|
Theory of Computation: Classical & Contemporary Approaches
This textbook has been written with the dual purpose to cover core material in the foundations of computing for graduate students in computer science, as well as to provide an introduction to some more advanced topics for those intending further study in the area. This book contains an invaluable collection of lectures for first-year graduates on the
theory of computation, focusing primarily on computational complexity theory. Topics and features include: organization into self-contained lectures of 3-7 pages; 41 primary lectures and a handful of supplementary lectures covering more specialized or advanced topics; and 12 homework sets and several miscellaneous homework exercises of varying levels of difficulty, many with hints and complete solutions. Aimed at advanced undergraduates and first-year graduates in Computer Science or Mathematics with an interest in the theory of computation and computational complexity, this book provides a thorough grounding the foundations of computational complexity theory.
Contents:
The Complexity of Computations.- Time and Space Complexity Classes and Savitch's Theorem.- Separation Results.- Logspace Computability.- The Circuit Value Problem.- The Knaster-Tarski Theorem.- Alternation.- The Polynomial-Time Hierarchy.- Parallel Complexity.- Probabilistic Complexity.- Chinese Remaindering.- Berlekamp's Algorithm.- Interactive Proofs.- Probabilistically Checkable Proofs.- Complexity of Decidable Theories.- Complexity of the Theory of Real Addition.- Lower Bound for the Theory of Real Addition.- Safra's Construction.- Relativized Complexity.- Nonexistence of Sparse Complete Sets.- Unique Satisfiability.- Toda's Theorem.- Lower Bounds for Constant Depth Circuits.- The Switching Lemma.- Tail Bounds.- Applications of the Recursion Theorem.- The Arithmetic Hierarchy.- Complete Problems in the Arithmetic Hierarchy.- Post's Problem.- The Friedberg - Muchnik Theorem.- The Analytic Hierarchy.- Kleene's Theorem.- Fair Termination and Harel's Theorem.- Exercises.- Hints and Solutions
|
