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Hilbert’s Tenth Problem and Paradigms of Computation

  • Yuri Matiyasevich
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3526)

Abstract

This is a survey of a century long history of interplay between Hilbert’s tenth problem (about solvability of Diophantine equations) and different notions and ideas from the Computability Theory.

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References

  1. 1.
    Adleman, L., Manders, K.: Computational complexity of decision procedures for polynomials. In: 16th Annual Symposium on Foundations of Computer Science, pp. 169–177 (1975)Google Scholar
  2. 2.
    Adleman, L., Manders, K.: Diophantine complexity. In: 17th Annual Symposium on Foundations of Computer Science, Houston, Texas, October 25-26, 1976, pp. 81–88. IEEE, Los Alamitos (1976)CrossRefGoogle Scholar
  3. 3.
    Blum, M.: A machine-independent theory of the complexity of recursive functions. Journal of the ACM 14(2), 322–336 (1967)zbMATHCrossRefGoogle Scholar
  4. 4.
    Chaitin, G.: Algorithmic Information Theory. Cambridge University Press, Cambridge (1987)CrossRefGoogle Scholar
  5. 5.
    Davis, M.: Arithmetical problems and recursively enumerable predicates (abstract). Journal of Symbolic Logic 15(1), 77–78 (1950)Google Scholar
  6. 6.
    Davis, M.: Arithmetical problems and recursively enumerable predicates. J. Symbolic Logic 18(1), 33–41 (1953)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Davis, M.: Speed-up theorems and Diophantine equations. In: Rustin, R. (ed.) Courant Computer Science Symposium 7: Computational Complexity, pp. 87–95. Algorithmics Press, New York (1973)Google Scholar
  8. 8.
    Davis, M.: Computability and Unsolvability. Dover Publications, New York (1982)zbMATHGoogle Scholar
  9. 9.
    Davis, M., Putnam, H., Robinson, J.: The decision problem for exponential Diophantine equations. Ann. Math. 74(2), 425–436 (1961) (Reprinted in The collected works of Julia Robinson, Feferman, S. (ed.) Collected Works, 6, 1996, xliv+p. 338. American Mathematical Society, Providence, RI, ISBN: 0-8218-0575-4)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Gödel, K.: Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme. I. Monatsh. Math. und Phys. 38(1), 173–198 (1931)CrossRefGoogle Scholar
  11. 11.
    Hack, M.: The equality problem for vector addition systems is undecidable. Theoretical Computer Science 2(1), 77–95 (1976)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Hilbert, D.: Mathematische Probleme. Vortrag, gehalten auf dem internationalen Mathematiker Kongress zu Paris 1900. Nachr. K. Ges. Wiss., Göttingen, Math.-Phys.Kl., 253–297 (1900); See also Hilbert, D.: Gesammelte Abhandlungen, vol. 3, p. 310. Springer, Berlin (1935) (Reprinted: New York : Chelsea (1965)); English translation: Bull. Amer. Math. Soc., 8, 437–479 (1901-1902); Reprinted in : Mathematical Developments arising from Hilbert problems. In: Browder, (ed.) Proceedings of symposia in pure mathematics, vol. 28, pp.1–34. American Mathematical Society (1976)Google Scholar
  13. 13.
    Hodgson, B.R., Kent, C.F.: A normal form for arithmetical representation of NP-sets. Journal of Computer and System Sciences 27(3), 378–388 (1983)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Jones, J.P., Matijasevič, J.V.: Exponential Diophantine representation of recursively enumerable sets. In: Stern, J. (ed.) Proceedings of the Herbrand Symposium: Logic Colloquium 1981. Studies in Logic and the Foundations of Mathematics, vol. 107, pp. 159–177. North Holland, Amsterdam (1982)CrossRefGoogle Scholar
  15. 15.
    Jones, J.P., Matijasevič, J.V.: A new representation for the symmetric binomial coefficient and its applications. Les Annales des Sciences Mathématiques du Québec 6(1), 81–97 (1982)zbMATHGoogle Scholar
  16. 16.
    Jones, J.P., Matijasevich, Y.V.: Direct translation of register machines into exponential Diophantine equations. In: Priese, L. (ed.) Report on the 1st GTI-workshop, vol. 13, pp. 117–130, Reihe Theoretische Informatik, Universität-Gesamthochschule Paderborn (1983)Google Scholar
  17. 17.
    Jones, J.P., Matijasevič, Y.V.: Register machine proof of the theorem on exponential Diophantine representation of enumerable sets. J. Symbolic Logic 49(3), 818–829 (1984)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Jones, J.P., Matijasevič, Y.V.: Proof of recursive unsolvability of Hilbert’s tenth problem. Amer. Math. Monthly 98(8), 689–709 (1991)zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Kent, C.F., Hodgsont, B.R.: An arithmetical characterization of NP. Theoretical Computer Science 21(3), 255–267 (1982)zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Ord, T., Kieu, T.D.: On the existence of a new family of Diophantine equations for Ω. Fundam. Inform. 56(3), 273–284 (2003)zbMATHMathSciNetGoogle Scholar
