Abstract Recursion and Intrinsic Complexity

Description

This book presents and applies a framework for studying the complexity of algorithms. It is aimed at logicians, computer scientists, mathematicians and philosophers interested in the theory of computation and its foundations, and it is written at a level suitable for non-specialists. Part I provides an accessible introduction to abstract recursion theory and its connection with computability and complexity. This part is suitable for use as a textbook for an advanced undergraduate or graduate course: all the necessary elementary facts from logic, recursion theory, arithmetic and algebra are included. Part II develops and applies an extension of the homomorphism method due jointly to the author and Lou van den Dries for deriving lower complexity bounds for problems in number theory and algebra which (provably or plausibly) restrict all elementary algorithms from specified primitives. The book includes over 250 problems, from simple checks of the reader's understanding, to current open problems.  Read more... Abstract......Page 1 Title page......Page 3 Contents......Page 5 Introduction......Page 9 1. Preliminaries......Page 15 Part I. Abstract (first order) recursion......Page 55 2. Recursive (McCarthy) programs......Page 57 3. Complexity theory for recursive programs......Page 111 Part II. Intrinsic complexity......Page 135 4. The homomorphism method......Page 137 5. Lower bounds from Presburger primitives......Page 167 6. Lower bounds from division with remainder......Page 179 7. Lower bounds from division and multiplication......Page 199 8. Non-uniform complexity in ℕ......Page 211 9. Polynomial nullity (0-testing)......Page 217 References......Page 237 Symbol index......Page 245 General index......Page 247