Koepke Machines and Satisfiability for Infinitary Propositional Languages

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10307)

Abstract

We consider complexity theory for Koepke machines, also known as Ordinal Turing Machines (OTMs), and define infinitary complexity classes \(\infty \)-\(\mathbf {P}\) and \(\infty {\text {-}}\mathbf {NP}\) and the OTM analogue of the satisfiability problem, denoted by \(\infty {\text {-}}\mathrm {SAT}\). We show that \(\infty {\text {-}}\mathrm {SAT}\) is in \(\infty {\text {-}}\mathbf {NP}\) and \(\infty {\text {-}}\mathbf {NP}\)-hard (i.e., the problem is \(\infty {\text {-}}\mathbf {NP}\)-complete), but not OTM decidable.

References

  1. 1.
    Carl, M.: Towards a Church-Turing-Thesis for infinitary computation (2013) preprint. arXiv:1307.6599
  2. 2.
    Carl, M.: Infinite time recognizability from random oracles and the recognizable jump operator. Computability (to appear)Google Scholar
  3. 3.
    Dawson, B.: Ordinal time Turing Computation. Ph.D. thesis, University of Bristol (2009)Google Scholar
  4. 4.
    Deolalikar, V., Hamkins, J.D., Schindler, R.: \(\mathbf{P}\ne \mathbf{NP}\cap \mathbf{co}{\text{-}}\mathbf{NP}\) for infinite time Turing machines. J. Log. Comput. 15(5), 577–592 (2005)Google Scholar
  5. 5.
    Hamkins, J.D., Lewis, A.: Infinite time turing machines. J. Symb. Log. 65(2), 567–604 (2000)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Hamkins, J.D., Welch, P.D.: \(\mathbf{P}^f \ne \mathbf{NP}^f\) for almost all \(f\). Math. Log. Q. 49(5), 536–540 (2003)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Jensen, R.B., Karp, C.: Primitive recursive set functions. In: Axiomatic Set Theory. Proceedings of the Symposium in Pure Mathematics of the American Mathematical Society held at the University of California, Los Angeles, California, 10 July–5 August, vol. XIII/I of Proceedings of Symposia in Pure Mathematics, pp. 143–176. American Mathematical Society (1971)Google Scholar
  8. 8.
    Kanamori, A.: The Higher Infinite. Large Cardinals in Set Theory from Their Beginnings. Springer Monographs in Mathematics, 2nd edn. Springer, Heidelberg (2003)MATHGoogle Scholar
  9. 9.
    Karp, C.: Languages with Expressions of Infinite Length. North-Holland, Amsterdam (1964)MATHGoogle Scholar
  10. 10.
    Koepke, P.: Turing computations on ordinals. Bull. Symb. Log. 11(3), 377–397 (2005)CrossRefMATHGoogle Scholar
  11. 11.
    Koepke, P.: Ordinal computability. In: Ambos-Spies, K., Löwe, B., Merkle, W. (eds.) CiE 2009. LNCS, vol. 5635, pp. 280–289. Springer, Heidelberg (2009). doi: 10.1007/978-3-642-03073-4_29 CrossRefGoogle Scholar
  12. 12.
    Koepke, P., Seyfferth, B.: Ordinal machines and admissible recursion theory. Ann. Pure Appl. Log. 160, 310–318 (2009)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Löwe, B.: Space bounds for infinitary computation. In: Beckmann, A., Berger, U., Löwe, B., Tucker, J.V. (eds.) CiE 2006. LNCS, vol. 3988, pp. 319–329. Springer, Heidelberg (2006). doi: 10.1007/11780342_34 CrossRefGoogle Scholar
  14. 14.
    Rin, B.: The computational strengths of \(\alpha \)-tape infinite time turing machines. Ann. Pure Appl. Log. 165(9), 1501–1511 (2014)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Schindler, R.: \(\mathbf{P}\ne \mathbf{NP}\) infinite time turing machines. Monatsh. Math. 139, 335–340 (2003)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Winter, J.: Space complexity in infinite time Turing machines. Master’s thesis, Universiteit van Amsterdam. ILLC Publications MoL-2007-14 (2007)Google Scholar
  17. 17.
    Winter, J.: Is P = PSPACE for Infinite time turing machines? In: Archibald, M., Brattka, V., Goranko, V., Löwe, B. (eds.) ILC 2007. LNCS, vol. 5489, pp. 126–137. Springer, Heidelberg (2009). doi: 10.1007/978-3-642-03092-5_10 CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Merlin Carl
    • 1
    • 2
  • Benedikt Löwe
    • 3
    • 4
    • 5
  • Benjamin G. Rin
    • 6
  1. 1.Fachbereich Mathematik und StatistikUniversität KonstanzKonstanzGermany
  2. 2.Fakultät für Informatik und MathematikUniversität PassauPassauGermany
  3. 3.Institute for Logic, Language and ComputationUniversiteit van AmsterdamAmsterdamThe Netherlands
  4. 4.Fachbereich MathematikUniversität HamburgHamburgGermany
  5. 5.Christ’s College, Churchill College, and Faculty of MathematicsUniversity of CambridgeCambridgeEngland
  6. 6.Departement Filosofie En ReligiewetenschapUniversiteit UtrechtUtrechtThe Netherlands

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