A COMPUTABLE UNIVERSE Understanding and Exploring Nature as Computation 8306.9789814374293-tp.indd 14/9/12 8:49 AM June 29, 2012 9:45 World Scientific Review Volume - 9in x 6in - 8306 A Computable Universe This page intentionally left blank AComputableUniverse A COMPUTABLE UNIVERSE Understanding and Exploring Nature as Computation Foreword by Sir Roger Penrose Editor Hector Zenil University of Sheffield, UK & Wolfram Research, USA World Scientific NEW JERSEY • 8306.9789814374293-tp.indd LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TA I P E I • CHENNAI 14/9/12 8:49 AM Published by World Scientific Publishing Co Pte Ltd Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library A COMPUTABLE UNIVERSE Understanding and Exploring Nature as Computation Copyright © 2013 by World Scientific Publishing Co Pte Ltd All rights reserved This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA In this case permission to photocopy is not required from the publisher ISBN 978-981-4374-29-3 Printed in Singapore ChinAng - A Computable Universe.pmd 9/4/2012, 8:56 AM June 29, 2012 9:45 World Scientific Review Volume - 9in x 6in - 8306 A Computable Universe To Elena v AComputableUniverse June 29, 2012 9:45 World Scientific Review Volume - 9in x 6in - 8306 A Computable Universe This page intentionally left blank AComputableUniverse August 30, 2012 13:35 World Scientific Review Volume - 9in x 6in - 8306 A Computable Universe AComputableUniverse Contents xiii Foreword R Penrose xxxvii Preface xliii Acknowledgements Introducing the Computable Universe H Zenil Historical, Philosophical & Foundational Aspects of Computation 21 Origins of Digital Computing: Alan Turing, Charles Babbage, & Ada Lovelace 23 D Swade Generating, Solving and the Mathematics of Homo Sapiens E Post’s Views on Computation 45 L De Mol Machines 63 R Turner vii August 30, 2012 13:35 World Scientific Review Volume - 9in x 6in - 8306 A Computable Universe viii AComputableUniverse Contents Effectiveness 77 N Dershowitz & E Falkovich Axioms for Computability: Do They Allow a Proof of Church’s Thesis? 99 W Sieg The Mathematician’s Bias — and the Return to Embodied Computation 125 S B Cooper Intuitionistic Mathematics and Realizability in the Physical World 143 A Bauer What is Computation? Actor Model versus Turing’s Model 159 C Hewitt Computation in Nature & the Real World 187 10 Reaction Systems: A Natural Computing Approach to the Functioning of Living Cells 189 A Ehrenfeucht, J Kleijn, M Koutny & G Rozenberg 11 Bacteria, Turing Machines and Hyperbolic Cellular Automata 209 M Margenstern 12 Computation and Communication in Unorganized Systems 231 C Teuscher 13 The Many Forms of Amorphous Computational Systems J Wiedermann 243 August 30, 2012 13:35 World Scientific Review Volume - 9in x 6in - 8306 A Computable Universe Contents 14 Computing on Rings AComputableUniverse ix 257 G J Mart´ınez, A Adamatzky & H V McIntosh 15 Life as Evolving Software 277 G J Chaitin 16 Computability and Algorithmic Complexity in Economics 303 K V Velupillai & S Zambelli 17 Blueprint for a Hypercomputer 333 F A Doria Computation & Physics & the Physics of Computation 345 18 Information-Theoretic Teleodynamics in Natural and Artificial Systems 347 A F Beavers & C D Harrison 19 Discrete Theoretical Processes (DTP) 365 E Fredkin 20 The Fastest Way of Computing All Universes 381 J Schmidhuber 21 The Subjective Computable Universe 399 M Hutter 22 What Is Ultimately Possible in Physics? 