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Computable Functions
A. Shen, Independent University of Moscow, Russia, and N. K. Vereshchagin, Moscow State Lomonosov University, Russia
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Student Mathematical Library
2003; 166 pp; softcover
Volume: 19
ISBN-10: 0-8218-2732-4
ISBN-13: 978-0-8218-2732-1
List Price: US$34
Institutional Members: US$27.20
All Individuals: US$27.20
Order Code: STML/19
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Basic Set Theory - A Shen and N K Vereshchagin

In 1936, before the development of modern computers, Alan Turing proposed the concept of a machine that would embody the interaction of mind, machine, and logical instruction. The idea of a "universal machine" inspired the notion of programs stored in a computer's memory. Nowadays, the study of computable functions is a core topic taught to mathematics and computer science undergraduates.

Based on the lectures for undergraduates at Moscow State University, this book presents a lively and concise introduction to the central facts and basic notions of the general theory of computation. It begins with the definition of a computable function and an algorithm and discusses decidability, enumerability, universal functions, numberings and their properties, \(m\)-completeness, the fixed point theorem, arithmetical hierarchy, oracle computations, and degrees of unsolvability. The authors complement the main text with over 150 problems. They also cover specific computational models, such as Turing machines and recursive functions.

The intended audience includes undergraduate students majoring in mathematics or computer science, and all mathematicians and computer scientists who would like to learn basics of the general theory of computation. The book is also an ideal reference source for designing a course.

Readership

Undergraduates, graduate students, research mathematicians, and computer scientists and programmers interested in the general theory of computation.

Reviews

"Material is a presented quite clearly and with a minimum of fuss ... an excellent text for a first course in computability."

-- Mathematical Reviews

Table of Contents

  • Computable functions, decidable and enumerable sets
  • Universal functions and undecidability
  • Numberings and operations
  • Properties of Gödel numberings
  • Fixed point theorem
  • \(m\)-reducibility and properties of enumerable sets
  • Oracle computations
  • Arithmetical hierarchy
  • Turing machines
  • Arithmeticity of computable functions
  • Recursive functions
  • Bibliography
  • Glossary
  • Index
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