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Hartmanis-Stearns Conjecture on Real Time and Transcendence

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Part of the Lecture Notes in Computer Science book series (LNTCS,volume 7160)

Abstract

Hartmanis-Stearns conjecture asserts that any number whose decimal expansion can be computed by a multitape Turing machine is either rational or transcendental. After half a century of active research by computer scientists and mathematicians the problem is still open but much more interesting than in 1965.

Keywords

  • Rational Number
  • Turing Machine
  • Continue Fraction
  • Algebraic Number
  • Irrational Number

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The research was supported by Agreement with the European Regional Development Fund (ERDF) 2010/0206/2DP/2.1.1.2.0/10/APIA/VIAA/011.

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References

  1. Ablayev, F.M., Freivalds, R.: Why Sometimes Probabilistic Algorithms Can Be More Effective. In: Wiedermann, J., Gruska, J., Rovan, B. (eds.) MFCS 1986. LNCS, vol. 233, pp. 1–14. Springer, Heidelberg (1986)

    CrossRef  Google Scholar 

  2. Adamczewski, B., Allouche, J.-P.: Reversals and palindromes in continued fractions. Theoretical Computer Science 380(3), 220–237 (2007)

    CrossRef  MathSciNet  MATH  Google Scholar 

  3. Adamczewski, B., Bugeaud, Y.: On the complexity of algebraic numbers I. Expansions in integer bases. Annals of Mathematics 165, 547–565 (2007)

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. Adamczewski, B., Bugeaud, Y.: On the complexity of algebraic numbers II. Continued fractions. Acta Mathematica 195(1), 1–20 (2005)

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. Adamczewski, B., Bugeaud, Y.: On the independence of expansions of algebraic numbers in an integer base. Bulletin London Mathematical Society 39(2), 283–289 (2007)

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. Adamczewski, B., Bugeaud, Y.: Mesures de transcendance et aspects quantitatifs de la mthode de Thue-Siegel-Roth-Schmidt. Proceedings of the London Mathematical Society 101(1), 1–26 (2010)

    CrossRef  MathSciNet  Google Scholar 

  7. Adamczewski, B., Bugeaud, Y., Luca, F.: Sur la complexité des nombres algébriques. Comptes Rendus de l’Académie des Sciences, Paris Ser. I 336, 11–14 (2004)

    Google Scholar 

  8. Agafonov, V.N.: Normal sequences and finite automata. Soviet Mathematics Doklady 9, 324–325 (1968)

    MathSciNet  MATH  Google Scholar 

  9. Allouche, J.-P.: Nouveaux résultats de transcendance de r\(\acute{e}\)els \(\grave{a}\) d\(\acute{e}\)veloppement non aléatoire. Gazette des Mathématiciens (84), 19–34 (2000)

    Google Scholar 

  10. Allouche, J.-P., Zamboni, L.Q.: Algebraic irrational binary numbers cannot be fixed points of non-trivial constant length or primitive morphisms. Journal of Number Theory 69, 119–124 (1998)

    CrossRef  MathSciNet  MATH  Google Scholar 

  11. Ambainis, A., Apsītis, K., Calude, C., Freivalds, R., Karpinski, M., Larfeldt, T., Sala, I., Smotrovs, J.: Effects of Kolmogorov Complexity Present in Inductive Inference as Well. In: Li, M. (ed.) ALT 1997. LNCS, vol. 1316, pp. 244–259. Springer, Heidelberg (1997)

    CrossRef  Google Scholar 

  12. Baker, A.: Linear forms in the logarithms of algebraic numbers I–III. Mathematika. A Journal of Pure and Applied Mathematics 13, 204–216 (1966); 14, 102–107 (1967); 14, 220–228 (1967)

    MATH  Google Scholar 

  13. van Aardenne-Ehrenfest, T., de Bruijn, N.G.: Circuits and trees in oriented linear graphs. Simon Stevin 28, 203–217 (1951)

    MathSciNet  MATH  Google Scholar 

  14. Becher, V., Figueira, S.: An example of a computable absolutely normal number. Theoretical Computer Science 270(1-2), 947–958 (2002)

    CrossRef  MathSciNet  MATH  Google Scholar 

  15. Borel, É.: Les probabilités dénombrables et leurs applications arithmétiques. Rendiconti del Circolo Matematico di Palermo 27, 247–271 (1909)

    CrossRef  MATH  Google Scholar 

  16. Borel, É.: Sur les chiffres décimaux de \(\sqrt{2}\) et divers problèmes en châine. Comptes Rendus de l’Académie des Sciences, Paris 230, 591–593 (1950)

    MATH  Google Scholar 

  17. Calude, C.: What is a random string? Journal of Universal Computer Science 1(1), 48–66 (1995)

    MATH  Google Scholar 

  18. Calude, C.: Borel normality and algorithmic randomness. In: Rozenberg, G., Salomaa, A. (eds.) Developments in Language Theory: At the Crossroads of Mathematics, Computer Science and Biology, pp. 113–119. World Scientific, Singapore (1994)

    Google Scholar 

  19. Calude, C.S., Zamfirescu, T.: Most numbers obey no probability laws. Publicationes Mathematicae Debrecen 54(Supplement), 619–623 (1999)

