3 Search Results for "Porter, Benjamin"

Document
DOI: 10.4230/LIPIcs.STACS.2022.57
Optimal Oracles for Point-To-Set Principles

Authors: D. M. Stull

Published in: LIPIcs, Volume 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)

Abstract
The point-to-set principle [Lutz and Lutz, 2018] characterizes the Hausdorff dimension of a subset E ⊆ ℝⁿ by the effective (or algorithmic) dimension of its individual points. This characterization has been used to prove several results in classical, i.e., without any computability requirements, analysis. Recent work has shown that algorithmic techniques can be fruitfully applied to Marstrand’s projection theorem, a fundamental result in fractal geometry. In this paper, we introduce an extension of point-to-set principle - the notion of optimal oracles for subsets E ⊆ ℝⁿ. One of the primary motivations of this definition is that, if E has optimal oracles, then the conclusion of Marstrand’s projection theorem holds for E. We show that every analytic set has optimal oracles. We also prove that if the Hausdorff and packing dimensions of E agree, then E has optimal oracles. Moreover, we show that the existence of sufficiently nice outer measures on E implies the existence of optimal Hausdorff oracles. In particular, the existence of exact gauge functions for a set E is sufficient for the existence of optimal Hausdorff oracles, and is therefore sufficient for Marstrand’s theorem. Thus, the existence of optimal oracles extends the currently known sufficient conditions for Marstrand’s theorem to hold. Under certain assumptions, every set has optimal oracles. However, assuming the axiom of choice and the continuum hypothesis, we construct sets which do not have optimal oracles. This construction naturally leads to a generalization of Davies' theorem on projections.
Cite as

D. M. Stull. Optimal Oracles for Point-To-Set Principles. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 57:1-57:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{stull:LIPIcs.STACS.2022.57,
  author =	{Stull, D. M.},
  title =	{{Optimal Oracles for Point-To-Set Principles}},
  booktitle =	{39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
  pages =	{57:1--57:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-222-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{219},
  editor =	{Berenbrink, Petra and Monmege, Benjamin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.57},
  URN =		{urn:nbn:de:0030-drops-158675},
  doi =		{10.4230/LIPIcs.STACS.2022.57},
  annote =	{Keywords: Algorithmic randomness, Kolmogorov complexity, geometric measure theory}
}
Document
DOI: 10.4230/DagSemProc.09291.26
Simulating Morphogenesis

Authors: Benjamin Porter

Published in: Dagstuhl Seminar Proceedings, Volume 9291, Computational Creativity: An Interdisciplinary Approach (2009)

Abstract
I am a graduate student at Monash University. My research is in the domain of computer graphics, where I am attempting to grow organic three dimensional forms using models inspired by real biological development. The presented animation demonstrated the simulated development of a virtual organism reminiscent of a starfish or sea urchin. As in a real developing organism the cells of the virtual organism divide and communicate, adding more complexity to the initially simple form. The behaviour of the cells result in the development of limbs, giving rise to the starfish-like appearance of the final organisms.
Cite as

Benjamin Porter. Simulating Morphogenesis. In Computational Creativity: An Interdisciplinary Approach. Dagstuhl Seminar Proceedings, Volume 9291, p. 1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{porter:DagSemProc.09291.26,
  author =	{Porter, Benjamin},
  title =	{{Simulating Morphogenesis}},
  booktitle =	{Computational Creativity: An Interdisciplinary Approach},
  pages =	{1--1},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2009},
  volume =	{9291},
  editor =	{Margaret Boden and Mark D'Inverno and Jon McCormack},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.09291.26},
  URN =		{urn:nbn:de:0030-drops-22106},
  doi =		{10.4230/DagSemProc.09291.26},
  annote =	{Keywords: Morphogenesis 3D modelling}
}
Document
DOI: 10.4230/DagSemProc.09291.28
Stimulating creative flow through computational feedback

Authors: Daniel Jones, Oliver Bown, Jon McCormack, Francois Pachet, Michael Young, Rodney Berry, Iris Asaf, and Benjamin Porter

Published in: Dagstuhl Seminar Proceedings, Volume 9291, Computational Creativity: An Interdisciplinary Approach (2009)

Abstract
This report summarises the discussion and experimental work produced by the authors at the 2009 symposium Computational Creativity: An Interdisciplinary Approach, Dagstuhl Leibniz-Zentrum für Informatik. It outlines the motivation for using computational techniques to stimulate human creativity, briefly summarising its historical context and predecessors, and describes two software studies produced by the group as base-line exemplars of these ideas.
Cite as

Daniel Jones, Oliver Bown, Jon McCormack, Francois Pachet, Michael Young, Rodney Berry, Iris Asaf, and Benjamin Porter. Stimulating creative flow through computational feedback. In Computational Creativity: An Interdisciplinary Approach. Dagstuhl Seminar Proceedings, Volume 9291, pp. 1-10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{jones_et_al:DagSemProc.09291.28,
  author =	{Jones, Daniel and Bown, Oliver and McCormack, Jon and Pachet, Francois and Young, Michael and Berry, Rodney and Asaf, Iris and Porter, Benjamin},
  title =	{{Stimulating creative flow through computational feedback}},
  booktitle =	{Computational Creativity: An Interdisciplinary Approach},
  pages =	{1--10},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2009},
  volume =	{9291},
  editor =	{Margaret Boden and Mark D'Inverno and Jon McCormack},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.09291.28},
  URN =		{urn:nbn:de:0030-drops-22232},
  doi =		{10.4230/DagSemProc.09291.28},
  annote =	{Keywords: Computational creativity}
}
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