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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: The Regular Post Embedding Problem extended with partial (co)directness is shown decidable and it is shown that combining directness and codirectness in Post Embedded problems leads to undecidability.
Abstract: The Regular Post Embedding Problem extended with partial (co)directness is shown decidable. This extends to universal and/or counting versions. It is also shown that combining directness and codirectness in Post Embedding problems leads to undecidability.
12 citations
TL;DR: This example by Nabutovsky and Weinberger is perhaps the first example of the use of c.e. sets to demonstrate specific mathematical or geometric complexity of a mathematical structure such as the depth and distribution of local minima.
Abstract: Topologists Nabutovsky and Weinberger discovered how to embed computably enumerable (c.e.) sets into the geometry of Riemannian metrics modulo diffeomorphisms. They used the complexity of the settling times of the c.e. sets to exhibit a much greater complexity of the depth and density of local minima for the diameter function than previously imagined. Their results depended on the existence of certain sequences of c.e. sets, constructed at their request by Csima and Soare, whose settling times had the necessary dominating properties. Although these computability results had been announced earlier, their proofs have been deferred until this paper. Computably enumerable sets have long been used to prove undecidability of mathematical problems such as the word problem for groups and Hilbert's Tenth Problem. However, this example by Nabutovsky and Weinberger is perhaps the first example of the use of c.e. sets to demonstrate specific mathematical or geometric complexity of a mathematical structure such as the depth and distribution of local minima. ?
12 citations
08 Jun 2005
TL;DR: This work shows that the convex hull, Voronoi diagram and Delaunay triangulation are Hausdorff and Lebesgue computable in Euclidean space given by its non-empty compact and convex subsets.
Abstract: We promote the concept of object directed computability in computational geometry in order to faithfully generalise the well-established theory of computability for real numbers and real functions. In object directed computability, a geometric object is computable if it is the effective limit of a sequence of finitary objects of the same type as the original object, thus allowing a quantitative measure for the approximation. The domain-theoretic model of computational geometry provides such an object directed theory, which supports two such quantitative measures, one based on the Hausdorff metric and one on the Lebesgue measure. With respect to a new data type for the Euclidean space, given by its non-empty compact and convex subsets, we show that the convex hull, Voronoi diagram and Delaunay triangulation are Hausdorff and Lebesgue computable.
12 citations
TL;DR: A new filled function is proposed that needs only one parameter and includes neither exponential terms nor logarithmic terms and is reasonable to expect that the proposed function has better computability than the previously reported ones.
12 citations
TL;DR: In a rather general setting, a number of basic theorems concerning computational complexity of derivability in adaptive logics are proved, and the corresponding completeness results are established in a very uniform way.
Abstract: In a rather general setting, we prove a number of basic theorems concerning computational complexity of derivability in adaptive logics. For that setting, the so-called standard format of adaptive logics is suitably adopted, and the corresponding completeness results are established in a very uniform way.
12 citations