417 S Wolfram 23 Universality, Turing Incompleteness and Observers K Sutner 435 August 30, 2012 13:35 World Scientific Review Volume - 9in x 6in - 8306 A Computable Universe 796 theory of everything, 412 asynchronous stationary amorphous computing system, 250 ATP batteries, 719 Atwood, B., 246 automaton theory, 731 Axiom of choice, 153 axiom systems, 711 Babbage, C., 24, 569 baby universes, 404 background set, 194 Banach, S., 157 Banach-Tarski paradox, 157 Bar-Hillel, Y., 354 Bar-Yehuda, R., 251 Barrow, J D., 415 Barwise, J., 354 basic representations, 510, 514 Batterman, R., 522 Bauer, A., xvii, xli Beal, J., 246 beam routing, 257 Beavers, A., xv, xli, 351 Bekenstein, J., 18 Bell experiments, 11 Bell states, 612 Bell’s inequality, 13 Bell’s theorem, 595, 699 Bell, J., 11, 576 Bell, J.S., 583 Benioff, P., 577 Bergson, H., 716 Bernoulli numbers, 569 BHK interpretation, 150 Bialek, W., 679 Big Bang, 527 big crunch, 404 billiard ball computer model, 675 Binary latching, 30 binary places, 774 binary sequence temporal, 402, 412 biological creativity, 277 biological differentiation, 682 AComputableUniverse Index Bishop, E., 143 Bit, 597 bit-string ontology, 402, 412 Black holes, 18 black matter, 527 black ravens, 401 Bletchley Park, 39 Blum, L., 522 Bohr, N., 14, 794 Bolognesi, T., xvi, xviii, xxii, Bombelli, L., 452 Boole, G., 570 boolean algebra, 718 Boolean logic, 570 Borel, E., 579 Borroughs, 25 Bostrom, N., 415 Brenner, S., 277 Breuer, T., xxiv, xli Bringsjord, S., xl, xliv broadcast protocol, 251 Brouwer, L.E.J., 58, 144 Brukner, Caslav, 584 Bullis, K., 246 Burgin, M., xl Bush, V., 38 Busy Beaver function, 279 Cabello, A., xxiv, xli, Calculating Space, Calude, C., xiv, xliii, 2, 4, 298, 631, 673 canonical form A, 51 canonical form C, 52 Cantor set, 696 Cantor set topology, 696 Cantor space, 697 Cantor, G., 481 Carnap, R., 354 Carriage of tens, 30 Cartesian space, 766 Cauchy, A., 146 causal set, 452 causal set program, 452 causality, 207, 777 August 30, 2012 13:35 World Scientific Review Volume - 9in x 6in - 8306 A Computable Universe Index causet, 452 Cawley, J., 715 cellular automata, 60, 257, 258, 559, 674, 678, 684, 695, 713, 765 rule 30, 700 cellular automaton, 440, 442, 575, 733 cellular automaton, elementary rule 110, 470 Chaitin’s Ω, 285 Chaitin, G., xvi, xxxvii, Chalmers, D.J., 71 Champernowne, D G number, 404 Chandio, F H., 246 Chomsky’s classification, 261 Chuang, I.A., 605 Church’s thesis, 47, 48, 53, 79, 80, 87, 674 Church’s thesis limit, 674 Church, A., xvii, 47, 48, 78, 79, 92, 108, 159, 257, 480, 571, 614, 787 Church-Turing thesis, 79, 92, 100, 172, 438, 480 Extended, 92 Cirac, I., 577 circuit, 353, 356, 361 circular Turing machines, 258 classical computer, 675 classification, 443 Client-cloud computing, 164 coarse-graining, 489 coding universe, 410 coding theorem, collapse wave-function, 402, 404 collision-based computing, 258 collisions, 257 Colmar (de), T, 24 Colossus, 38 combinatorial circuits, 254 communication graph, 250 compact representation, 402 compactification, 404 Compatible experiments, 596 AComputableUniverse 797 complete theory of everything, 406, 409, 414 completeness, 172 complex adaptive system, 355 complexity, 568 Kolmogorov, 407 parameter, 412 program, 407 theory of everything, 407 complexity theory, 355, 615 compression, 401 compression algorithm, 703 comptometers, 25 computability assumption, 413 computable, 436 observer process, 413 universe, 400, 404, 413 computable processes, computable universe hypothesis, 8, 483 computation, 63, 347–350, 356 computation-universality, 683 computational philosophy, 349, 360 Computational Representation Theorem, 159 computational universality, 247 computationalism, xxxviii, computer, 436 computor, 106 Comrie, L., 26 conditions initial, 402, 406 configuration, 442 confluence, 444 consciousness, 617 conservation principles, 683 consistency, 610 mathematical, 407 constructive mathematics, 143 Constructor functions, 84–86, 90, 91 content-addressable memory, 352 context sequence, 194 context-independent process, 195 Contextual computer, 597 Continuity principle, 154 August 30, 2012 13:35 World Scientific Review Volume - 9in x 6in - 8306 A Computable Universe 798 continuous universe, 412 contracting universe, 