ABSTRACT
We promote the theory of computational complexity on metric spaces: as natural common generalization of (i) the classical discrete setting of integers, binary strings, graphs etc. as well as of (ii) the bit-complexity theory on real numbers and functions according to Friedman, Ko (1982ff), Cook, Braverman et al.; as (iii) resource-bounded refinement of the theories of computability on, and representations of, continuous universes by Pour-El&Richards (1989) and Weihrauch (1993ff); and as (iv) computational perspective on quantitative concepts from classical Analysis: Our main results relate (i.e. upper and lower bound) Kolmogorov's entropy of a compact metric space X polynomially to the uniform relativized complexity of approximating various families of continuous functions on X. The upper bounds are attained by carefully crafted oracles and bit-cost analyses of algorithms perusing them. They all employ the same representation (i.e. encoding, as infinite binary sequences, of the elements) of such spaces, which thus may be of own interest. The lower bounds adapt adversary arguments from unit-cost Information-Based Complexity to the bit model. They extend to, and indicate perhaps surprising limitations even of, encodings via binary string functions (rather than sequences) as introduced by Kawamura&Cook (SToC'2010, §3.4). These insights offer some guidance towards suitable notions of complexity for higher types.
- {ACN07} K. Aehlig, S. Cook, P. Nguyen: "Relativizing Small Complexity Classes and their Theories", pp. 374--388 in Proc. 21st Int. Workshop on Computer Science Logic, 16th Ann. Conf. EACSL, LNCS vol. 4646 (2007). Google ScholarDigital Library
- {Brat02} V. Brattka: "Some Notes on Fine Computability", J. Universal Computer Science vol.8:3 (2002).Google Scholar
- {BrCo06} M. Braverman, S.A. Cook: "Computing over the Reals: Foundations for Scientific Computing", pp. 318--329 in Notices of the AMS vol.53:3 (2006).Google Scholar
- {Buss88} J.F. Buss: "Relativized Alternation and Space-Bounded Computation", pp. 351--378 in Journal of Computer and System Sciences vol.36 (1988). Google ScholarDigital Library
- {dBYa10} M. de Brecht, A. Yamamoto: "Topological properties of concept spaces", pp. 327--340 in Information and Computation vol.208:4 (2010). Google ScholarDigital Library
- {CSV13} S. Chaudhuri, S. Sankaranarayanan, M.Y. Vardi: "Regular Real Analysis", pp. 509--518 in Proc. 28th Ann. IEEE Symposium on Logic in Computer Science (LiCS2013). Google ScholarDigital Library
- {DaRo09} N. Danner, J.S. Royer: "Ramified Structural Recursion and Corecursion", arXiv:1201.4567 (2012).Google Scholar
- {Esca13} M.H. Escardó: "Algorithmic Solution of Higher-Type Equations", pp. 839--854 in Journal of Logic and Computation vol.23:4 (2013).Google ScholarCross Ref
- {FeZi15} H. Férée, M. Ziegler: "On the Computational Complexity of Positive Linear Functionals on C{0; 1}", presented at CCA2015; extended abstract in Proc. 6th Int. Conf. on Math. Aspects of Computer and Information Sciences (2015).Google Scholar
- {FeHo13} H. Férée: "Higher-order Complexity in Analysis", in Proc. 10th International Conference on Computability and Complexity in Analysis (CCA2013); Inria hal-00915973.Google Scholar
- {FGH14} H. Férée, W. Gomaa, M. Hoyrup: "Analytical properties of resource-bounded real functionals", pp. 647--671 in Journal of Complexity vol.30:5 (2014).Google ScholarCross Ref
- {FHHP15} H. Férée, E. Hainry, M. Hoyrup, R. Péchoux: "Characterizing Polynomial Time Complexity of Stream Programs using Interpretations", pp. 41--54 in Theoretical Computer Science vol. 595 (2015). Google ScholarDigital Library
- {FlGr06} J. Flum, M. Grohe: Parameterized Complexity Theory, Springer (2006). Google ScholarDigital Library
- {GrKi14} V. Gregoriades, T. Kihara: "Recursion and Effectivity in the Decomposability Conjecture", submitted; arXiv:1410.1052 (2014).Google Scholar
- {Hert99} P. Hertling: "A Real Number Structure that is Effectively Categorical", pp. 147--182 in Mathematical Logic Quarterly vol.45:2 (1999).Google ScholarCross Ref
- {Hert04} P. Hertling: "A BanachMazur computable but not Markov computable function on the computable real numbers", pp. 227--246 in Ann. Pure and Applied Logic vol.132 (2004).Google Scholar
- {IRK01} R.J. Irwin, J.S. Royer, B.M. Kapron: "On Characterizations of the Basic Feasible Functionals (Part I)", pp. 117--153 in J. Func. Program. vol.11:1 (2001). Google ScholarDigital Library
- {KaCo96} B.M. Kapron, S.A. Cook: "A New Characterization of Type-2 Feasibility", pp. 117--132 in SIAM Journal on Computing vol.25:1 (1996). Google ScholarDigital Library
