TY - JOUR

T1 - Computational benefit of smoothness

T2 - Parameterized bit-complexity of numerical operators on analytic functions and Gevrey's hierarchy

AU - Kawamura, Akitoshi

AU - Müller, Norbert

AU - Rösnick, Carsten

AU - Ziegler, Martin

N1 - Publisher Copyright:
© 2015 The Authors.
Copyright:
Copyright 2018 Elsevier B.V., All rights reserved.

PY - 2015/10/1

Y1 - 2015/10/1

N2 - Abstract The synthesis of (discrete) Complexity Theory with Recursive Analysis provides a quantitative algorithmic foundation to calculations over real numbers, sequences, and functions by approximation up to prescribable absolute error 1/2n (roughly corresponding to n binary digits after the radix point). In this sense Friedman and Ko have shown the seemingly simple operators of maximization and integration 'complete' for the standard complexity classes NP and #P - even when restricted to smooth (=C∞) arguments. Analytic polynomial-time computable functions on the other hand are known to get mapped to polynomial-time computable functions: non-uniformly, that is, disregarding dependences other than on the output precision n. The present work investigates the uniform parameterized complexity of natural operators Λ on subclasses of smooth functions: evaluation, pointwise addition and multiplication, (iterated) differentiation, integration, and maximization. We identify natural integer parameters k=k(f) which, when given as enrichment to approximations to the function argument f, permit to computably produce approximations to Λ(f); and we explore the asymptotic worst-case running time sufficient and necessary for such computations in terms of the output precision n and said k. It turns out that Maurice Gevrey's 1918 classical hierarchy climbing from analytic to (just below) smooth functions provides for a quantitative gauge of the uniform computational complexity of maximization and integration that, non-uniformly, exhibits the phase transition from tractable (i.e. polynomial-time) to intractable (in the sense of NP-'hardness'). Our proof methods involve Hard Analysis, Approximation Theory, and an adaptation of Information-Based Complexity to the bit model.

AB - Abstract The synthesis of (discrete) Complexity Theory with Recursive Analysis provides a quantitative algorithmic foundation to calculations over real numbers, sequences, and functions by approximation up to prescribable absolute error 1/2n (roughly corresponding to n binary digits after the radix point). In this sense Friedman and Ko have shown the seemingly simple operators of maximization and integration 'complete' for the standard complexity classes NP and #P - even when restricted to smooth (=C∞) arguments. Analytic polynomial-time computable functions on the other hand are known to get mapped to polynomial-time computable functions: non-uniformly, that is, disregarding dependences other than on the output precision n. The present work investigates the uniform parameterized complexity of natural operators Λ on subclasses of smooth functions: evaluation, pointwise addition and multiplication, (iterated) differentiation, integration, and maximization. We identify natural integer parameters k=k(f) which, when given as enrichment to approximations to the function argument f, permit to computably produce approximations to Λ(f); and we explore the asymptotic worst-case running time sufficient and necessary for such computations in terms of the output precision n and said k. It turns out that Maurice Gevrey's 1918 classical hierarchy climbing from analytic to (just below) smooth functions provides for a quantitative gauge of the uniform computational complexity of maximization and integration that, non-uniformly, exhibits the phase transition from tractable (i.e. polynomial-time) to intractable (in the sense of NP-'hardness'). Our proof methods involve Hard Analysis, Approximation Theory, and an adaptation of Information-Based Complexity to the bit model.

UR - http://www.scopus.com/inward/record.url?scp=84938749973&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84938749973&partnerID=8YFLogxK

U2 - 10.1016/j.jco.2015.05.001

DO - 10.1016/j.jco.2015.05.001

M3 - Article

AN - SCOPUS:84938749973

VL - 31

SP - 689

EP - 714

JO - Journal of Complexity

JF - Journal of Complexity

SN - 0885-064X

IS - 5

M1 - 1247

ER -