682 Conway’s game of Life, 266, 470 Conway, J., 469 Cook, M., 440 Cooper, B., xvii, xxxviii, xli Coore, D., 244, 246 Copeland, J., 70 correspondence theory, 410 cosmogony problem, 689 cosmological models, 404, 405 cosmological arrow, 682 cosmology, 680 Costa, J.F., xl countable model, 412 universe, 412 CP-violation, 681 Crick, F., 6, 277, 719 Csirel’son, B.S., 613 CSP, 167 Curry-Howard isomorphism, 720, 724 cybernetics, 785 cyclic tag systems, 258 cyclotrons, 258 C#, 164 Darwin evolution, 693 Darwin’s theory of evolution, 278 Darwin, C., 7, 277, 684 Darwinian evolution, 7, 716 data, 401 Davies, P C W., 673 Davies, P C W., 415 Davis, M., 49, 55, 439 de Bruijn diagrams, 258 De Mol, L., xli de Morgan, A., 570 de Ruyter, R., 673, 679 Deacon, T., 348, 349, 356 decay, 204 decision making, 401 decoherence, 404, 716 AComputableUniverse Index deductive reasoning, 401 degree, 439 intermediate, 439 Delahaye, J.-P., xliii DeMol, L., xvii, xxx Dennett, D., 361 dequantizing algorithms, 682 Dershowitz, N., xli description, 401 description size, determinism, 779 deterministic model, 402 theory, 402 deterministic chaos, 466 Deutsch, D., xvii, xli, 3, 577, 615, 689 Deutsch-Jozsa algorithm, 615 diagonalisation, 605 diagonalization method, xxxiv Difference Engine, 28 difference engine, 569 differential equations, 739 digital, 348 digital computer, 736 digital computers, 734 digital information, digital particle, 748 digital software, 278 digitalization, 742 Dijkstra, E., 167 diophantine equation, 674 Diophantine equations, 77 direct observation, 410 discovery scientific, 401 discrimination of states, 608, 612 dissipative cellular automata algorithms, 683 dissipative lattice gas algorithms, 683 distinguishable systems, 483 diversity measure, 355 DNA, 277, 281, 297, 719 DNA replication, Dolnick, E., 415 August 30, 2012 13:35 World Scientific Review Volume - 9in x 6in - 8306 A Computable Universe Index domain overlap, 412 double Johnson noise, 707 Dretske, F., 354 Dubucs, J., xliv duration, 204 duration function, 204 duration sequence, 205 dynamic associative network, 352 dynamic associative network (DAN), 351–356, 361 Dyson, F., 794 ECA with memory, 258 Eckert, J.P., 573 Eckert, P., 38 EDSAC, 38 EDVAC, 38 Effective method, 78, 79, 81, 84, 86, 88, 89, 92, 92 effective product, 514 egocentric model, 403 Ehrenfeucht, A., xxxiii Einstein, A., 11, 13, 527, 576 Eisenstat, D., 248 electronuclear forces, elementary particles, elementary substructure, 443 enabled reaction, 192 ENIAC, 38 ensembles non-statistical, 492 statistical, 491 entity, 194 entropy, 350, 352–356, 358, 359 Entscheidungsproblem, 47, 50, 57, 78, 435 epistemology, 402 information theory, 403 EPR experiment, 11 erratum sheet story, 707 Ethernet, 683 Euclidean geometry, 556 Everett’s many-worlds interpretation, 404 AComputableUniverse 799 Everett-Wheeler interpretation, 681 Everitt, H., 404 evo-devo, 297 evolution state, 406 exhaustive search, 283 experience, 411 explanation, 73, 401 extensions theory of everything, 411 falsifiability, 414 Popper, K., 407 Feynman, R., 1, 560, 578, 675, 683 finite automaton, 261 finiteness problem, 50, 53 first-order logic, 47, 50 fitness, 281 flip-flops, 770 Floridi, L., 351, 360 flow arc, 202 flying amorphous computer, 252 formal language, 347, 348 formalists’ motto, 353 FORTRAN, 280 foundation science, 414 Fran¸cois, B., xliv Fredkin, E., 365 Fredkin, Ed, xvi, xxiv, 3, 266, 442, 452, 474, 482, 483, 575, 631, 673, 675, 683, 792 free will, 13 Frege, G., 570 Friedman, H.M., 80 functional description, 72 fundamental theory, 412 future observations, 411 fuzzy logic, 626 Gă odel, xxix Gă odels incompleteness theorem, 677, 711, 715 August 30, 2012 13:35 World Scientific Review Volume - 9in x 6in - 8306 A Computable Universe 800 Gă odel, K., 47, 78, 79, 91, 108, 159, 436, 570, 614, 719, 723 Gă odelization, xxxi G acs, P., 