Alan Turing and the Origins of Complexity

The 75th anniversary of Turing’s seminal paper and his cente nnial year anniversary occur in 2011 and 2012, respectively. It is natural to review and assess Turing’s co ntributions in diverse fields in the light of new developments that his thoughts has triggered in many scientific comm unities. Here, the main idea is to discuss how the work of Turing allows us to change our views on the foundation s f Mathematics, much like quantum mechanics changed our conception of the world of Physics. Basic notion s like computability and universality are discussed in a broad context, making special emphasis on how the notion f complexity can be given a precise meaning after Turing, i.e., not just qualitative but also quantitat ive. Turing’s work is given some historical perspective with respect to some of his precursors, contemporaries and m athematicians who took up his ideas farther.

[1]  K. Gödel.  Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I , 1931 .

[2]  T. Rado.  On non-computable functions , 1962 .

[3]  Gregory J. Chaitin,et al.  On the Length of Programs for Computing Finite Binary Sequences , 1966, JACM.

[4]  Ming Li,et al.  Kolmogorov Complexity and its Applications , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[5]  G. Leibniz,et al.  Discours de Metaphysique , 2013 .

[6]  A. Kolmogorov.  Three approaches to the quantitative definition of information , 1968 .

[7]  H. Briegel,et al.  Quantum algorithms for spin models and simulable gate sets for quantum computation , 2008, 0805.1214.

[8]  P. Boltuć,et al.  APA NEWSLETTER ON Philosophy and Computers , 2007 .

[9]  A. Turing.  On Computable Numbers, with an Application to the Entscheidungsproblem. , 1937 .

[10]  B. J. Copeland,et al.  Alan Turing’s Forgotten Ideas in Computer Science , 1999 .

[11]  Cristian Claude,et al.  Information and Randomness: An Algorithmic Perspective , 1994 .

[12]  B. Kraus,et al.  QUANTUM KOLMOGOROV COMPLEXITY AND ITS APPLICATIONS , 2006 .

[13]  Claude E. Shannon,et al.  A Universal Turing Machine with Two Internal States , 1956 .

[14]  W. Dur,et al.  Quantum algorithms for classical lattice models , 2011, 1104.2517.

[15]  G.E. Moore,et al.  Cramming More Components Onto Integrated Circuits , 1998, Proceedings of the IEEE.

[16]  Péter Gács,et al.  Quantum algorithmic entropy , 2000, Proceedings 16th Annual IEEE Conference on Computational Complexity.

[17]  M. A. Martin-Delgado,et al.  On Quantum Effects in a Theory of Biological Evolution , 2011, Scientific Reports.

[18]  Marvin Minsky,et al.  Size and structure of universal Turing machines using Tag systems , 1962 .

[19]  Martin D. Davis,et al.  Computability and Unsolvability , 1959, McGraw-Hill Series in Information Processing and Computers.

[20]  Marcus Hutter,et al.  Algorithmic Information Theory , 1977, IBM J. Res. Dev..

[21]  Umesh V. Vazirani,et al.  Quantum complexity theory , 1993, STOC.

[22]  Sophie Laplante,et al.  Quantum Kolmogorov complexity , 2000, Proceedings 15th Annual IEEE Conference on Computational Complexity.

[23]  Michael Meyling,et al.  Axiomatic Set Theory , 2008 .

[24]  K. Hepp.  I) Philosophy of Mathematics and Natural Science , 2010 .

[25]  Karl Svozil.  Halting probability amplitude of quantum computers , 1996 .

[26]  Yurii Rogozhin,et al.  Small Universal Turing Machines , 1996, Theor. Comput. Sci..

[27]  Tien D. Kieu,et al.  Quantum Algorithm for Hilbert's Tenth Problem , 2001, ArXiv.

[28]  R. Feynman.  Simulating physics with computers , 1999 .

[29]  Thomas M. Cover,et al.  Elements of Information Theory , 2005 .

[30]  Gregory J. Chaitin,et al.  Exploring RANDOMNESS , 2001, Discrete Mathematics and Theoretical Computer Science.

[31]  Karl Svozil.  Halting Probability Amplitude Quantum Computers , 1995, J. Univers. Comput. Sci..

[32]  H. Briegel,et al.  Algorithmic complexity and entanglement of quantum states. , 2005, Physical review letters.

[33]  Per Martin-Löf,et al.  The Definition of Random Sequences , 1966, Inf. Control..

[34]  Paul M. B. Vitányi,et al.  Three approaches to the quantitative definition of information in an individual pure quantum state , 1999, Proceedings 15th Annual IEEE Conference on Computational Complexity.

[35]  A. M. Turing,et al.  The Essential Turing: Seminal Writings in Computing, Logic, Philosophy, Artificial Intelligence, and Artificial Life plus The Secrets of Enigma , 2004 .

[36]  A. Galindo,et al.  Information and computation: Classical and quantum aspects , 2001, quant-ph/0112105.

[37]  C. Luther,et al.  INDEPENDENCE OF THE CONTINUUM HYPOTHESIS , 2009 .

[38]  Harold Marston Morse.  Recurrent geodesics on a surface of negative curvature , 1921 .

[39]  Donald E. Knuth,et al.  Selected papers on analysis of algorithms , 2000, CSLI lecture notes series.

[40]  G. Chaitin.  To a Mathematical Theory of Evolution and Biological Creativity , 2011 .

[41]  Joan B. Quick,et al.  Philosophy of Mathematics and Natural Science , 1950 .

[42]  D. Dugué,et al.  Les nombres inaccessibles , 1952 .

[43]  Gregory J. Chaitin,et al.  A recent technical report , 1974, SIGA.

[44]  Cristian S. Calude.  Information and Randomness , 1994, Monographs in Theoretical Computer Science An EATCS Series.

[45]  M. Borel.  Les probabilités dénombrables et leurs applications arithmétiques , 1909 .

[46]  Ray J. Solomonoff,et al.  A Formal Theory of Inductive Inference. Part I , 1964, Inf. Control..

[47]  Gregory J. Chaitin,et al.  The Limits of Mathematics , 1995, J. Univers. Comput. Sci..

[48]  Cristian S. Calude,et al.  Computing a Glimpse of Randomness , 2002, Exp. Math..

[49]  Jim Law.  Selected papers on analysis of algorithms , 2001, SOEN.

[50]  C. Mora,et al.  ALGORITHMIC COMPLEXITY OF QUANTUM STATES , 2004 .

[51]  C. Torrance.  Review: Kurt Gödel, The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis with the Axioms of Set Theory , 1941 .

[52]  Mary Tiles,et al.  Georg Cantor: His Mathematics and Philosophy of the Infinite. , 1982 .

[53]  Scott Aaronson,et al.  The Complexity Zoo , 2008 .

[54]  Cristian S. Calude,et al.  Coins, Quantum Measurements, and Turing's Barrier , 2002, Quantum Inf. Process..