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This volume, with a foreword by Sir Roger Penrose, discusses the foundations of computation in relation to nature.

It focuses on two main questions:

  • What is computation?
  • How does nature compute?

The contributors are world-renowned experts who have helped shape a cutting-edge computational understanding of the universe. They discuss computation in the world from a variety of perspectives, ranging from foundational concepts to pragmatic models to ontological conceptions and philosophical implications.

The volume provides a state-of-the-art collection of technical papers and non-technical essays, representing a field that assumes information and computation to be key in understanding and explaining the basic structure underpinning physical reality. It also includes a new edition of Konrad Zuse's “Calculating Space” (the MIT translation), and a panel discussion transcription on the topic, featuring worldwide experts in quantum mechanics, physics, cognition, computation and algorithmic complexity.

The volume is dedicated to the memory of Alan M Turing — the inventor of universal computation, on the 100th anniversary of his birth, and is part of the Turing Centenary celebrations.

    • Foreword (R Penrose)
    • Preface
    • Acknowledgements
    • Introducing the Computable Universe (H Zenil)
  • Historical, Philosophical & Foundational Aspects of Computation:
    • Origins of Digital Computing: Alan Turing, Charles Babbage, & Ada Lovelace (D Swade)
    • Generating, Solving and the Mathematics of Homo Sapiens. E Post's Views on Computation (L De Mol)
    • Machines (R Turner)
    • Effectiveness (N Dershowitz & E Falkovich)
    • Axioms for Computability: Do They Allow a Proof of Church's Thesis? (W Sieg)
    • The Mathematician's Bias — and the Return to Embodied Computation (S B Cooper)
    • Intuitionistic Mathematics and Realizability in the Physical World (A Bauer)
    • What is Computation? Actor Model versus Turing's Model (C Hewitt)
  • Computation in Nature & the Real World:
    • Reaction Systems: A Natural Computing Approach to the Functioning of Living Cells (A Ehrenfeucht, J Kleijn, M Koutny & G Rozenberg)
    • Bacteria, Turing Machines and Hyperbolic Cellular Automata (M Margenstern)
    • Computation and Communication in Unorganized Systems (C Teuscher)
    • The Many Forms of Amorphous Computational Systems (J Wiedermann)
    • Computing on Rings (G J Martínez, A Adamatzky & H V McIntosh)
    • Life as Evolving Software (G J Chaitin)
    • Computability and Algorithmic Complexity in Economics (K V Velupillai & S Zambelli)
    • Blueprint for a Hypercomputer (F A Doria)
  • Computation & Physics & the Physics of Computation:
    • Information-Theoretic Teleodynamics in Natural and Artificial Systems (A F Beavers & C D Harrison)
    • Discrete Theoretical Processes (DTP) (E Fredkin)
    • The Fastest Way of Computing All Universes (J Schmidhuber)
    • The Subjective Computable Universe (M Hutter)
    • What Is Ultimately Possible in Physics? (S Wolfram)
    • Universality, Turing Incompleteness and Observers (K Sutner)
    • Algorithmic Causal Sets for a Computational Spacetime (T Bolognesi)
    • The Computable Universe Hypothesis (M P Szudzik)
    • The Universe is Lawless or “Pantôn chrêmatôn metron anthrôpon einai” (C S Calude, F W Meyerstein & A Salomaa)
    • Is Feasibility in Physics Limited by Fantasy Alone? (C S Calude & K Svozil)
  • The Quantum, Computation & Information:
    • What is Computation? (How) Does Nature Compute? (D Deutsch)
    • The Universe as Quantum Computer (S Lloyd)
    • Quantum Speedup and Temporal Inequalities for Sequential Actions (M Żukowski)
    • The Contextual Computer (A Cabello)
    • A Gödel-Turing Perspective on Quantum States Indistinguishable from Inside (T Breuer)
    • When Humans Do Compute Quantum (P Zizzi)
  • Open Discussion Section:
    • Open Discussion on A Computable Universe (A Bauer, T Bolognesi, A Cabello, C S Calude, L De Mol, F Doria, E Fredkin, C Hewitt, M Hutter, M Margenstern, K Svozil, M Szudzik, C Teuscher, S Wolfram & H Zenil)
  • Live Panel Discussion (transcription):
    • What is Computation? (How) Does Nature Compute? (C S Calude, G J Chaitin, E Fredkin, A J Leggett, R de Ruyter, T Toffoli & S Wolfram)
  • Zuse's Calculating Space:
    • Calculating Space (Rechnender Raum) (K Zuse)
    • Afterword to Konrad Zuse's Calculating Space (A German & H Zenil)

Readership: Graduate students who are specialized researchers in computer science, information theory, quantum theory and modern philosophy and the general public who are interested in these subject areas.

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