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A Parameterized Halting Problem

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The Multivariate Algorithmic Revolution and Beyond

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7370))

Abstract

The parameterized problem \(p\textsc{-Halt}\) takes as input a nondeterministic Turing machine \(\mathbb{M}\) and a natural number n, the size of \(\mathbb{M}\) being the parameter. It asks whether every accepting run of \(\mathbb{M}\) on empty input tape takes more than n steps. This problem is in the class XPuni, the class “uniform XP,” if there is an algorithm deciding it, which for fixed machine \(\mathbb{M}\) runs in time polynomial in n. It turns out that various open problems of different areas of theoretical computer science are related or even equivalent to \(p{\rm \textsc{-Halt}\in{XP}_{uni}}\). Thus this statement forms a bridge which allows to derive equivalences between statements of different areas (proof theory, complexity theory, descriptive complexity, …) which at first glance seem to be unrelated. As our presentation shows, various of these equivalences may be obtained by the same method.

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References

  1. Aumann, Y., Dombb, Y.: Fixed Structure Complexity. In: Grohe, M., Niedermeier, R. (eds.) IWPEC 2008. LNCS, vol. 5018, pp. 30–42. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  2. Buhrman, H., Hitchcock, J.M.: NP-hard sets are exponentially dense unless coNP ⊆ NP/poly. In: Proceedings of the 23rd Annual IEEE Conference on Computational Complexity (CCC 2008), Electronic Colloquium on Computational Complexity (ECCC 2008), Report TR08 022, pp. 1–7 (2008), http://eccc.hpi-web.de/eccc-local/Lists/TR-2008.html

  3. Buss, S., Chen, Y., Flum, J., Friedman, S., Müller, M.: Strong isomorphism reductions in complexity theory. The Journal of Symbolic Logic 76(4), 1381–1402 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cai, L., Chen, J., Downey, R., Fellows, M.: the parameterized complexity of short computation and factorization. Archive for Mathematical Logic 36, 321–337 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cesati, M.: The Turing way to parameterized complexity. Journal of Computer and System Sciences 67, 654–685 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cesati, M., Di Ianni, M.: Computation models for parameterized complexity. Mathematicall Logical Quarterly 43, 179–202 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chandra, A.K., Kozen, D., Stockmeyer, L.J.: Alternation. Journal of the ACM 28, 114–133 (1981); 77–90 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen, Y., Flum, J.: On the complexity of Gödel’s proof predicate. The Journal of Symbolic Logic 75, 239–254 (2009)

    Article  MATH  Google Scholar 

  9. Chen, Y., Flum, J.: A Logic for PTIME and a Parameterized Halting Problem. In: Blass, A., Dershowitz, N., Reisig, W. (eds.) Gurevich Festschrift. LNCS, vol. 6300, pp. 251–276. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  10. Chen, Y., Flum, J.: On p-Optimal Proof Systems and Logics for PTIME. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010, Part II. LNCS, vol. 6199, pp. 321–332. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  11. Chen, Y., Flum, J.: On Slicewise Monotone Parameterized Problems and Optimal Proof Systems for TAUT. In: Dawar, A., Veith, H. (eds.) CSL 2010. LNCS, vol. 6247, pp. 200–214. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  12. Chen, Y., Flum, J.: Listings and logics. In: Proceedings of the 26th Annual IEEE Symposium on Logic in Computer Science (LICS 2011), pp. 165–174 (2011)

    Google Scholar 

  13. Cook, S.: The complexity of theorem proving procedures. In: Proceedings of the Third Annual ACM Symposium on Theory of Computing, pp. 151–158 (1971)

    Google Scholar 

  14. Cook, S., Reckhow, R.: The relative efficiency of propositional proof systems. The Journal of Symbolic Logic 44, 36–50 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  15. Downey, R., Fellows, M.: Fixed-parameter tractability and commpleteness III: Some structurl aspects of the W-hierarchy. In: Ambos-Spies, K., et al. (eds.) Complexity Theory, pp. 166–191 (1993)

    Google Scholar 

  16. Downey, R., Fellows, M.: Parameterized Complexity. Springer (1999)

    Google Scholar 

  17. Downey, R.: Private communication

    Google Scholar 

  18. Fagin, R.: Generalized first–order spectra and polynomial–time recognizable sets. In: Karp, R.M. (ed.) Complexity of Computation. SIAM-AMS Proceedings, vol. 7, pp. 43–73 (1974)

