This textbook is uniquely written with dual purpose. It covers core material in the foundations of computing for graduate students in computer science and also provides an introduction to some more advanced topics for those intending further study in the area. The book contains an invaluable collection of lectures for first-year graduates on the theory of computation, focusing primarily on computational complexity theory. It also deals with the classification of computational problems in terms of their inherent complexity.

It incorporates rigorous treatment of computational models, such as deterministic, nondeterministic, and alternating Turing machines; circuits; probabilistic machines; interactive proof systems; automata on infinite objects; and logical formalisms. Features include more than 40 lectures for first year graduate students, and a dozen homework sets and exercises. The book is aimed at advanced undergraduates and first-year graduates in Computer Science or Mathematics.

Lectures.- The Complexity of Computations.- Time and Space Complexity Classes and Savitch's Theorem.- Separation Results.- The Immerman-Szelepcsényi Theorem.- Logspace Computability.- The Circuit Value Problem.- The Knaster-Tarski Theorem.- Alternation.- Problems Complete for PSPACE.- The Polynomial-Time Hierarchy.- More on the Polynomial-Time Hierarchy.- Parallel Complexity.- Relation of NC to Time-Space Classes.- Probabilistic Complexity.- BPP ?2P ? ?2P.- Chinese Remaindering.- Complexity of Primality Testing.- Berlekamp's Algorithm.- Interactive Proofs.- PSPACE IP.- IP PSPACE.- Probabilistically Checkable Proofs.- NP PCP(n3, 1).- More on PCP.- A Crash Course in Logic.- Complexity of Decidable Theories.- Complexity of the Theory of Real Addition.- Lower Bound for the Theory of Real Addition.- Lower Bound for Integer Addition.- Automata on Infinite Strings and S1S.- Determinization of ?-Automata.- Safra's Construction.- Relativized Complexity.- Nonexistence of Sparse Complete Sets.- Unique Satisfiability.- Toda's Theorem.- Circuit Lower Bounds and Relativized PSPACE = PH.- Lower Bounds for Constant Depth Circuits.- The Switching Lemma.- Tail Bounds.- The Gap Theorem and Other Pathology.- Partial Recursive Functions and Gödel Numberings.- Applications of the Recursion Theorem.- Abstract Complexity.- 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.- Homework 1.- Homework 2.- Homework 3.- Homework 4.- Homework 5.- Homework 6.- Homework 7.- Homework 8.- Homework 9.- Homework 10.- Homework 11.- Homework 12.- Miscellaneous Exercises.- Hints and Solutions.- Homework 1Solutions.- Homework 2 Solutions.- Homework 3 Solutions.- Homework 4 Solutions.- Homework 5 Solutions.- Homework 6 Solutions.- Homework 7 Solutions.- Homework 8 Solutions.- Homework 9 Solutions.- Homework 10 Solutions.- Homework 11 Solutions.- Homework 12 Solutions.- Hints for Selected Miscellaneous Exercises.- Solutions to Selected Miscellaneous Exercises.