Abstract
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.
Supported in part by the Marie Curie International Research Staff Exchange Scheme Fellowship 294962 within the 7th European Community Framework Programme and by the German Research Foundation (DFG) with project Zi 1009/4-1. We acknowledge seminal discussions with Vassilis Gregoriades, Thorsten Kräling, and Hermann K.-G. Walter.
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References
Agrawal, M.: The Isomorphism Conjecture for \(\mathcal{NP}\). In: Cooper, S.B., Sorbi, A. (eds.) Computability in Context, pp. 19–48. World Scientific (2009)
Bürgisser, P., Clausen, M., Shokrollahi, M.A.: Algebraic Complexity Theory. Springer (1997)
Beyersdorff, O., Köbler, J., Müller, S.: Proof Systems that Take Advice. Proof Systems that Take Advice 209(3), 320–332 (2011)
Brattka, V.: Recursive Characterization of Computable Real-Valued Functions and Relations. Theoretical Computer Science 162, 45–77 (1996)
Brattka, V.: Computable Invariance. Theoretical Computer Science 210, 3–20 (1999)
Braverman, M.: On the Complexity of Real Functions. In: Proc. 46th Annual IEEE Symposium on Foundations of Computer Science, pp. 155–164
Braverman, M., Cook, S.A.: Computing over the Reals: Foundations for Scientific Computing. Notices of the Americal Mathematical Society 53(3), 318–329 (2006)
Brattka, V., Pauly, A.M.: Computation with Advice. In: Electronic Proceedings in Theoretical Computer Science, vol. 24 (June 2010)
Brandt, U., Walter, H.K.-G.: Cohesiveness in Promise Problems. Presented at the 64th GI Workshop on Algorithms and Complexity (2012)
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)
Even, S., Selman, A.L., Yacobi, Y.: Hard-Core Theorems for Complexity Classes. Journal of the ACM 32(1), 205–217 (1985)
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)
Goldreich, O.: Computational Complexity: A Conceptual Perspective. Cambridge University Press (2008)
Grzegorczyk, A.: On the Definitions of Computable Real Continuous Functions. Fundamenta Mathematicae 44, 61–77 (1957)
Hemaspaandra, L.A., Torenvliet, L.: Theory of Semi-Feasible Algorithms. Springer Monographs in Theoretical Computer Science (2003)
Hertling, P.: Topological Complexity of Zero Finding with Algebraic Operations. Journal of Complexity 18(4), 912–942 (2002)
Kawamura, A.: Lipschitz Continuous Ordinary Differential Equations are Polynomial-Space Complete. Computational Complexity 19(2), 305–332 (2010)
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)
Kawamura, A., Cook, S.A.: Complexity Theory for Operators in Analysis. ACM Transactions in Computation Theory 4(2), article 5 (2012)
Ko, K.-I., Friedman, H.: Computational Complexity of Real Functions. Theoretical Computer Science 20, 323–352 (1982)
Ko, K.-I.: Complexity Theory of Real Functions. Birkhäuser (1991)
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)
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)
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)
Li, M., Vitányi, P.: An Introduction to Kolmogorov Complexity and Its Applications, 2nd edn. Springer (1997)
Luckhardt, H.: A Fundamental Effect in Computations on Real Numbers. Theoretical Computer Science 5, 321–324 (1977)
Lynch, N.: On Reducibility to Complex or Sparse Sets. Journal of the ACM 22(3), 341–345 (1975)
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)
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)
Müller, N.T.: Uniform Computational Complexity of Taylor Series. In: Ottmann, T. (ed.) ICALP 1987. LNCS, vol. 267, pp. 435–444. Springer, Heidelberg (1987)
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)
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)
Müller, N.T., Moiske, B.: Solving Initial Value Problems in Polynomial Time. In: Proc. 22nd JAIIO-PANEL, pp. 283–293 (1993)
Müller, N.T., Zhao, X.: Complexity of Operators on Compact Sets. Electronic Notes Theoretical Computer Science 202, 101–119 (2008)
Pour-El, M.B., Richards, J.I.: Computability in Analysis and Physics. Springer (1989)
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)
Traub, J.F., Wasilkowski, G.W., Woźniakowski, H.: Information-Based Complexity. Academic Press (1988)
Weihrauch, K.: Computable Analysis. Springer (2000)
Ziegler, M., Brattka, V.: Computability in Linear Algebra. Theoretical Computer Science 326, 187–211 (2004)
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)
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Ambos-Spies, K., Brandt, U., Ziegler, M. (2013). Real Benefit of Promises and Advice. In: Bonizzoni, P., Brattka, V., Löwe, B. (eds) The Nature of Computation. Logic, Algorithms, Applications. CiE 2013. Lecture Notes in Computer Science, vol 7921. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39053-1_1
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