458 Gandy machine, 441 Gandy machines, 112 Gandy, R., 48, 159, 441 Gardner, M., 415 Gell-Mann, M., 415, 687, 707 general recursive function, 48 general relativity, 1, 677, 740 general-purpose computation, 33 generalization by postulation, 51 genetic regulatory network, 195 genomics, 719 geocentric model, 403 German, A., xxxvii, 673, 729 Gibson, W., 570 Goldreich, O., 251 Goodstein’s theorem, xxxi gravitation, 527, 742 gravitational path integral, 453 Greif, I., 165 grid points, 782 Grim, P., 351, 360 GRWP, 716 Gră unwald, P D., 415 Gurevich, Y., 81, 88 Halting problem, 152, 300, 439, 572 Hanson, Ch., 244, 246 hardware random generator, 707 Harper, C L., 415 Harrison, C., xli Harrison, E., 415 Harvard Mark I, 38 Haugeland, J., 348, 353 Hayes, B., 676, 692, 714 heliocentric model, 403, 404 heliocentric model, 404 Herbrand, J., 159 Herschel, 28 Hewitt, C., xxiii, xli Heyting, A., 144 AComputableUniverse Index hidden variables, Higgs boson, 725 Hilbert space, 627 Hilbert’s program, 723 Hilbert’s programme, 787 Hilbert’s Second Problem, 77 Hilbert’s Tenth Problem, 77 Hilbert, D., 77, 78, 435, 493, 554, 559, 570, 724 Hintikka, J., 354 Hoare, T., 169 Hofstadter, D., 166, 676 Holevo, A.S., 606 Hollerith, H., 24 holographic principle, 18, 475 Homsy, G., 244, 246 Hoyle, F., 244 human mind, 617 Hutter, M., 8, 415, 522 hybrid computation, 682 Hypercomputation, 93 hypercomputation, xxxiv, xxxviii IBM, 24 idealism, 412, 712 implementation, 63, 66 incompleteness theorem, 570 inconsistency-robust Logic Programming, 167 Independence number, 603 induction, 401 universal, 414 inductive reasoning, 401 inertia, 683 inference map, 607 infinite universe, 412 infinitesimal, 146 infinitesimal calculus, 553 inflation model, 405 information algorithmic, 407 information theory, 350, 354, 402, 414 algorithmic, 401 August 30, 2012 13:35 World Scientific Review Volume - 9in x 6in - 8306 A Computable Universe Index epistemology, 403 information-processing machine, 734 information-theoretic teleodynamics, 353 information-theoretic teleodynamics (ITT), 350, 351 inhibitor, 191 inhibitor arc, 202 initial conditions, 402, 406 initial state, 194 input alphabet, 200 integer base, 705 Intel, 708 intensional, 66 intentional stance, 75, 361 interactive process, 194, 204 internal scaling dimension, 468 interpretation many-worlds, 404 interval arithmetic, 517 intuitionistic logic, 144 Inverse Relationship Principle (IRP), 358 Inverse Relationship Principle (IRP), 358 Inverse Relationship Principle (IRP), 354, 357, 359 invertible cellular automata, 683 it from bit, 15 Itai, A., 251 Jacquard loom, 569 Java, 164 JavaScript, 164 Johnson G., 673 joint complexity, 299 Joosten, J.J., xliv Jozsa, R., 615 jump to universality, 440 Kahn, J M., 246 Kant E., 559 Kari, 683 Katz, R H., 246 Kelley, J.L., 83 AComputableUniverse 801 Kelly, K., 522 Klaus, S., xli Kleene tree, 155 Kleene, S., xvii, xxxviii, 48, 79, 151, 155, 159, 257, 614, 787 Knight, T F., Jr., 244, 246 knowledge objective, 415 subjective, 415 theory, 403, 412 Knuth, D.E., 80, 83 Kochen and Specker theorem, 15 Kolmogorov complexity, 407 Kolmogorov complexity, Kolmogorov, A., 144, 724 Kolmogorov-Chaitin complexity, Kreisel’s criterion, 517 Kreisel, G., 480 Kronecker, 713 Kronheimer, E., xxiv Kurzweil, R., 246 lambda calculus, 720 typed, 720 Lancaster, D., 705 Landauer’s principle, 18 Landauer, R., 18, 438, 709 Lange, T., 352 language program, 409 Lardner, D., 27 Last, M., 244, 246 Lattimore, T., 415 law of nature, 401 law of cancellation, 148 Law of excluded middle, 144 lawfulness hypothesis, 529 lawless, Universe, 526 lawlessness hypothesis, 530, 535 Laws for Actors, 165 laws of computation, 553 laws of the Universe, 528 Lawvere, W.F., 147 LCAU21, 264 August 30, 2012 13:35 World Scientific Review Volume - 9in x 6in - 8306 A Computable Universe 802 learning machine, 401 Leggett, A., xxxvii, 