Skip to main content
SpringerLink
Log in

Hilbert’s Tenth Problem and Paradigms of Computation

  • Conference paper
New Computational Paradigms (CiE 2005)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3526))

Included in the following conference series:

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  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. 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)

    Chapter  Google Scholar 

  3. Blum, M.: A machine-independent theory of the complexity of recursive functions. Journal of the ACM 14(2), 322–336 (1967)

    Article  MATH  Google Scholar 

  4. Chaitin, G.: Algorithmic Information Theory. Cambridge University Press, Cambridge (1987)

    Book  Google Scholar 

  5. Davis, M.: Arithmetical problems and recursively enumerable predicates (abstract). Journal of Symbolic Logic 15(1), 77–78 (1950)

    Google Scholar 

  6. Davis, M.: Arithmetical problems and recursively enumerable predicates. J. Symbolic Logic 18(1), 33–41 (1953)

    Article  MATH  MathSciNet  Google Scholar 

  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. Davis, M.: Computability and Unsolvability. Dover Publications, New York (1982)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  10. Gödel, K.: Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme. I. Monatsh. Math. und Phys. 38(1), 173–198 (1931)

    Article  Google Scholar 

  11. Hack, M.: The equality problem for vector addition systems is undecidable. Theoretical Computer Science 2(1), 77–95 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  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. 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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    MATH  Google Scholar 

  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. 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)

    Article  MATH  MathSciNet  Google Scholar 

  18. Jones, J.P., Matijasevič, Y.V.: Proof of recursive unsolvability of Hilbert’s tenth problem. Amer. Math. Monthly 98(8), 689–709 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  19. Kent, C.F., Hodgsont, B.R.: An arithmetical characterization of NP. Theoretical Computer Science 21(3), 255–267 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  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. Lambek, J.: How to program an infinite abacus. Canad. Math. Bull. 4, 295–302 (1961)

    Article  MathSciNet  Google Scholar 

  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)

    Chapter  Google Scholar 

  24. Manders, K.L., Adleman, L.: NP-complete decision problems for binary quadratics. J.Comput. System Sci. 16(2), 168–184 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  25. Martin-Löf, P.: Notes on Constructive Mathematics. Almqvist & Wikseil, Stockholm (1970)

    Google Scholar 

  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. 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. 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. 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. Matiysevich, Y.: A direct method for simulating partial recursive functions by Diophantine equations. Annals Pure Appl. Logic 67, 325–348 (1994)

    Article  Google Scholar 

  31. Matiyasevich, Y.: Hilbert’s tenth problem: what was done and what is to be done. Contemporary mathematics 270, 1–47 (2000)

    MathSciNet  Google Scholar 

  32. Melzak, Z.A.: An informal arithmetical approach to computability and computation. Canad. Math. Bull. 4, 279–294 (1961)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  34. Minsky, M.L.: Computation: Finite and Infinite Machines. Prentice Hall, Englewood Cliffs (1967)

    MATH  Google Scholar 

  35. Pollett, C.: On the Bounded Version of Hilbert’s Tenth Problem. Arch. Math. Logic 42(5), 469–488 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  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. Shepherdson, J.C., Sturgis, H.E.: Computability of recursive functions. J. ACM 10(2), 217–255 (1963)

    Article  MATH  MathSciNet  Google Scholar 

  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 Mathematics

    Google Scholar 

  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. 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. 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. 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. http://logic.pdmi.ras.ru/Hilbert10

Download references

Author information

Authors and Affiliations

  1. Steklov Institute of Mathematics Laboratory of Mathematical Logic, 27, Fontanka, St.Petersburg, 191023, Russia

    Yuri Matiyasevich

Authors
  1. Yuri Matiyasevich
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. School of Mathematics, University of Leeds, LS2 9JT, Leeds, U.K.

    S. Barry Cooper

  2. Department Mathematik, Universität Hamburg, Bundesstrasse 55, 20146, Hamburg, Germany

    Benedikt Löwe

  3. Universiteit van Amsterdam,  

    Leen Torenvliet

Rights and permissions

Reprints and permissions

Copyright information

© 2005 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Matiyasevich, Y. (2005). Hilbert’s Tenth Problem and Paradigms of Computation. In: Cooper, S.B., Löwe, B., Torenvliet, L. (eds) New Computational Paradigms. CiE 2005. Lecture Notes in Computer Science, vol 3526. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11494645_39

Download citation

  • DOI: https://doi.org/10.1007/11494645_39

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-26179-7

  • Online ISBN: 978-3-540-32266-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation