## Computability and Complexity TheoryThis volume introduces materials that are the core knowledge in the theory of computation. The book is self-contained, with a preliminary chapter describing key mathematical concepts and notations and subsequent chapters moving from the qualitative aspects of classical computability theory to the quantitative aspects of complexity theory. Dedicated chapters on undecidability, NP-completeness, and relative computability round off the work, which focuses on the limitations of computability and the distinctions between feasible and intractable. Topics and features: *Concise, focused materials cover the most fundamental concepts and results in the field of modern complexity theory, including the theory of NP-completeness, NP-hardness, the polynomial hierarchy, and complete problems for other complexity classes *Contains information that otherwise exists only in research literature and presents it in a unified, simplified manner; for example, about complements of complexity classes, search problems, and intermediate problems in NP *Provides key mathematical background information, including sections on logic and number theory and algebra *Supported by numerous exercises and supplementary problems for reinforcement and self-study purposes With its accessibility and well-devised organization, this text/reference is an excellent resource and guide for those looking to develop a solid grounding in the theory of computing. Beginning graduates, advanced undergraduates, and professionals involved in theoretical computer science, complexity theory, and computability will find the book an essential and practical learning tool. |

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### Contents

Chapter 1 Preliminaries | 1 |

Chapter 2 Introduction to Computability | 23 |

Chapter 3 Undecidability | 41 |

Chapter 4 Introduction to Complexity Theory | 74 |

Chapter 5 Basic Results of Complexity Theory | 81 |

Chapter 6 Nondeterminism and NPCompleteness | 123 |

Chapter 7 Relative Computability | 145 |

Chapter 8 Nonuniform Complexity | 181 |

Chapter 9 Parallelism | 200 |

Chapter 10 Probabilistic Complexity Classes | 225 |

Chapter 11 Introduction to Counting Classes | 247 |

Chapter 12 Interactive Proof Systems | 261 |

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### Common terms and phrases

accepting computation accepting configuration algorithm alternating Turing machine assume belongs to NP Boolean formula bounded cells clean(k co-NP complexity classes complexity theory computably enumerable computation tree Corollary define Definition denote deterministic Turing machine effectively presentable encoding example exists family of circuits finite function f halts Hamiltonian circuit Homework ifand induction input of length input string input word interactive proof system language Lemma logn logspace machine that accepts moves multitape Turing machine natural numbers nondeterministic Turing machine NP-complete Observe one-tape oracle Turing machine output P/poly partial computable function partial function Pm-complete polynomial hierarchy polynomial-time positive integer probabilistic procedure proof of Theorem Proposition protocol prove PSPACE quantifiers query random recursive reducibility result satisfying assignments scanned simulation space-bounded Turing machine space-constructible steps subset Suppose symbol tape time-bounded Turing machine total computable function undecidable variables verifier vertex cover words of length