695, 716 Leibnitz, G., 24 Leibniz, G., 1, 146 length minimize, 411 Lerman, M., 441 Levin’s universal distribution, Levin, L., Levin, L.A., 458 Li, M., 415 Liebowitz, B., 244, 246 limited observer, 410 Link, J R., 246 Lloyd, S., xviii, xxi, xli, 3, 452, 577, 687, 694, 718 locality, 683 localization observer, 403, 405, 406, 408, 413, 414 within universe, 409 localizer, 244 Loewenheim-Skolem theorem, 412 Logic Programming, 167 Loll, R., xxxvii Lorentz contraction hypothesis, 740 Lorentz transformations, 768 Lov´ asz number, 603 Lovelace, A., xvii, 36, 569 Lucas, J.R., xl machine learning, 401 Turing, 409 macroscopic quantum coherence, 699 many-worlds interpretation, 404 many-worlds interpretation, 681 Marconi, 701 Margenstern, M., xvi, xxiii, xxxiii, xli marking, 203 Markov, A.A (the younger), 79 Marshall, J., xliv AComputableUniverse Index Martnez, G.J., xvi Martin, A.M., 673 Martin-Lă of random sequence, 530 Martinez, G.J., xxxix materialism, 712 Mathematica, 685, 705–707, 721, 722 mathematical consistency, 407 Mathematical Structure Hypothesis, xxxix Mauchly, J., 573 Maximally incompatible experiments, 597 Maxwell equations, 740 McAllister, J., McCarthy, J., 677 McClelland, J., 352 MDL principle, 401, 414 meaningful theory, 414 measurement from inside, 610, 612 measurement function, 207 measurement problem, mechanical universe, 400 mechanics Newton, I., 402 mechanistic theories, 480 Meinberg, F., xliii memory, 352, 356 Menabrea, L., 39 Mendel, G., Mermin, N D., 590 metabiology, 278 Meyer, D., 452 Microprogramming, 30 Milner, R., 171 minimize length, 411 minimum-size program, 282 Minkowski, H space, 412 Minsky, M., 53, 55, 675, 677 mitochondria, 719 MML August 30, 2012 13:35 World Scientific Review Volume - 9in x 6in - 8306 A Computable Universe Index principle, 401, 414 model, 401 countable, 412 deterministic, 402 egocentric, 403 geocentric, 403 heliocentric, 403, 404 inflation, 405 standard, 407 model selection rational criterion, 403 models all-a-carte, 404 bogus, 403 cosmological, 404, 405 for our world, 403 perfect, 404, 406 physics, 402 speculative, 403 module, 207 molecular communication, 253 molecular computation, 718 molecular computing, 718 Monte Carlo simulations, 706 Monte-Carlo method, 781 Moore’s law, 714 morphogenesis, 277 Mosconi, J., xliv MSDOS, 264 multiple theories, 412 multiverse theories, 404 universal, 404 universe, 415 mutation, 281 mutation distance, 282 Myrheim-Meyer dimension, 456 Nagpal, R., 244, 246 nano-bots, 246 nano-machines, 253 Napier’s Bones, 24 Napier, J., 569 natural computing, 189 Natural Deduction, 183 AComputableUniverse 803 natural selection, nature, 347–350, 356, 361 law, 401 Newman, M., 38 Newton, I, 146 Newton, I., 1, 527, 693 mechanics, 402 Newtonian mechanics, 441 NextStep, 264 Nielsen, M.A., 605 Nikoletseas, S., 244 NKS, 675, 686, 701, 704 noise random, 406 nomological, 348, 349 non-commuting operators, 13 Non-Computable Universe Hypothesis, xl Non-contextual computer, 597 non-materialistic view, 712 non-standard analysis, 147, 697 non-standard arithmetics, 711 non-Turing computable functions, 556 Non-Turing Computable Universe Hypothesis, xxxix non-uniform models, 254 noncontinuous processes, 735 Nondeterministic Turing Machine, 162 normal number, 404 normal form, 52 normal form theorem, 53 normal proof, 720 NP complete, 674 NP completeness, 674 number Champernowne, D G., 404 normal, 404 number of planar triangulations, 475 Nygaard, K., 165 objective knowledge, 415 theory of everything, 406, 409 world, 412, 415 August 30, 2012 13:35 World Scientific Review Volume - 9in x 6in - 8306 A Computable Universe 804 Objective C, 164 observation, 401 direct, 410 process, 410, 414 string, 402 time-series, 402 observation language, 445 observations future, 411 observer, 445 limited, 410 localization, 403, 405, 406, 408, 413, 414 perception ability, 410 