a false proof is bad for your brain.

We require that all homework solutions be typed up. We will provide LaTeX shells for you to flesh out with your solutions, but you do not need to use them. Hand-drawn diagrams are permitted.

We encourage collaboration. In fact, we regard it as essential to passing the class. We do not expect you to be able to solve every homework problem on your own. We do expect you to write down your solutions on your own, however. To repeat: each person must write up their solutions separately. Also, please note the people with whom you worked. If you consult any reference material other than the textbook, please note on your homework which sources you used for each problem.

The quizzes will be held on:

• Day 10
  
• Day 22
  
• Day 32

There will be no homework due on the day in which a quiz is given.

The final has yet to be scheduled.