The basic feasible functionals in computable analysis

Lambov, Branimir

doi:10.1016/j.jco.2006.06.005

Cited by 10 publications

(5 citation statements)

References 5 publications

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“…A proof that this second-order representation induces the established notions of computability and polynomial-time computability of reals and real functions (i.e. the notions from [Ko91] or [Wei00]) can be found in [Lam06]. It is computably equivalent to the Cauchy representation of the reals from Definition 1.2 if the standard enumeration of the dyadic numbers is chosen as dense sequence.…”

Section: Basic Notational Conventionsmentioning

confidence: 97%

Complexity theory for spaces of integrable functions

Steinberg¹

2017

Logical Methods in Computer Science ; Volume 13

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This paper investigates second-order representations in the sense of Kawamura and Cook for spaces of integrable functions that regularly show up in analysis. It builds upon prior work about the space of continuous functions on the unit interval: Kawamura and Cook introduced a representation inducing the right complexity classes and proved that it is the weakest second-order representation such that evaluation is polynomial-time computable.The first part of this paper provides a similar representation for the space of integrable functions on a bounded subset of Euclidean space: The weakest representation rendering integration over boxes is polynomial-time computable. In contrast to the representation of continuous functions, however, this representation turns out to be discontinuous with respect to both the norm and the weak topology.The second part modifies the representation to be continuous and generalizes it to Lp-spaces. The arising representations are proven to be computably equivalent to the standard representations of these spaces as metric spaces and to still render integration polynomial-time computable. The family is extended to cover Sobolev spaces on the unit interval, where less basic operations like differentiation and some Sobolev embeddings are shown to be polynomial-time computable.Finally as a further justification quantitative versions of the Arzelà-Ascoli and Fréchet-Kolmogorov Theorems are presented and used to argue that these representations fulfill a minimality condition. To provide tight bounds for the Fréchet-Kolmogorov Theorem, a form of exponential time computability of the norm of L p is proven.

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Section: Basic Notational Conventionsmentioning

confidence: 97%

Complexity theory for spaces of integrable functions

Steinberg¹

2017

Logical Methods in Computer Science ; Volume 13

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“…Moreover, we obtain a reasonable notion of polynomial-time computability of real functions f on R (not just [0, 1]) without additional work: (ρ R , ρ R )-FP turns out to be a reasonable class that coincides with the one by Hoover [7], who required that the 2 −m -approximation of the value f (t) should be delivered within time polynomial in m and log|t| (this equivalence has been essentially observed by Lambov [18]). Another equivalent definition appears in Takeuti [21], inspired by Pour-El's approach to computable analysis.…”

Section: Computation On Real Numbersmentioning

confidence: 99%

Complexity theory for operators in analysis

Kawamura

Cook

2010

Proceedings of the Forty-Second ACM Symposium on Theory of Computing

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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.

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“…We refer to the space Rc := (R, ξ Rc ), as the represented space of Cauchy reals. The representation ξ Rc is used throughout literature with great confidence that it induces the right notion of complexity for real numbers and there are many results supporting this: The functions that have a polytime computable realiser are exactly those that are polytime computable in the sense of Ko [Ko91] as proved by Lambov [Lam06]. It is well known that Ko's notion can be reproduced in Weihrauch's type two theory of effectivity [Wei00].…”

Section: Notations and Complexity On The Realsmentioning

confidence: 99%

“…It consists of a rigorous specification of input and output, so that it precisely describes the steps that have to be taken to obtain the desired result to a given accuracy. Software packages based on computable analysis include iRRAM [Mül], Ariadne [BBC + 08], AERN [Kon], and RealLib [Lam06].…”

Section: Introductionmentioning

confidence: 99%

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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The basic feasible functionals in computable analysis

Cited by 10 publications

References 5 publications

Complexity theory for spaces of integrable functions

Complexity theory for spaces of integrable functions

Complexity theory for operators in analysis

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

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