observer process computable, 413 Ockham’s razor justification, 414 principle, 407, 413 science, 407, 413 validity, 414 ontology bit-string, 402, 412 OpenStep, 264 oracle, 300 oracle machines, xxxiv oracles, 80, 84–86, 89–92, 278 oracles for real numbers, 486, 507 complete, 507 nested, 507 standard decimal, 486 ordering fraction, 456 Ortiz, G., xxxvii, 673, 700 oscillating universe, 404 Overflow, 30 overlap domain, 412 P vs NP problem, 714 Pachner move, 464 Packard, N., 676 Page, S., 355 paradigm shift, 568 parameter complexity, 412 AComputableUniverse Index parametric theory, 412 Park, M., 246 Parkinson’s law, 724 partial theory, 411 particle physics, 681 Pascaline, 24 Peano arithmetic, xxxi, 711 Peano, G., 722 Penrose, R., xxxviii, xl, xliii, 529, 559 Pentium chip, 707, 717 perception ability, 406 observer, 410 perfect models, 404, 406 perfect randomness, 707 Petr˚ u, L., 248–253 Petri net, 202 Phdrus, 709 phasespace, 443 philosophy of computation, 348 philosophy of mind, 618 physical computation, 64 physical laws, 528 physical machines, 63 physical models, 479 basic representations of, 514 computable, 482 determined by partial recursive functions, 504 epimorphic, 505 faithful, 480, 506, 514 induced by a function, 516 isomorphic, 503 non-negative integer, 504 normal forms of, 513 observationally equivalent, 505 reduced, 505 physical reality, 11 physics models, 402 Piccinini, G., 66 pinball machines, 441 Pipelining, 30 Pirsig, R., 709 August 30, 2012 13:35 World Scientific Review Volume - 9in x 6in - 8306 A Computable Universe Index Pister, K S J., 244, 246 place, 202 Planck area, 18 Plank, M., 14 plankalkă ul, 791 Plato, 529, 709 pluralism, 412 Podolsky, B., 11 Poincar´e, H., 531 Polling, 30 polynomial primitives, 674 polynomial time, 674 Popper, K falsifiability, 407 population protocols, 248 Post production systems, 52 Post tag systems, 787 Post’s formulation 1, 54, 56 Post’s problem, 439 Post’s thesis I, 49, 50 Post’s thesis II, 49 Post, E.L., 45, 79, 84, 159, 257, 571, 787 postdictive success, 407 power predictive, 403, 405, 406 PowerPC, 695 pre-Socratics, 688 precision amplitudes, 674 predicate logic, 436 prediction, 401 predictive indistinguishable, 408 power, 403, 405, 406 Presburger arithmetic, 437 Prevost, H., 36 Principia Mathematica, 53, 722 principle MDL, 401, 414 MML, 401, 414 Ockham’s razor, 407, 413 simplicity, 407 Principle of Comput Equivalence, 440 principle of micro-affinity, 147 AComputableUniverse 805 principle of sufficient reason, 698 priority argument, 439 probabilistic theory, 411 probability subjectivist interpretation of, 492 process, 437 complete, 446 intermediate, 446 observation, 410, 414 undecidable, 446 Process calculi, 171 program complexity, 407 language, 409 program extraction from proofs, 151 program-size complexity, 299 Projective measurement, 596 proof normalization, 720 proof systems, 722 proof theory, 435, 437 protein folding, Pulse-shaping, 30 punched card, 788 Putnam, H, 69 Pythagoras, 568 Pythagoras syndrome, 568 quantum universes, 404 quantum algorithms, 682 quantum cellular automata, 559 quantum computation, 555, 568, 718 quantum computer, 4, 577, 674, 675, 687 quantum computing, 679, 707 quantum decoherence, 19 quantum entanglement, 613 Quantum Field Theory, 618 quantum gravity, 17 Quantum Information, 617 quantum information, 583, 679, 682 quantum measurement, 607 quantum measurement paradox, 681 quantum mechanics, 1, 576, 623, 674, 677, 699, 707 August 30, 2012 13:35 World Scientific Review Volume - 9in x 6in - 8306 A Computable Universe 806 ensemble interpretation of, 496 many-worlds interpretation of, 490 Quantum Meta-Language, 617 Quantum Object-Language, 617 quantum randomness, 706 quantum reality, 11 quantum system, 675 quantum teleportation, 615 quantum theory, 623 qubit, 583, 597 quit, 627 quorum sensing, 253 radioactive decay, 707 RAND