Average-Case Polynomial-Time Computability of Hamiltonian Dynamics

Kawamura, Akitoshi; Thies, Holger; Ziegler, Martin

doi:10.4230/lipics.mfcs.2018.30

Cited by 4 publications

(2 citation statements)

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“…A variation of the Ko-Friedman style definition is precisely given in the textbook by Weihrauch [15]. An equivalent definition is given in [6]. Definition 3.…”

Section: Computable Real Numbers and Computable Real Functionsmentioning

confidence: 99%

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Solovay reduction and continuity

Kumabe,

Miyabe,

Mizusawa

et al. 2019

Preprint

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The objective of this study is a better understanding of the relationships between reduction and continuity. Solovay reduction is a variation of Turing reduction based on the distance of two real numbers. We characterize Solovay reduction by the existence of a certain real function that is computable (in the sense of computable analysis) and Lipschitz continuous. We ask whether there exists a reducibility concept that corresponds to Hölder continuity. The answer is affirmative. We introduce quasi Solovay reduction and characterize this new reduction via Hölder continuity. In addition, we separate it from Solovay reduction and Turing reduction and investigate the relationships between complete sets and partial randomness.

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“…A variation of the Ko-Friedman style definition is precisely given in the textbook by Weihrauch [15]. An equivalent definition is given in [6]. Definition 3.…”

Section: Computable Real Numbers and Computable Real Functionsmentioning

confidence: 99%

“…Definition 4. [6] Suppose that f is a partial function from R to R. The function f is computable if there exists an oracle Turing machine M with the following property. For each x in the domain of f and for each name ϕ of x, M ϕ computes a name of f (x).…”

Section: Computable Real Numbers and Computable Real Functionsmentioning

confidence: 99%

Solovay reduction and continuity

Kumabe,

Miyabe,

Mizusawa

et al. 2019

Preprint

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Parametrised second-order complexity theory with applications to the study of interval computation

Neumann¹,

Steinberg²

2020

Theoretical Computer Science

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We extend the framework for complexity of operators in analysis devised by Kawamura and Cook (2012) to allow for the treatment of a wider class of representations. The main novelty is to endow represented spaces of interest with an additional function on names, called a parameter, which measures the complexity of a given name. This parameter generalises the size function which is usually used in second-order complexity theory and therefore also central to the framework of Kawamura and Cook. The complexity of an algorithm is measured in terms of its running time as a second-order function in the parameter, as well as in terms of how much it increases the complexity of a given name, as measured by the parameters on the input and output side.As an application we develop a rigorous computational complexity theory for interval computation. In the framework of Kawamura and Cook the representation of real numbers based on nested interval enclosures does not yield a reasonable complexity theory. In our new framework this representation is polytime equivalent to the usual Cauchy representation based on dyadic rational approximation. By contrast, the representation of continuous real functions based on interval enclosures is strictly smaller in the polytime reducibility lattice than the usual representation, which encodes a modulus of continuity. Furthermore, the function space representation based on interval enclosures is optimal in the sense that it contains the minimal amount of information amongst those representations which render evaluation polytime computable.

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A Universal Ordinary Differential Equation

Bournez¹,

Pouly²

2020

Logical Methods in Computer Science ; Volume 16

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Average-Case Polynomial-Time Computability of Hamiltonian Dynamics

Cited by 4 publications

References 0 publications

Solovay reduction and continuity

Solovay reduction and continuity

Parametrised second-order complexity theory with applications to the study of interval computation

A Universal Ordinary Differential Equation

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