  21. 21.
    Kreisel, G.: Davis, Martin; Putnam, Hilary; Robinson, Julia. The decision problem for exponential Diophantine equations. Mathematical Reviews 24#A3061, 573 (1962)Google Scholar
  22. 22.
    Lambek, J.: How to program an infinite abacus. Canad. Math. Bull. 4, 295–302 (1961)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Lipmaa, H.: On Diophantine Complexity and Statistical Zero-Knowledge Arguments. In: Laih, C.-S. (ed.) ASIACRYPT 2003. LNCS, vol. 2894, pp. 398–415. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  24. 24.
    Manders, K.L., Adleman, L.: NP-complete decision problems for binary quadratics. J.Comput. System Sci. 16(2), 168–184 (1978)zbMATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Martin-Löf, P.: Notes on Constructive Mathematics. Almqvist & Wikseil, Stockholm (1970)Google Scholar
  26. 26.
    Matiyasevich, Y.V.: Diofantovost’ perechislimykh mnozhestv. Dokl. AN SSSR 191(2), 278–282 (1970); Translated in: Soviet Math. Doklady 11(2), 354-358 (1970)Google Scholar
  27. 27.
    Matiyasevich, Y.V.: Sushchestvovanie neèffektiviziruemykh otsenok v teorii èksponentsial’no diofantovykh uravneniĭ. Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR (LOMI) 40, 77–93 (1974); Translated in: Journal of Soviet Mathematics 8(3), 299–311 (1977)Google Scholar
  28. 28.
    Matiyasevich, Y.: Novoe dokazatel’stvo teoremy ob èksponentsial´no diofantovom predstavlenii perechislimykh predikatov. Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR (LOMI) 60, 75–92 (1976); Translated in: Journal of Soviet Mathematics 14(5), 1475–1486 (1980)Google Scholar
  29. 29.
    Matiyasevich, Y.: Desyataya Problema Gilberta, Moscow, Fizmatlit (1993); English translation: Hilbert’s tenth problem. MIT Press, Cambridge (1993); French translation: Le dixième problème de Hilbert, Masson (1995), http://logic.pdmi.ras.ru/~yumat/H10Pbook, mirrored at, http://www.informatik.uni-stuttgart.de/ifi/ti/personen/Matiyasevich/H10Pbook
  30. 30.
    Matiysevich, Y.: A direct method for simulating partial recursive functions by Diophantine equations. Annals Pure Appl. Logic 67, 325–348 (1994)CrossRefGoogle Scholar
  31. 31.
    Matiyasevich, Y.: Hilbert’s tenth problem: what was done and what is to be done. Contemporary mathematics 270, 1–47 (2000)MathSciNetGoogle Scholar
  32. 32.
    Melzak, Z.A.: An informal arithmetical approach to computability and computation. Canad. Math. Bull. 4, 279–294 (1961)zbMATHMathSciNetCrossRefGoogle Scholar
  33. 33.
    Minsky, M.L.: Recursive unsolvability of Post’s problem of “tag” and other topics in the theory of Turing machines. Ann. of Math. 74(2), 437–455 (1961)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Minsky, M.L.: Computation: Finite and Infinite Machines. Prentice Hall, Englewood Cliffs (1967)zbMATHGoogle Scholar
  35. 35.
    Pollett, C.: On the Bounded Version of Hilbert’s Tenth Problem. Arch. Math. Logic 42(5), 469–488 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  36. 36.
    Post, E.L.: Formal reductions of the general combinatorial decision problem. Amer. J. Math. 65, 197–215 (1943); Reprinted in “The collected works of E.L. Post”, Davis, M. (ed.) Birkhäuser, Boston (1994)Google Scholar
  37. 37.
    Shepherdson, J.C., Sturgis, H.E.: Computability of recursive functions. J. ACM 10(2), 217–255 (1963)zbMATHMathSciNetCrossRefGoogle Scholar
  38. 38.
    Tseitin, G.S.: Odin sposob izlozheniya teorii algorifmov i perechislimykh mnozhestv. Trudy Matematicheskogo instituta im. V. A. Steklova 72, 69–99 (1964); English translation in Proceedings of the Steklov Institute of MathematicsGoogle Scholar
  39. 39.
    van Emde Boas, P.: Dominos are forever. In: Priese, L. (ed.) Report on the 1st GTI-workshop, vol. 13, pp. 75–95, Reihe Theoretische Informatik, Universität-Gesamthochschule Paderborn (1983)Google Scholar
  40. 40.
    Vinogradov, A.K., Kosovskiĭ, N.K.: Ierarkhiya diofantovykh predstavleniĭ primitivno rekursivnykh predikatov. In: Vychislitel’naya tekhnika i voprosy kibernetiki, vol. 12, pp. 99–107. Lenigradskiĭ Gosudarstvennyĭ Universitet, Leningrad (1975)Google Scholar
  41. 41.
    Yukna, S.: Arifmeticheskie predstavleniya klassov mashinnoĭ slozhnosti. In: Matematicheskaya logika i eë primeneniya, vol. 2, pp. 92–107. Institut Matematiki i Kibernetiki Akademii Nauk Litovskoĭ SSR, Vil’nyus (1982)Google Scholar
  42. 42.
    Yukna, S.: Ob arifmetizatsii vychisleniĭ. In: Matematicheskaya logika i eë primeneniya, vol. 3, pp. 117–125. Institut Matematiki i Kibernetiki Akademii Nauk Litovskoĭ SSR, Vil’nyus (1983)Google Scholar
  43. 43.

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Yuri Matiyasevich
    • 1
  1. 1.Steklov Institute of Mathematics Laboratory of Mathematical LogicSt.PetersburgRussia

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