Real Benefit of Promises and Advice

  • Klaus Ambos-Spies
  • Ulrike Brandt
  • Martin Ziegler
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7921)


Promises are a standard way to formalize partial algorithms; and advice quantifies nonuniformity. For decision problems, the latter is captured in common complexity classes such as \(\mathcal{P}/\operatorname{poly}\), that is, with advice growing in size with that of the input. We advertise constant-size advice and explore its theoretical impact on the complexity of classification problems – a natural generalization of promise problems – and on real functions and operators. Specifically we exhibit problems that, without any advice, are decidable/computable but of high complexity while, with (each increase in the permitted size of) advice, (gradually) drop down to polynomial-time.


Turing Machine Multivalued Mapping Hard Core Kolmogorov Complexity Circuit Family 
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.


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  1. 1.
    Agrawal, M.: The Isomorphism Conjecture for \(\mathcal{NP}\). In: Cooper, S.B., Sorbi, A. (eds.) Computability in Context, pp. 19–48. World Scientific (2009)Google Scholar
  2. 2.
    Bürgisser, P., Clausen, M., Shokrollahi, M.A.: Algebraic Complexity Theory. Springer (1997)Google Scholar
  3. 3.
    Beyersdorff, O., Köbler, J., Müller, S.: Proof Systems that Take Advice. Proof Systems that Take Advice 209(3), 320–332 (2011)zbMATHGoogle Scholar
  4. 4.
    Brattka, V.: Recursive Characterization of Computable Real-Valued Functions and Relations. Theoretical Computer Science 162, 45–77 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Brattka, V.: Computable Invariance. Theoretical Computer Science 210, 3–20 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Braverman, M.: On the Complexity of Real Functions. In: Proc. 46th Annual IEEE Symposium on Foundations of Computer Science, pp. 155–164Google Scholar
  7. 7.
    Braverman, M., Cook, S.A.: Computing over the Reals: Foundations for Scientific Computing. Notices of the Americal Mathematical Society 53(3), 318–329 (2006)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Brattka, V., Pauly, A.M.: Computation with Advice. In: Electronic Proceedings in Theoretical Computer Science, vol. 24 (June 2010)Google Scholar
  9. 9.
    Brandt, U., Walter, H.K.-G.: Cohesiveness in Promise Problems. Presented at the 64th GI Workshop on Algorithms and Complexity (2012)Google Scholar
  10. 10.
    Even, S., Selman, A.L., Yacobi, Y.: The Complexity of Promise Problems with Applications to Public-Key Cryptography. Inform. and Control 61, 159–173 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Even, S., Selman, A.L., Yacobi, Y.: Hard-Core Theorems for Complexity Classes. Journal of the ACM 32(1), 205–217 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Goldreich, O.: On Promise Problems: A Survey. In: Goldreich, O., Rosenberg, A.L., Selman, A.L. (eds.) Shimon Even Festschrift. LNCS, vol. 3895, pp. 254–290. Springer, Heidelberg (2006)Google Scholar
  13. 13.
    Goldreich, O.: Computational Complexity: A Conceptual Perspective. Cambridge University Press (2008)Google Scholar
  14. 14.
    Grzegorczyk, A.: On the Definitions of Computable Real Continuous Functions. Fundamenta Mathematicae 44, 61–77 (1957)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Hemaspaandra, L.A., Torenvliet, L.: Theory of Semi-Feasible Algorithms. Springer Monographs in Theoretical Computer Science (2003)Google Scholar
  16. 16.
    Hertling, P.: Topological Complexity of Zero Finding with Algebraic Operations. Journal of Complexity 18(4), 912–942 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Kawamura, A.: Lipschitz Continuous Ordinary Differential Equations are Polynomial-Space Complete. Computational Complexity 19(2), 305–332 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Kawamura, A., Cook, S.A.: Complexity Theory for Operators in Analysis. In: Proc. 42nd Ann. ACM Symp. on Theory of Computing (STOC 2010), pp. 495–502 (2010)Google Scholar
  19. 19.
