The Theory of Computation Bernard M.E. Moret

TABLE OF CONTENTS

1. Introduction

1.1 Motivation and Overview
1.2 History

2. Preliminaries

2.1 Numbers and their Representation
2.2 Problems, Instances, and Solutions
2.3 Asymptotic Notation
2.4 Graphs
2.5 Alphabets, Strings, and Languages
2.6 Functions and Infinite Sets
2.7 Pairing Functions
2.8 Cantor's Proof: the Technique of Diagonalization
2.9 Implications for Computability
2.10 Exercises
2.11 Bibliography

3. Finite Automata and Regular Languages

3.1 Introduction

3.1.1 States and Automata

3.1.2 Finite Automata as Language Acceptors

3.1.3 Determinism and Nondeterminism

3.1.4 Checking vs. Computing
3.2 Properties of Finite Automata

3.2.1 Equivalence of Finite Automata

3.2.2 epsilon Transitions
3.3 Regular Expressions

3.3.1 Definitions and Examples

3.3.2 Regular Expressions and Finite Automata

3.3.3 Regular Expressions from Deterministic Finite Automata

  The Base Case

  The Inductive Step

  Completing the Proof

  Reviewing the Construction of Regular Expressions from Finite Automata
3.4 The Pumping Lemma and Closure Properties

3.4.1 The Pumping Lemma

3.4.2 Closure Properties of Regular Languages

3.4.3 Ad Hoc Closure Properties
3.5 Conclusion
3.6 Exercises
3.7 Bibliography

4. Universal Models of Computation

4.1 Encoding Instances
4.2 Choosing a Model of Computation

4.2.1 Issues of Computability

4.2.2 The Turing Machine

4.2.3 Multitape Turing Machines

4.2.4 The Register Machine

4.2.5 Translation Between Models
4.3 Model Independence
4.4 Turing Machines as Enumerators and Acceptors
4.5 Exercises
4.6 Bibliography

5. Computability Theory

5.1 Primitive Recursive Functions

5.1.1 Defining Primitive Recursive Functions

5.1.2 Ackermann's Function and the Grzegorczyk Hierarchy
5.2 Partial Recursive Functions
5.3 Arithmetization: Encoding a Turing Machine
5.4 Programming Systems
5.5 Recursive and R.E. Sets
5.6 Rice's Theorem and the Recursion Theorem
5.7 Degrees of Unsolvability
5.8 Exercises
5.9 Bibliography

6. Complexity Theory: Foundations

6.1 Reductions

6.1.1 Reducibility Between Problems

6.1.2 Reductions and Complexity Classes
6.2 Classes of Complexity

6.2.1 Hierarchy Theorems

6.2.2 Model-Independent Complexity Classes

  Deterministic complexity classes

  Certificates and nondeterminism
6.3 Complete Problems

6.3.1 NP-Completeness: Cook's Theorem

6.3.2 Space Completeness

  Polynomial space

  Polylogarithmic space

6.3.3 Provably Intractable Problems
6.4 Exercises
6.5 Bibliography

7. Proving Problems Hard

7.1 Some Important NP-Complete Problems
7.2 Some P-Completeness Proofs
7.3 From Decision to Optimization and Enumeration

7.3.1 Turing Reductions and Search Problems

7.3.2 The Polynomial Hierarchy

7.3.3 Enumeration Problems
7.4 Exercises
7.5 Bibliography

8. Complexity Theory in Practice

8.1 Circumscribing Hard Problems

8.1.1 Restrictions of Hard Problems

8.1.2 Promise Problems
8.2 Strong NP-Completeness
8.3 The Complexity of Approximation

8.3.1 Definitions

8.3.2 Constant-Distance Approximations

8.3.3 Approximation Schemes

8.3.4 Fixed-Ratio Approximations and the Class OptNP

8.3.5 No Guarantee Unless P equals NP
8.4 The Power of Randomization
8.5 Exercises
8.6 Bibliography

9. Complexity Theory: The Frontier

9.1 Introduction
9.2 The Complexity of Specific Instances
9.3 Average-Case Complexity
9.4 Parallelism and Communication

9.4.1 Parallelism

9.4.2 Models of Parallel Computation

9.4.3 When Does Parallelism Pay?

9.4.4 Communication and Complexity
9.5 Interactive Proofs and Probabilistic Proof Checking

9.5.1 Interactive Proofs

9.5.2 Zero-Knowledge Proofs

9.5.3 Probabilistically Checkable Proofs
9.6 Complexity and Constructive Mathematics
9.7 Bibliography

References

A. Proofs

A.1 Quod Erat Demonstrandum, or What is a Proof?
A.2 Proof Elements
A.3 Proof Techniques

A.3.1 Construction: Linear Thinking

A.3.2 Contradiction: Reductio ad Absurdum

A.3.3 Induction: the Domino Principle

A.3.4 Diagonalization: Putting it all Together
A.4 How to Write a Proof
A.5 Practice