corporation, 707 random noise, 406 random number, 691 random number generator, 691 random numbers, 706 Rathmanner, S., 415 rationalism, 702 Rauch, E., 244, 246 RCA, 708 reachability, 443 reachability problem, 52 reactant, 191 reaction, 191 enabled, 192 result, 192 reaction system, 194 real algebra, 722 real-world realizer, 151 realism, 412 reality, 712 realizability, 150 realizability relation, 150 realizer, 150 reasoning deductive, 401 inductive, 401 Rechnender Raum, recursion, 356, 359 Recursive functions, 78–80, 787 Oracular, 90, 92 Partial, 79, 90–92 AComputableUniverse Index reductionism, 412 regular expressions, 260 Reisig, W., 86 Repeatable experiment, 596 representation compact, 402 result of reaction, 192 result sequence, 194 reversibility, 683 Riemann hypothesis, 699 RNA, 701 Robinson, A., 147 Rojas, R., 787 Rosen, N., 11 Rosen, R., 481, 503 Rosser, J.B., 48, 614, 615 Rowland, T., xxxvii, xliv Rozenberg, G., xxxiii, xxxix rule, 401 Rumelhart, D., 352 Ruppert, E., 248 Russell’s system, 179 Russell, B., 570, 712, 720, 722, 724 Sailor, M J., 246 Salomaa, A., xxxviii, xl Schă onfinkel, M., 787 Schickard, W., 24 Schmidhuber, J., xxi, xxiv, xxxvii, 3, 415, 452, 787 Schră odingers cat, 597, 716 Schră odinger, E., 493 science foundation, 414 Ockham’s razor, 407, 413 scientific discovery, 401 Searle, J., xl, 444 second law of thermodynamics, 685 selection principle theory of everything, 408, 410, 415 self-assembling, self-sampling anthropic, 414 assumption, 414 universal, 414 August 30, 2012 13:35 World Scientific Review Volume - 9in x 6in - 8306 A Computable Universe Index Seligman, J., 354 semantic, 354, 360 semantic information, 351, 354 semantics, 159 semidecidable, 436 separable space, 412 sequence binary, 402, 412 Sequential postulates, 81, 89 SETI, 701, 703, 716 Shah, M J., 246 Shannon, C., 350, 573 Sharp measurement, 596 Shoenfield, J.R., 79 Shor’s algorithm, 717 Shor, P., 577 Shore, R.A., 80 Sieg, W., xvii, xxx, xli, xliv, 48, 159 simple mapping account (SMA), 69 simplicity principle, 407 simulation, 91, 92, 675 singularity, 246 SKI combinators, 787 smart dust, 246 Smolin’s baby universe theory, 404 Smolin, L., 404, 415 SMP, 685 Soare, R.I., 47 solipsism, 412 Sorkin, R.D., 452 space Minkowski, H., 412 separable, 412 space-time, 149 speckled computing, 246 Sprevak, M., 72 sprinkling, 454 SSEM, 38 standard model, 407 Standish, R., 415 state, 194 evolution, 406 initial, 194 AComputableUniverse 807 transition, 192 State machine, 165 state sequence, 194 statistical mechanics, 680 Statman, R., 521 steampunk, 570 step, 203 step sequence, 203 Sterling, B., 570 Stove, D C., 415 string observation, 402 theory, 404, 407 Structures, first order, 81–83 subjective knowledge, 415 theory of everything, 406, 410 universe, 400 success postdictive, 407 super-Turing computing power, 254 Susskind, L., 475 Sussman, G J., 244, 246 Sutner, K., xvi Svozil, K., 631, 673 Swade, D., xvii, xli switching circuit, 197 synchronous amorphous computing system, 249 synthetic differential geometry, 146 Szil´ ard, L., 18 Szudzik, M., xxii, xliii, 8, 479 T-cells, 720 tabulator, 24 tag system, 51 Tarski’s result, 674 Tarski, A., 157, 624, 674 Tegmark, M., 415, 452 teleodynamic, 348, 349, 351, 356, 360, 361 teleodynamic game page, 360 teleological, 348, 350, 360 teleonomic, 349, 351, 360, 361 temporal binary sequence, 402, 412 August 30, 2012 13:35 World Scientific Review Volume - 9in x 6in - 8306 A Computable Universe 808 Tesla, 702 Teuscher, C, xli Teuscher, C., xvii, xxiii The Algorithmic Information Hypothesis, xl the Bombe, 38 The Cellular Automaton Hypothesis, xxxix The Computable Universe Hypothesis, xxxviii The Computational Pragmatic Hypothesis, xxxix The Hyper-computable Hypothesis, xl The Informational Universe Hypothesis, xxxix The Non-Turing Computable Universe Hypothesis, xl The Random Universe Hypothesis, xl