    Kawamura, A., Cook, S.A.: Complexity Theory for Operators in Analysis. ACM Transactions in Computation Theory 4(2), article 5 (2012)Google Scholar
  20. 20.
    Ko, K.-I., Friedman, H.: Computational Complexity of Real Functions. Theoretical Computer Science 20, 323–352 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Ko, K.-I.: Complexity Theory of Real Functions. Birkhäuser (1991)Google Scholar
  22. 22.
    Ko, K.-I.: Polynomial-Time Computability in Analysis. In: Ershov, Y.L., et al. (eds.) Handbook of Recursive Mathematics, vol. 2, pp. 1271–1317 (1998)Google Scholar
  23. 23.
    Kawamura, A., Ota, H., Rösnick, C., Ziegler, M.: Computational Complexity of Smooth Differential Equations. In: Rovan, B., Sassone, V., Widmayer, P. (eds.) MFCS 2012. LNCS, vol. 7464, pp. 578–589. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  24. 24.
    Kreisel, G., Macintyre, A.: Constructive Logic versus Algebraization I. In: Troelstra, A.S., van Dalen, D. (eds.) Proc. L.E.J. Brouwer Centenary Symposium, pp. 217–260. North-Holland (1982)Google Scholar
  25. 25.
    Li, M., Vitányi, P.: An Introduction to Kolmogorov Complexity and Its Applications, 2nd edn. Springer (1997)Google Scholar
  26. 26.
    Luckhardt, H.: A Fundamental Effect in Computations on Real Numbers. Theoretical Computer Science 5, 321–324 (1977)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Lynch, N.: On Reducibility to Complex or Sparse Sets. Journal of the ACM 22(3), 341–345 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Michaux, C.: \(\mathcal{P}\neq\mathcal{NP}\) over the Nonstandard Reals Implies \(\mathcal{P}\neq\mathcal{NP}\) over ℝ. Theoretical Computer Science 133, 95–104 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Müller, N.T.: Subpolynomial Complexity Classes of Real Functions and Real Numbers. In: Kott, L. (ed.) ICALP 1986. LNCS, vol. 226, pp. 284–293. Springer, Heidelberg (1986)CrossRefGoogle Scholar
  30. 30.
    Müller, N.T.: Uniform Computational Complexity of Taylor Series. In: Ottmann, T. (ed.) ICALP 1987. LNCS, vol. 267, pp. 435–444. Springer, Heidelberg (1987)CrossRefGoogle Scholar
  31. 31.
    Müller, N.T.: Constructive Aspects of Analytic Functions. In: Proc. Workshop on Computability and Complexity in Analysis (CCA), InformatikBerichte FernUniversität Hagen, vol. 190, pp. 105–114 (1995)Google Scholar
  32. 32.
    Müller, N.T.: The iRRAM: Exact Arithmetic in C++. In: Blank, J., Brattka, V., Hertling, P. (eds.) CCA 2000. LNCS, vol. 2064, pp. 222–252. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  33. 33.
    Müller, N.T., Moiske, B.: Solving Initial Value Problems in Polynomial Time. In: Proc. 22nd JAIIO-PANEL, pp. 283–293 (1993)Google Scholar
  34. 34.
    Müller, N.T., Zhao, X.: Complexity of Operators on Compact Sets. Electronic Notes Theoretical Computer Science 202, 101–119 (2008)CrossRefGoogle Scholar
  35. 35.
    Pour-El, M.B., Richards, J.I.: Computability in Analysis and Physics. Springer (1989)Google Scholar
  36. 36.
    Specker, E.: The Fundamental Theorem of Algebra in Recursive Analysis. In: Dejon, B., Henrici, P. (eds.) Constructive Aspects of the Fundamental Theorem of Algebra, pp. 321–329. Wiley-Interscience (1969)Google Scholar
  37. 37.
    Traub, J.F., Wasilkowski, G.W., Woźniakowski, H.: Information-Based Complexity. Academic Press (1988)Google Scholar
  38. 38.
    Weihrauch, K.: Computable Analysis. Springer (2000)Google Scholar
  39. 39.
    Ziegler, M., Brattka, V.: Computability in Linear Algebra. Theoretical Computer Science 326, 187–211 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Ziegler, M.: Real Computation with Least Discrete Advice: A Complexity Theory of Nonuniform Computability. Annals of Pure and Applied Logic 163(8), 1108–1113 (2012)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Klaus Ambos-Spies
    • 1
  • Ulrike Brandt
    • 2
  • Martin Ziegler
    • 3
  1. 1.Department of Mathematics and Computer ScienceRuprecht-Karls-Universität HeidelbergHeidelbergGermany
  2. 2.Department of Computer ScienceTechnische Universität DarmstadtDarmstadtGermany
  3. 3.Department of MathematicsTechnische Universität DarmstadtDarmstadtGermany

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