scispace - formally typeset
Search or ask a question
Topic

Computability

About: Computability is a research topic. Over the lifetime, 2829 publications have been published within this topic receiving 85162 citations.


Papers
More filters
Journal ArticleDOI
TL;DR: In this article, the Henkin quantifier is used to prove the relative computability of real relations in the context of the Weierstrass Theorem (i.e., every real function f :[0, 1]→ℝ is computable relative to some oracle).
Abstract: A type-2 computable real function is necessarily continuous; and this remains true for relative, i.e. oracle-based, computations. Conversely, by the Weierstrass Approximation Theorem, every continuous f :[0,1]→ℝ is computable relative to some oracle. In their search for a similar topological characterization of relatively computable multi- valued functions f :[0,1]⇒ℝ (aka relations), Brattka and Hertling (1994) have considered two notions: weak continuity (which is weaker than relative computability) and strong continuity (which is stronger than relative computability). Observing that uniform continuity plays a crucial role in the Weierstrass Theorem, we propose and compare several notions of uniform continuity for relations. Here, due to the additional quantification over values y ∈ f ( x ), new ways arise of (linearly) ordering quantifiers — yet none turns out as satisfactory. We are thus led to a concept of uniform continuity based on the Henkin quantifier ; and prove it necessary for relative computability of compact real relations. In fact iterating this condition yields a strict hierarchy of notions each necessary — and the ω-th level also sufficient — for relative computability.

30 citations

Journal ArticleDOI
TL;DR: It is shown that the forward secret key Capacity with a noisy public channel is not computable and consequently, there is no algorithm that can simulate or compute the secret key capacity; even if there are no limitations on computational complexity and computing power.
Abstract: Secret key generation refers to the problem of generating a common secret key without revealing any information about it to an eavesdropper. All users observe correlated components of a common source and can further use a noisy public channel for discussion, which is open to eavesdroppers. A related problem is that of secure authentication, which has structural similarities and connections to the first problem. For authentication, users need to be enrolled and securely authenticated while minimizing the privacy leakage rate. This paper studies the algorithmic computability of the forward secret key capacity and the secure authentication capacity. For the algorithmic computability, the concept of a Turing machine is used as it provides fundamental performance limits for today's digital computers. In this paper, it is shown that the forward secret key capacity with a noisy public channel is not computable and consequently, there is no algorithm that can simulate or compute the secret key capacity; even if there are no limitations on computational complexity and computing power. On the other hand, if the public channel is noiseless so that there are no rate constraints on the public communication, the secret key capacity is a computable continuous function, which is the strongest form of computability. A similar behavior is subsequently observed for the authentication problem: The secure authentication capacity under storage rate and privacy leakage rate constraints is not computable, while the case without privacy leakage rate constraints is computable.

30 citations

Posted Content
TL;DR: In Computable Economics an eclectic approach is adopted where the main criterion is numerical content for economic entities and both the computable and the constructive traditions are freely and indiscriminately invoked in the formaliza- tion of economic entities.
Abstract: Computability theory came into being as a result of Hilbert's attempts to meet Brouwer's challenges, from an intuitionistc and constructive standpoint, to formalism as a foundation for mathematical practice. Viewed this way, constructive mathematics should be one vision of computability theory. However, there are fundamental differences between computability theory and constructive mathematics: the Church-Turing thesis is a disciplining criterion in the former and not in the latter; and classical logic - particularly, the law of the excluded middle - is not accepted in the latter but freely invoked in the former, especially in proving universal negative propositions. In Computable Economic an eclectic approach is adopted where the main criterion is numerical content for economic entities. In this sense both the computable and the constructive traditions are freely and indiscriminately invoked and utilised in the formalization of economic entities. Some of the mathematical methods and concepts of computable economics are surveyed in a pedagogical mode. The context is that of a digital economy embedded in an information society.

30 citations

Book ChapterDOI
01 Jan 1973

30 citations

Journal ArticleDOI
TL;DR: In this paper, an automatic method for the computer generation of random variables with a characteristic function satisfying certain regularity conditions was developed. But the method is based upon a generalization of the rejection method and exploits the duality between densities and their Fourier transforms.
Abstract: An automatic method is developed for the computer generation of random variables with a characteristic function satisfying certain regularity conditions. The method is based upon a generalization of the rejection method and exploits the duality between densities and their Fourier transforms. It takes finite time almost surely, does not use approximations or inversions, and does not require explicit knowledge of the characteristic function (only its computability is assumed—hence the adjective “automatic”). As a by-product, we show how the sum of n independent random variables with common density f can be generated in time essentially independent of n, at least when its characteristic function satisfies the above mentioned regularity conditions.

30 citations


Network Information
Related Topics (5)
Finite-state machine
15.1K papers, 292.9K citations
86% related
Mathematical proof
13.8K papers, 374.4K citations
86% related
Model checking
16.9K papers, 451.6K citations
85% related
Time complexity
36K papers, 879.5K citations
85% related
Concurrency
13K papers, 347.1K citations
85% related
Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202344
2022119
202189
202098
2019111
201897