Computability and analysis: the legacy of Alan Turing

Avigad, Jeremy; Brattka, Vasco

doi:10.1017/cbo9781107338579.002

Cited by 78 publications

(26 citation statements)

References 192 publications

(172 reference statements)

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“…Matematik Antik Mısır ve Babil'den bu yana kendi tarihinin büyük bölümünde algoritmik bir doğaya sahipti. Al-Khwarizmi'nin (El Harezmi) 9. yüzyıldaki çalışmalarından Cardano ve Lagrange'a cebir de temelde denklemlerin çözülmesi için geliştirilmiş algoritmalar yolıuyla ilerlemiştir (Avigad & Brattka, 2012). Bu noktada algoritmik matematik kavramını kendi karşıtı olarak tanımlanan diyalektik matematik kavramı ile birlikte ele almak uygun olur.…”

Section: Algoritmik Ve Hesaplanabilir Matematik Ve İktisada Olası Yansımalarıunclassified

İktisadın Formalist Aksiyomatik Bir Disipline Dönüşüm Sürecinde Hilbert – Bourbaki Yerine Poincare – Brouwer Yaklaşımı Benimsenebilir miydi?

Öğüt

2019

Yildiz Social Science Review

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To end the debate on the foundations of mathematics Hilbert attempted to axiomatize mathematics with formalist method. In the same period, physicists who reached to the limits of experimental progress applied the same method to quantum physics. In 1930's Wald and von Neumann in 1950's Debreu implemented similar techniques on economics through the General Equilibrium Theory. In this process both of the physics and economics began to use high abstraction level mathematical methods like topology and group theory instead of differential equations.

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Section: Algoritmik Ve Hesaplanabilir Matematik Ve İktisada Olası Yansımalarıunclassified

İktisadın Formalist Aksiyomatik Bir Disipline Dönüşüm Sürecinde Hilbert – Bourbaki Yerine Poincare – Brouwer Yaklaşımı Benimsenebilir miydi?

Öğüt

2019

Yildiz Social Science Review

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“…They are further equivalent to the von Neumann-architecture without hardware limitations and the theory of recursive functions, cf. also [13][14][15][16][17]. Accordingly, Turing machines provide fundamental performance limits for today's digital computers.…”

Section: Introductionmentioning

confidence: 99%

Communication Over Block Fading Channels – An Algorithmic Perspective On Optimal Transmission Schemes

Boche

Schaefer

Poor

2021

ICASSP 2021 - 2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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Wireless channels are considered that change over time but remain constant for a certain (coherence) period. This behavior is perfectly captured by block fading channels and affects the performance of the corresponding wireless communication systems. Desired closedform characterizations of optimal transmission schemes remain unknown in many cases. This paper approaches this issue from a fundamental, algorithmic point of view by studying whether or not it is in principle possible to construct or find such optimal transmission schemes algorithmically (without putting any constraints on the computational complexity of such algorithms). To this end, the concept of averaged channels is considered as a model for block fading and it is shown that, although the averaged channel itself is computable, the corresponding capacity need not be computable, i.e., there exists no (universal) algorithm that takes the channel as an input and computes the corresponding capacity expression. Subsequently, examples of block fading channels are presented for which it is even impossible to find an algorithm that computes for every blocklength the corresponding optimal transmission scheme.

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“…In his paper Borel claims that a real x is 'computable' if, given any natural number n, we can obtain a rational q within 1 n of x [10]. 3 What Borel means by 'computable' is uncertain, particularly since it would be another 20 years before any formal notion of computation emerged. We also hesitate to speculate what Borel intended when he spoke of 'obtaining' a rational close to x.…”

Section: Defining a Computable Realmentioning

confidence: 99%

Aspects of Computable Analysis

Porter¹

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<p>Computable analysis has been well studied ever since Turing famously formalised the computable reals and computable real-valued function in 1936. However, analysis is a broad subject, and there still exist areas that have yet to be explored. For instance, Sierpinski proved that every real-valued function ƒ : ℝ → ℝ is the limit of a sequence of Darboux functions. This is an intriguing result, and the complexity of these sequences has been largely unstudied. Similarly, the Blaschke Selection Theorem, closely related to the Bolzano-Weierstrass Theorem, has great practical importance, but has not been considered from a computability theoretic perspective. The two main contributions of this thesis are: to provide some new, simple proofs of fundamental classical results (highlighting the role of ∏0/1 classes), and to use tools from effective topology to analyse the Darboux property, particularly a result by Sierpinski, and the Blaschke Selection Theorem. This thesis focuses on classical computable analysis. It does not make use of effective measure theory.</p>

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Computability and analysis: the legacy of Alan Turing

Cited by 78 publications

References 192 publications

İktisadın Formalist Aksiyomatik Bir Disipline Dönüşüm Sürecinde Hilbert – Bourbaki Yerine Poincare – Brouwer Yaklaşımı Benimsenebilir miydi?

İktisadın Formalist Aksiyomatik Bir Disipline Dönüşüm Sürecinde Hilbert – Bourbaki Yerine Poincare – Brouwer Yaklaşımı Benimsenebilir miydi?

Communication Over Block Fading Channels – An Algorithmic Perspective On Optimal Transmission Schemes

Aspects of Computable Analysis

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