The Standard Quantum Universe Hypothesis, xli theorem Loewenheim-Skolem, 412 theories multiple, 412 multiverse, 404 theory, 72 approximate, 411, 412 correspondence, 410 deterministic, 402 fundamental, 412 knowledge, 403, 412 meaningful, 414 of nothing, 404, 414 parametric, 412 partial, 411 probabilistic, 411 string, 404, 407 theory of automatons, 732 theory of everything, xxxix, 4, 401, 414 assumptions, 412 best complete, 411 complete, 406, 409, 414 complexity, 407 conventional, 414 extensions, 411 AComputableUniverse Index formal quest, 414 objective, 406, 409 selection principle, 408, 410, 415 subjective, 406, 410 universal, 404, 408 thermodynamic, 349, 353 thermodynamical equilibrium, 700 thermodynamics, 623, 681, 701 time-series observation, 402 timed probabilistic automaton, 253 Toffoli gate, 683 Toffoli, T., xxxvii, 266, 673, 676, 700 token, 347, 348 topologies computable, 507 effective, 507 touring ant, 462 transducer, 442 transition, 202 transition function, 200 transition system, 200 transitive percolation, 456 trinet, 464 trinet mobile automata, 464 true randomness, 13 Turing barrier, 438, 682 Turing computable functions, 559 Turing machine, 5, 47–49, 54, 56, 59, 60, 542, 571, 674 universal, 409 Turing machine, elementary, 461 Turing machines, 79, 80, 88, 92 Oracular, 85, 86, 92 Turing universality, xxxviii Turing’s Model, 159 Turing’s thesis, 47, 48, 79 Turing, A., 719, 723 Turing, A.M., 23, 47, 48, 78, 79, 85, 90, 92, 112, 159, 257, 277, 347, 352, 353, 436, 480, 558, 568, 614, 615 Turing, Alan, 583 Turing-complete, 352 turmite, 470 Turner, R, xxii Turner, R., xli August 30, 2012 13:35 World Scientific Review Volume - 9in x 6in - 8306 A Computable Universe AComputableUniverse Index TX2, 706 type-two effectivity, 507 typing monkeys, 579 Ulam, S., 575 Unbounded Nondeterminism, 167 unconventional computation, 718 unconventional computing, 257 undecidability, 172 uniformity, 683 universal induction, 414 multiverse, 404 self-sampling, 414 theory of everything, 404, 408 Turing machine, 409 universal computer, 437 universal programming language, 297 universal Turing computer, universal Turing machine, 674 universal-computing CA, 266 universe coding, 410 computable, 400, 404, 413 continuous, 412 countable, 412 infinite, 412 mechanical, 400 multiverse, 415 oscillating, 404 subjective, 400 visible, 405 universes baby, 404 quantum, 404 validity Ockham’s razor, 414 van Leeuwen, J., 254 variables inter-universes, 404 Velupillai, K., xiv Velupillai, V., xxxix, xli Vinge, V., 245 visible universe, 405 809 Vit´ anyi, P M B., 415 von Neumann’s normalisation, 546 von Neumann, J., xxvi, 15, 33, 436, 570, 623, 678, 684, 724, 790 Wallace, C S., 415 Wang, H., 439 Warneke, B., 244, 246 Watson, J., 6, 277, 719 wave-function collapse, 402, 404 Weierstrass, K., 146, 154 Weihrauch, K., 522 Weihs, H., 12 Weinfurter, Harald, 590 Weiss, R., 244, 246 Weyl, H., 49 Wheeler, J.A., 1, 404 Whitehead, 722, 724 Wiedermann, J., xvi, xxiii, xli, 248, 249, 251–254 Wiener, N., 354 wireless sensory network, 248 Wittgenstein, L., 178 Wolfram Principle of computational equivalence, 674, 701, 716 Wolfram rule 30, Wolfram’s rule 110, 266, 440 Wolfram, S., xvi, xxiv, 3, 259, 415, 440, 442, 451, 458, 482, 483, 522, 575, 631, 673, 676 Wong, K J., 246 world models, 403 objective, 412, 415 worlds many, 404 XOR, 678 Z-series machines, 38 Zambelli, S., xxxix, xli Zeilinger, A., 12 Zenil, H., xvi, xxiv, xxxiii, xl, 522, 534, 631, 673, 700, 729, 790 August 30, 2012 13:35 World Scientific Review Volume - 9in x 6in - 8306 A Computable Universe 810 Zeno, 555 Zeno machine, 542 Zeno of Elea, 554 Zermelo-Fraenkel set theory, 723 Zizzi, P., xxix, xxxiii, xxxviii Zoller, P., 577 AComputableUniverse Index Zukowski, M., xxi Zuse, H., 790 Zuse, K., xvi, xxxix, 38, 415, 442, 452, 482, 483, 572, 787 Calculating Space, 729 Rechnender Raum, 729