A Tutorial on Computable Analysis

Brattka, Vasco; Hertling, Peter; Weihrauch, Klaus

doi:10.1007/978-0-387-68546-5_18

Cited by 153 publications

(160 citation statements)

References 78 publications

Supporting

Mentioning

159

Contrasting

Order By: Relevance

“…for any t ∈ I we can compute an approximation of y(t) with precision 2 −n in time polynomial in n -see e.g. [8]). On the other hand, this system can be solved explicitly and yields: One immediately sees that y d being a tower of exponentials prevents y from being polynomial time (precision-)computable over R, for any reasonable notion, although y d (and y) is polynomial time (precision-)computable over any fixed compact.…”

Section: A Note On Rescalingmentioning

confidence: 99%

Computational complexity of solving polynomial differential equations over unbounded domains

Pouly

Graça²

2016

Theoretical Computer Science

View full text Add to dashboard Cite

In this paper we investigate the computational complexity of solving ordinary differential equations (ODEs) y = p(y) over unbounded time domains, where p is a vector of polynomials. Contrarily to the bounded (compact) time case, this problem has not been well-studied, apparently due to the "intuition" that it can always be reduced to the bounded case by using rescaling techniques. However, as we show in this paper, rescaling techniques do not seem to provide meaningful insights on the complexity of this problem, since the use of such techniques introduces a dependence on parameters which are hard to compute.We present algorithms which numerically solve these ODEs over unbounded time domains. These algorithms have guaranteed accuracy, i.e. given some arbitrarily large time t and error bound ε as input, they will output a valueỹ which satisfies y(t) −ỹ ≤ ε. We analyze the complexity of these algorithms and show that they computeỹ in time polynomial in several quantities including the time t, the accuracy of the output ε and the length of the curve y from 0 to t, assuming it exists until time t. We consider both algebraic complexity and bit complexity.

show abstract

Section: A Note On Rescalingmentioning

confidence: 99%

Computational complexity of solving polynomial differential equations over unbounded domains

Pouly

Graça²

2016

Theoretical Computer Science

View full text Add to dashboard Cite

show abstract

“…An answer can be given only relatively to computability concepts on the measures and functions under consideration, which must be defined in advance. For studying computability on general spaces we use the representation approach for computable analysis (TTE, Type Two theory of Effectivity) [KW85,Wei00,BHW08]. In TTE computability on finite words w ∈ Σ * and infinite sequences p ∈ Σ N is defined explicitly, for example by Turing machines, and then such finite or infinite sequences are used as names of abstract objects.…”

Section: Introductionmentioning

confidence: 99%

Computability of the Radon-Nikodym Derivative

Hoyrup

Rojas

Weihrauch

2011

Models of Computation in Context

Self Cite

View full text Add to dashboard Cite

We study the computational content of the Radon-Nokodym theorem from measure theory in the framework of the representation approach to computable analysis. We define computable measurable spaces and canonical representations of the measures and the integrable functions on such spaces. For functions f, g on represented sets, f is W-reducible to g if f can be computed by applying the function g at most once. Let RN be the Radon-Nikodym operator on the space under consideration and let EC be the non-computable operator mapping every enumeration of a set of natural numbers to its characteristic function. We prove that for every computable measurable space, RN is W-reducible to EC, and we construct a computable measurable space for which EC is W-reducible to RN.

show abstract

“…Instead of the machine in TTE that converted infinite strings to infinite strings, we use an oracle Turing machine (henceforth just "machine") to convert regular functions to regular functions: 3 The original theorem is stated with a condition slightly weaker than Lipschitz continuity. 4 Ko's formulation [12] already uses string functions instead of infinite strings of TTE, but it does not make full use of this extension.…”

Section: Computation Over Regular Functionsmentioning

confidence: 99%

Complexity Theory for Operators in Analysis

Kawamura

Cook

2012

ACM Trans. Comput. Theory

View full text Add to dashboard Cite

We propose a new framework for discussing computational complexity of problems involving uncountably many objects, such as real numbers, sets and functions, that can be represented only by approximation. The key idea is to use a certain class of string functions, which we call regular functions, as names representing these objects. These are more expressive than infinite sequences, which served as names in prior work that formulated complexity in more restricted settings. An important advantage of using regular functions is that we can define their size in the way inspired by highertype complexity theory. This enables us to talk about computation on regular functions whose time or space is bounded polynomially in the input size, giving rise to more general analogues of the classes P, NP, and PSPACE. We also define NP-and PSPACEcompleteness under suitable many-one reductions.Because our framework separates machine computation and semantics, it can be applied to problems on sets of interest in analysis once we specify a suitable representation (encoding). As prototype applications, we consider the complexity of functions (operators) on real numbers, real sets, and real functions. The latter two cannot be represented succinctly using existing approaches based on infinite sequences, so ours is the first treatment of them. As an interesting example, the task of numerical algorithms for solving the initial value problem of differential equations is naturally viewed as an operator taking real functions to real functions. As there was no complexity theory for operators, previous results could only state how complex the solution can be. We now reformulate them to show that the operator itself is polynomial-space complete. Categories and Subject Descriptors General TermsTheory This is a preprint.

show abstract

A Tutorial on Computable Analysis

Cited by 153 publications

References 78 publications

Computational complexity of solving polynomial differential equations over unbounded domains

Computational complexity of solving polynomial differential equations over unbounded domains

Computability of the Radon-Nikodym Derivative

Complexity Theory for Operators in Analysis

Contact Info

Product

Resources

About