Theory of Computation

ISBN-13: 9781849965712 / Angielski / Miękka / 2010 / 418 str.

Dexter C. Kozen

Opis

This textbook is uniquely written with dual purpose. It cover cores 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. This innovative text focuses primarily on computational complexity theory: the classification of computational problems in terms of their inherent complexity. The book contains an invaluable collection of lectures for first-year graduates on the theory of computation. Topics and features include more than 40 lectures for first year graduate students, and a dozen homework sets and exercises.

Recenzje

From the reviews:

"This book represents the lecture notes of Dexter Kozen for the first-year graduate students in computer science at Cornell University. The book contains 41 primary lectures and 10 supplementary lectures covering more specialized and advanced topics. There are also 12 homework sets and several miscellaneous homework exercises ... many with hints and complete solutions. ... there is a bibliography of 127 titles. The book contains a very useful list of notations and abbreviations and an index." (Daniela Marinescu, Zentralblatt MATH, Vol. 1102 (4), 2007)

"The book is a collection of lecture notes based on a one-semester course for first-year graduate students in computer science at Cornell ... . The course serves a dual purpose: to cover material in the foundations of computing for graduate students in computer science preparing for their Ph.D. qualifying exams, and to provide an introduction to some more advanced topics in the theory of computational complexity for those intending to pursue further study in the area." (Ulrich Tamm, Mathematical Reviews, Issue 2007 f)

"This textbook covers topics essential to the theory of computation. ... In short, this is an interesting and enjoyable book that is strongly recommended to people who appreciate accuracy and concision. It will surely be an important textbook on the theory of computation for years to come. The intended audience is advanced undergraduate and first-year graduate students in computer science. It could also be useful to computer scientists with an interest in the foundation of computing and computational complexity." (G. Ciobanu, Computing Reviews, Vol. 49 (5), May, 2008)

"Kozen does a great job of explaining the material... The book is a continuation of the author's excellent work in the field... The 12 homework sets, along with several miscellaneous problem sets, make this book an excellent pedagogical option for the classroom." (Hector Zenil, ACM Computing Reviews, Vol. 49 (9), September 2008)

Spis treści

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 1 Solutions.- 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.

Blurb

In these early years of the 21^st Century, researchers in the field of computing are delving ever further into the new possibilities of the science and to the primary tools that form its foundations. The theory behind computation has never been more important.

Theory of Computation is a unique textbook that serves the dual purposes of covering core material in the foundations of computing, as well as providing an introduction to some more advanced contemporary topics. This innovative text focuses primarily, although by no means exclusively, on computational complexity theory: 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. Although the complexity universe stops at polynomial space in most treatments, this work also examines higher complexity levels all the way up through primitive and partial recursive functions and the arithmetic and analytic hierarchies.

Topics and features:

• Provides in-depth coverage of both classical and contemporary approaches in one useful, concise volume

• Organized into readily applicable, self-contained primary and secondary lectures

• Contains more than 180 homework exercises of varying difficulty levels, many with hints and solutions

• Includes approximation and inapproximation results, and some lower bounds

• Treats complexity theory and classical recursion theory in a unified framework

Advanced undergraduates and first-year graduates in Computer Science or Mathematics will receive a thorough grounding in the core theory of computation and computational complexity, as well as an introduction to advanced contemporary topics for further study. Computing professionals and other scientists interested in learning more about these topics will also find this text extremely useful.

 Prof. Dexter Kozen teaches at Cornell University, Ithaca, New York, and has comprehensively class-tested this book’s content. He authored the highly successful Automata and Computability, which offers an introduction to the basic theoretical models of computability, and The Design and Analysis of Algorithms.