    MathSciNet  MATH  Google Scholar 

  20. Champernowne, D.G.: The construction of decimals normal in the scale of ten. The Journal of the London Mathematical Society 8, 254–260 (1933)

    CrossRef  MathSciNet  MATH  Google Scholar 

  21. Cobham, A.: On the base-dependence of sets of numbers recognizable by finite automata. Mathematical Systems Theory 3, 186–192 (1969)

    CrossRef  MathSciNet  MATH  Google Scholar 

  22. Cobham, A.: Uniform tag sequences. Mathematical Systems Theory 6, 164–192 (1972)

    CrossRef  MathSciNet  MATH  Google Scholar 

  23. Copeland, A.H., Erdös, P.: Note on normal numbers. Bulletin of the American Mathematical Society 52, 857–860 (1946)

    CrossRef  MathSciNet  MATH  Google Scholar 

  24. Eilenberg, S.: Automata, Languages and Machines, vol. A, B. Academic Press, New York (1974)

    MATH  Google Scholar 

  25. Ferenczi, S., Mauduit, C.: Transcendence of numbers with a low complexity expansion. Journal of Number Theory 67(2), 146–161 (1997)

    CrossRef  MathSciNet  MATH  Google Scholar 

  26. Freivalds, R.: Amount of nonconstructivity in deterministic finite automata. Theoretical Computer Science 411(38-39), 3436–3443 (2010)

    CrossRef  MathSciNet  MATH  Google Scholar 

  27. Freivalds, R., Kinber, E.B., Wiehagen, R.: How inductive inference strategies discover their errors. Information and Computation 118(2), 208–226 (1995)

    CrossRef  MathSciNet  MATH  Google Scholar 

  28. Gold, E.M.: Language identification in the limit. Information and Control 10(5), 447–474 (1967)

    CrossRef  MathSciNet  MATH  Google Scholar 

  29. Hartmanis, J., Stearns, R.E.: On the computational complexity of algorithms. Transactions of the American Mathematical Society 117, 285–306 (1965)

    CrossRef  MathSciNet  MATH  Google Scholar 

  30. Hurwitz, A.: Über die angenäherte Darstellung der Irrationalzahlen durch rationale Brüche. Mathematische Annalen 39(2), 279–284 (1891)

    CrossRef  MathSciNet  Google Scholar 

  31. Kaneps, J., Freivalds, R.: Minimal Nontrivial Space Complexity of Probabilistic One-Way Turing Machines. In: Rovan, B. (ed.) MFCS 1990. LNCS, vol. 452, pp. 355–361. Springer, Heidelberg (1990)

    CrossRef  Google Scholar 

  32. Khinchin, A.Y.: Continued Fractions. Dover Publications (2007) (translation from Russian original, GITTL, 1949)

    Google Scholar 

  33. Kolmogorov, A.N., Uspensky, V.A.: Algorithms and randomness. Theory of Probability and Its Applications 32, 389–412 (1987)

    CrossRef  MathSciNet  MATH  Google Scholar 

  34. Liouville, J.: Sur des classes très-étendues de quantités dont la valeur n’est ni algébrique, ni même réductible à des irrationnelles algébriques. Journal de Mathématiques Pures et Appliquées 16(1), 133–142 (1851)

    Google Scholar 

  35. Loxton, J.H., van der Poorten, A.J.: Transcendence and algebraic independence by a method of Mahler. In: Baker, A., Masser, D.W. (eds.) Transcendence Theory: Advances and Applications. ch. 15, pp. 211–226. Academic Press, London (1977)

    Google Scholar 

  36. Roth, K.F.: Rational approximations to algebraic numbers. Mathematika. A Journal of Pure and Applied Mathematics 2, 1–20, 168 (1955)

    MathSciNet  MATH  Google Scholar 

  37. Schlickewei, H.P.: The 2-adic Thue-Siegel-Roth-Schmidt theorem. Archiv der Mathematik 29(1), 267–270 (1977)

    CrossRef  MathSciNet  MATH  Google Scholar 

  38. Schmidt, W.: The subspace theorem in diophantine approximation. Compositio Mathematica, Tome 69(2), 121–173 (1989)

    MathSciNet  MATH  Google Scholar 

  39. Sierpiński, W.: Démonstration élémentaire d’un théorème de M. Borel sur les nombres absolument normaux et détermination effective d’un tel nombre. Bulletin de la Société Mathématique de France 45, 125–144 (1917)

    MathSciNet  MATH  Google Scholar 

  40. Shallit, J., Breitbart, Y.: Automaticity I: Properties of a measure of descriptional complexity. Journal of Computer and Systems Science 53(1), 10–25 (1996)

    CrossRef  MathSciNet  MATH  Google Scholar 

  41. Weisstein, E.W.: Transcendental number. From MathWorld–A Wolfram Web Resource, http://mathworld.wolfram.com/TranscendentalNumber.html

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Freivalds, R. (2012). Hartmanis-Stearns Conjecture on Real Time and Transcendence. In: Dinneen, M.J., Khoussainov, B., Nies, A. (eds) Computation, Physics and Beyond. WTCS 2012. Lecture Notes in Computer Science, vol 7160. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27654-5_9

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  • DOI: https://doi.org/10.1007/978-3-642-27654-5_9

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