New Computational Paradigms

1. Front Matter

   Pages i-xiii

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2. The Turing Model of Computation and its Applications to Logic, Mathematics, Philosophy, and Computer Science

   1. Alan Turing, Logical and Physical

      • Andrew Hodges

      Pages 3-15

   2. Computability and Numberings

      • Serikzhan Badaev, Sergey Goncharov

      Pages 19-34

   3. Computation as Conversation

      • Johan van Benthem

      Pages 35-58

   4. Computation Paradigms in Light of Hilbert's Tenth Problem

      • Yuri Matiyasevich

      Pages 59-85

   5. Elementary Algorithms and Their Implementations

      • Yiannis N. Moschovakis, Vasilis Paschalis

      Pages 87-118

   6. Applications of the Kleene–Kreisel Density Theorem to Theoretical Computer Science

      • Dag Normann

      Pages 119-138

   7. Church Without Dogma: Axioms for Computability

      • Wilfried Sieg

      Pages 139-152

   8. Computability on Topological Spaces via Domain Representations

      • Viggo Stoltenberg-Hansen, John V. Tucker

      Pages 153-194

   9. On the Power of Broadcasting in Mobile Computing

      • Jiří Wiedermann, Dana Pardubská

      Pages 195-209

3. Logic, Algorithms and Complexity

   1. The Computational Power of Bounded Arithmetic from the Predicative Viewpoint

      • Samuel R. Buss

      Pages 213-222

   2. Effective Uniform Bounds from Proofs in Abstract Functional Analysis

      • Ulrich Kohlenbach

      Pages 223-258

   3. Effective Fractal Dimension in Algorithmic Information Theory

      • Elvira Mayordomo

      Pages 259-285

   4. Metamathematical Properties of Intuitionistic Set Theories with Choice Principles

      • Michael Rathjen

      Pages 287-312

   5. New Developments in Proofs and Computations

      • Helmut Schwichtenberg

      Pages 313-340

4. Models of Computation from Nature

   1. From Cells to (Silicon) Computers, and Back

      • Gheorghe Păun

      Pages 343-371

   2. Computer Science, Informatics, and Natural Computing—Personal Reflections

      • Grzegorz Rozenberg

      Pages 373-379

5. Computable Analysis and Real Computation

   1. A Survey on Continuous Time Computations

      • Olivier Bournez, Manuel L. Campagnolo

      Pages 383-423

   2. A Tutorial on Computable Analysis

      • Vasco Brattka, Peter Hertling, Klaus Weihrauch

      Pages 425-491

   3. A Continuous Derivative for Real-Valued Functions

      • Abbas Edalat

      Pages 493-519

In recent years, classical computability has expanded beyond its original scope to address issues related to computability and complexity in algebra, analysis, and physics. The deep interconnection between "computation" and "proof" has originated much of the most significant work in constructive mathematics and mathematical logic of the last 70 years. Moreover, the increasingly compelling necessity to deal with computability in the real world (such as computing on continuous data, biological computing, and physical models) has brought focus to new paradigms of computation that are based on biological and physical models. These models address questions of efficiency in a radically new way and even threaten to move the so-called Turing barrier, i.e. the line between the decidable and the un-decidable.

This book examines new developments in the theory and practice of computation from a mathematical perspective, with topics ranging from classical computability to complexity, from biocomputing to quantum computing. The book opens with an introduction by Andrew Hodges, the Turing biographer, who analyzes the pioneering work that anticipated recent developments concerning computation’s allegedly new paradigms. The remaining material covers traditional topics in computability theory such as relative computability, theory of numberings, and domain theory, in addition to topics on the relationships between proof theory, computability, and complexity theory. New paradigms of computation arising from biology and quantum physics are also discussed, as well as the computability of the real numbers and its related issues.

This book is suitable for researchers and graduate students in mathematics, philosophy, and computer science with a special interest in logic and foundational issues. Most useful to graduate students are the survey papers on computable analysis and biological computing. Logicians and theoretical physicists will also benefit from this book.

• Analysis
• algorithms
• complexity
• complexity theory
• computability theory
• computer
• computer science
• information theory
• logic
• mathematical logic
• model theory
• philosophy
• proof theory

From the reviews:

“It is addressed to researcher and graduate students … . All contributions to the book have been rigorously refereed, and the standards with respect to layout, references … are high. … This is a piece of excellent pedagogical work. The paper is hereby recommended. … I personally find very readable and informative. … I enjoyed reading these papers, and I assume they are all right when we take them for what they are … .”­­­ (Lars Kristiansen, Studia Logica, Vol. 97, 2011)

• University of Leeds, UK

  S. Barry Cooper

• University of Amsterdam, Netherlands

  Benedikt Löwe

• Università di Siena, Italy

  Andrea Sorbi

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