Second-Order Linear-Time Computability with Applications to Computable Analysis

Kawamura, Akitoshi; Steinberg, Florian; Thies, Holger

doi:10.1007/978-3-030-14812-6_21

Cited by 2 publications

(3 citation statements)

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“…We note that Kawamura, Steinberg, and Thies [KST19] have defined linear-time computability for functions that are not necessarily Lipschitz. The restriction of their definition to the Lipschitz case is equivalent to our notion.…”

Section: Transducer Functionsmentioning

confidence: 99%

“…Clearly, every pointwise linear-time computable function is linear-time computable; our next goal is to explain why not every linear-time computable function f [D 2 ∩ [0, 1]] ⊆ Q is pointwise linear-time computable. As mentioned above, Kawamura, Steinberg, and Thies [KST19] have defined linear time computability for not necessarily Lipschitz functions. The idea is that the computation of f (x) with precision 2 −n uses only O(n) bits of x and takes only O(n) steps.…”

Section: Transducer Functionsmentioning

confidence: 99%

“…The idea is that the computation of f (x) with precision 2 −n uses only O(n) bits of x and takes only O(n) steps. This idea is formalised using secondorder polynomials; we omit the formal definition (see [KST19]). The crucial difference with pointwise linear time computability is that f (d) only has to be computed with a good enough precision, even for a dyadic rational d. So, for instance, if f (0) = 0.000 .…”

Section: Transducer Functionsmentioning

confidence: 99%

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Computable classifications of continuous, transducer, and regular functions

Franklin¹,

Hölzl²,

Melnikov³

et al. 2020

Preprint

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We develop a systematic algorithmic framework that unites global and local classification problems for functional separable spaces and apply it to attack classification problems concerning the Banach space C[0,1] of real-valued continuous functions on the unit interval. We prove that the classification problem for continuous (binary) regular functions among almost everywhere linear, pointwise linear-time Lipshitz functions is Σ 0 2 -complete. We show that a function f ∶ [0, 1] → R is (binary) transducer if and only if it is continuous regular; interestingly, this peculiar and nontrivial fact was overlooked by experts in automata theory. As one of many consequences, our Σ 0 2 -completeness result covers the class of transducer functions as well. Finally, we show that the Banach space C[0, 1] of real-valued continuous functions admits an arithmetical classification among separable Banach spaces. Our proofs combine methods of abstract computability theory, automata theory, and functional analysis. Contents1. Introduction 1 2. Background on regular functions 5 2.1. Regular functions 6 2.2. Transducer functions 10 2.3. Linear time computability 11 3. Proof of Theorem 1.1 12 3.1. The technical propositions 14 4. Proof of Theorem 1.2 20 5. Proof of Theorem 1.3 21 References 25Note that computability implies continuity. An adaptation of polynomial-time computability to continuous functions has previously been studied; the foundations can be found in [Ko91]. However, it is not necessarily clear that polynomial-time computability is the right notion of efficiency when it comes to infinite objects; see [BDKM19, Rou19] for a discussion. It is thus not surprising that other models of efficient computability for members of C[0, 1] have been tested as well; we will be focused on two such notions.

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Section: Transducer Functionsmentioning

confidence: 99%

Section: Transducer Functionsmentioning

confidence: 99%

Section: Transducer Functionsmentioning

confidence: 99%

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