    Google Scholar 

  19. Flum, J., Grohe, M.: Fixed-parameter tractability, definability, and model checking. SIAM Journal on Computing 31, 113–145 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  20. Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer (2006)

    Google Scholar 

  21. Fortnow, L., Grochow, J.: Complexity classes of equivalence problems revisited, arXiv:0907.4775v1, [cs.CC] (2009)

    Google Scholar 

  22. Hartmanis, J., Hemachandra, L.: Complexity classes without machines: On complete languages for UP. Theoretical Computer Science 58, 129–142 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  23. Immerman, N.: Relational queries computable in polynomial time. Information and Control 68, 86–104 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  24. Köbler, J., Messner, J.: Complete problems for promise classes by optimal proof systems for test sets. In: Proceedings of the 13th IEEE Conference on Computational Complexity (CCC 1998), pp. 132–140 (1998)

    Google Scholar 

  25. Köbler, J., Messner, J., Torán, J.: Optimal proof systems imply complete sets for promise classes. Information and Computation 184, 71–92 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kowalczyk, W.: Some Connections Between Presentability of Complexity Classes and the Power of Formal Systems of Reasonning. In: Chytil, M.P., Koubek, V. (eds.) MFCS 1984. LNCS, vol. 176, pp. 364–369. Springer, Heidelberg (1984)

    Chapter  Google Scholar 

  27. Krajíček, J., Pudlák, P.: Propositional proof systems, the consistency of first order theories and the complexity of computations. The Journal of Symbolic Logic 54, 1063–1088 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  28. Levin, L.: Universal search problems. Problems of Information Transmission 9(3), 265–266 (1973) (in Russian); (english translation) Trakhtenbrot, B.A.: A survey of Russian approaches to perebor (brute-force search) algorithms. Annals of the History of Computing 6(4), 384–400 (1984)

    Google Scholar 

  29. Messner, J., Torán, J.: Optimal Proof Systems for Propositional Logic and Complete Sets. In: Meinel, C., Morvan, M. (eds.) STACS 1998. LNCS, vol. 1373, pp. 477–487. Springer, Heidelberg (1998)

    Chapter  Google Scholar 

  30. Monroe, H.: Speedup for natural problems and noncomputability. Theoretical Computer Science 412(4-5), 478–481 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  31. Nash, A., Remmel, J.B., Vianu, V.: PTIME Queries Revisited. In: Eiter, T., Libkin, L. (eds.) ICDT 2005. LNCS, vol. 3363, pp. 274–288. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  32. Sadowski, Z.: On an optimal propositional proof system and the structure of easy subsets. Theoretical Computer Science 288(1), 181–193 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  33. Turing, A.: On computable numbers, with an application to the Entscheidungsproblem. Proc. London Math. Society 2, 230–265 (1936)

    MathSciNet  MATH  Google Scholar 

  34. Vardi, M.Y.: The complexity of relational query languages. In: Proceedings of the 14th Annual ACM Symposium on Theory of Computing (STOC 1982), pp. 137–146 (1982)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Shanghai Jiaotong University, China

    Yijia Chen

  2. Albert-Ludwigs-Universität Freiburg, Germany

    Jörg Flum

Authors
  1. Yijia Chen
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  2. Jörg Flum
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Editor information

Editors and Affiliations

  1. Department of Information and Computing Sciences, Utrecht University, P.O. Box 80.089, 3508 TB, Utrecht, The Netherlands

    Hans L. Bodlaender

  2. School of Mathematics, Statistics and Operations Research, Victoria University, P.O. Box 600, Wellington, New Zealand

    Rod Downey

  3. Institute of Informatics, University of Bergen, Postboks 7800, 5020, Bergen, Norway

    Fedor V. Fomin

  4. Computer and Automation Research Institute, Hungarian Academy of Sciences (MTA SZTAKI), Pf. 63, 1518, Budapest, Hungary

    Dániel Marx

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Chen, Y., Flum, J. (2012). A Parameterized Halting Problem. In: Bodlaender, H.L., Downey, R., Fomin, F.V., Marx, D. (eds) The Multivariate Algorithmic Revolution and Beyond. Lecture Notes in Computer Science, vol 7370. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30891-8_17

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  • DOI: https://doi.org/10.1007/978-3-642-30891-8_17

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-30890-1

  • Online ISBN: 978-3-642-30891-8

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