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Computability
About: Computability is a research topic. Over the lifetime, 2829 publications have been published within this topic receiving 85162 citations.
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TL;DR: It turns out that any f(\colon R to R) computable in the first or second sense is still necessarily continuous whereas the third type of hypercomputation provides the required power to evaluate for instance the discontinuous Heaviside function.
Abstract: By the sometimes so-called 'Main Theorem' of Recursive Analysis, every computable real function is necessarily continuous. We wonder whether and which kinds of HYPERcomputation allow for the effective evaluation of also discontinuous f:R->R. More precisely the present work considers the following three super-Turing notions of real function computability:
* relativized computation; specifically given oracle access to the Halting Problem 0' or its jump 0'';
* encoding real input x and/or output y=f(x) in weaker ways also related to the Arithmetic Hierarchy;
* non-deterministic computation.
It turns out that any f:R->R computable in the first or second sense is still necessarily continuous whereas the third type of hypercomputation does provide the required power to evaluate for instance the discontinuous sign function.
39 citations
TL;DR: In this paper, a modified metric based on the Hilbert-Schmidt norm is proposed to define a rescaled version of the geometric measure of quantum discord, which does not suffer from the pathological dependence on state purity.
Abstract: We propose a modified metric based on the Hilbert-Schmidt norm and adopt it to define a rescaled version of the geometric measure of quantum discord. Such a measure is found not to suffer from the pathological dependence on state purity. Although the employed metric is still noncontractive under quantum operations, we show that the resulting indicator of quantum correlations is in agreement with other bona fide discord measures in a number of physical examples. We present a critical assessment of the requirements of reliability versus computability when approaching the task of quantifying, or measuring, general quantum correlations in a bipartite state.
39 citations
TL;DR: This paper gives exact recursion-theoretical characterization of the computational power of this kind of fuzzy Turing machines and shows that fuzzy languages accepted by these machines with a computable t-norm correspond exactly to the union Σ10 ∪ Π10 of recursively enumerable languages and their complements.
39 citations
01 Jan 2002
TL;DR: Some recent work of the author with Hirschfeldt, Laforte, and Nies concerning methods of calibrating randomness is discussed.
Abstract: We study computably enumerable reals (i.e. their left cut is computably enumerable) in terms of their spectra of representations and presentations. Then we study such objects in terms of algorithmic randomness, culminating in some recent work of the author with Hirschfeldt, Laforte, and Nies concerning methods of calibrating randomness. x
39 citations
01 Nov 1986
TL;DR: The first elements of such a theory are presented and constitute a significant step towards establishing a complete methodology that determines systolic array implementations for a very general class of coupled systems of recurrence equations.
Abstract: : Many important algorithms in signal and image processing, speech and pattern recognition of matrix computations consist of coupled systems of recurrence equations. Systolic arrays are regular networks of tightly coupled simple processors with limited storage that provide cost effective high throughput implementations of many such algorithms. While there are some mathematical techniques for finding efficient schedules for uniform recurrence equations, there is no general theory for more general systems of recurrence equations. The first elements of such a theory are presented in this paper and constitute a significant step towards establishing a complete methodology that determines systolic array implementations for a very general class of coupled systems of recurrence equations; these implementations exhibit provably optimal computation time while satisfying various user-specified constraints.
38 citations