- {KaCo10} A. Kawamura, S.A. Cook: "Complexity Theory for Operators in Analysis", pp. 495--502 in Proc. 42nd Ann. ACM Symp. on Theory of Computing (STOC 2010); full version in ACM Transactions in Computation Theory vol.4:2 (2012), article 5. Google ScholarDigital Library
- {KaOt14} A. Kawamura, H. Ota: "Small Complexity Classes for Computable Analysis", pp. 432--444 in Proc. 39th Int. Symp. on Math. Found. Computer Science (MFCS2014), Springer LNCS vol.8635.Google Scholar
- {KaPa14} A. Kawamura, A. Pauly: "Function Spaces for Second-Order Polynomial Time", pp. 245--254 in Proc. 10th Conf. on Computability in Europe, Springer LNCS vol.8493 (2014).Google Scholar
- {KaPa15} A. Kawamura, A. Pauly: "Function Spaces for Second-Order Polynomial Time", arXiv:1401.2861v2 (2015).Google Scholar
- {KMRZ15} A. Kawamura, N. Müller, C. Rösnick, M. Ziegler: "Computational Benefit of Smoothness: Parameterized Bit-Complexity of Numerical Operators on Analytic Functions and Gevrey's Hierarchy", pp. 689--714 in Journal of Complexity vol.31:5 (2015). Google ScholarDigital Library
- {Ko91} K.-I. Ko: Computational Complexity of Real Functions, Birkhäuser (1991).Google Scholar
- {Kohl08} U. Kohlenbach: Applied Proof Theory, Springer (2008).Google Scholar
- {KORZ12} A. Kawamura, H. Ota, C. Rösnick, M. Ziegler: "Computational Complexity of Smooth Differential Equations", pp. 578--589 in Proc. 37th Int.Symp. Math. Found. Comp. Sci. (MFCS'2012), Springer LNCS vol.7464; full version in Logical Methods in Computer Science vol.10:1 (2014). Google ScholarDigital Library
- {KoTi59} A.N. Kolmogorov, V.M. Tikhomirov: "ϵ-Entropy and ϵ-Capacity of Sets in Functional Spaces", pp. 3--86 in Uspekhi Mat. Nauk vol.14:2 (1959); also pp.86--170 in Selected Works of A.N. Kolmogorov vol.III (A.N. Shiryayev Edt.), Nauka (1987) and Springer (1993).Google Scholar
- {KRC00} V. Kabanets, C. Rackoff, S.A. Cook: "Efficiently Approximable Real-Valued Functions", ECCC Report No.34 (2000).Google Scholar
- {KSZ14} A. Kawamura, F. Steinberg, M. Ziegler: "Complexity of Laplace's and Poisson's Equation", abstract p.231 in Bulletin of Symbolic Logic vol.20:2 (2014); full version to appear in Mathem. Structures in Computer Science (2016).Google Scholar
- {Lamb06} B. Lambov: "The basic feasible functionals in computable analysis", pp. 909--917 in Journal of Complexity vol.22:6 (2006). Google ScholarDigital Library
- {Mehl76} K. Mehlhorn: "Polynomial and Abstract Subrecursive Classes", pp. 147--178 in Journal of Computer and System Sciences vol.12:2 (1976). Google ScholarDigital Library
- {Nied06} R. Niedermeier: Invitation to Fixed-Parameter Algorithms, Oxford University Press (2006).Google Scholar
- {PaZi13} A. Pauly, M. Ziegler: "Relative Computability and Uniform Continuity of Relations", vol.5 in the Journal of Logic and Analysis (2013).Google Scholar
- {PER89} M.B. Pour-El, I. Richards: Computability in Analysis and Physics, Springer (1989).Google Scholar
- {Rett12} R. Rettinger: "On computable approximations of Landau's constant", Logical Methods in Computer Science vol.8(4) (2012).Google Scholar
- {Schr95} M. Schröder: "Topological Spaces Allowing Type-2 Complexity Theory", in Workshop on Computability and Complexity in Analysis, Informatik-Berichte 190, FernUniversität Hagen (1995).Google Scholar
- {Schr04} M. Schröder: "Spaces Allowing Type-2 Complexity Theory Revisited", pp. 443--459 in Mathematical Logic Quarterly vol.50 (2004).Google Scholar
- {Schr06} M. Schröder: "Admissible Representations in Computable Analysis", pp. 471--480 in Proc. 2nd Conf. on Computability in Europe (CiE'06), LNCS vol.3988. Google ScholarDigital Library
- {Tima63} A.F. Timan: Theory of Approximation of Functions of a Real Variable, Pergamon (1963).Google Scholar
- {Weih00} K. Weihrauch: Computable Analysis, Springer (2000). Google ScholarDigital Library
- {Weih03} K. Weihrauch: "Computational Complexity on Computable Metric Spaces", pp. 3--21 in Mathematical Logic Quarterly vol.49:1 (2003).Google Scholar
- {WeZh02} K. Weihrauch, N. Zhong: "Is Wave Propagation Computable or Can Wave Computers Beat the Turing Machine?", pp. 312--332 in Proc. London Mathem. Society vol.85:2 (2002).Google Scholar
- {Wils88} C.B. Wilson: "A measure of relativized space which is faithful with respect to depth", pp. 303--312 in J. Comput. System Sci. vol.36 (1988). Google ScholarDigital Library
- {Zieg07} M. Ziegler: "Real Hypercomputation and Continuity", pp. 177--206 in Theory of Computing Systems vol.41 (2007). Google ScholarDigital Library
Index Terms
- Complexity Theory of (Functions on) Compact Metric Spaces
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