# Computable Operations on Compact Subsets of Metric Spaces with Applications to Fréchet Distance and Shape Optimization

@article{Park2017ComputableOO, title={Computable Operations on Compact Subsets of Metric Spaces with Applications to Fr{\'e}chet Distance and Shape Optimization}, author={Chan-Bong Park and Ji-won Park and Sewon Park and Dongseong Seon and Martin Ziegler}, journal={ArXiv}, year={2017}, volume={abs/1701.08402} }

We extend the Theory of Computation on real numbers, continuous real functions, and bounded closed Euclidean subsets, to compact metric spaces $(X,d)$: thereby generically including computational and optimization problems over higher types, such as the compact 'hyper' spaces of (i) nonempty closed subsets of $X$ w.r.t. Hausdorff metric, and of (ii) equicontinuous functions on $X$. The thus obtained Cartesian closure is shown to exhibit the same structural properties as in the Euclidean case…

## 5 Citations

Effective local compactness and the hyperspace of located sets

- MathematicsArXiv
- 2019

It is shown that effective local compactness suffices to ensure that the hyperspace of closed-and-overt sets (aka located sets, aka closed sets with full information) is computably compact and computably metrizable.

On the computability of the Fr\'echet distance of surfaces in the bit-model of real computation.

- Mathematics, Computer Science
- 2017

We show that the Fr\'echet distance of two-dimensional parametrised surfaces in a metric space is computable in the bit-model of real computation. An analogous result in the real RAM model for…

The Fréchet distance of surfaces is computable

- Mathematics, Computer ScienceArXiv
- 2017

We show that the Fr\'echet distance of two-dimensional parametrised surfaces in a metric space is computable. This settles a long-standing open question in computational geometry.

Continuous Team Semantics

- Computer ScienceTAMC
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It is shown how to define approximate versions of the usual independence/dependence atoms for restricted classes of formulae, and it is shown that one can assume w.l.o.g. that teams are closed sets to import techniques from computable analysis to study the complexity of formula satisfaction and model checking.

Theory and Applications of Models of Computation

- EngineeringLecture Notes in Computer Science
- 2019

This paper looks at ways of scheduling workloads over the multiplexed batteries to maximize the overall efficiency and considers two ways to model the efficiency and give efficient solutions to the same.

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