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**Published in:** Dagstuhl Seminar Proceedings, Volume 6341, Computational Structures for Modelling Space, Time and Causality (2007)

From 20.08.06 to 25.08.06, the Dagstuhl Seminar 06341 ``Computational Structures for Modelling Space, Time and Causality'' was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available.

Ralph Kopperman, Prakash Panangaden, Michael B. Smyth, and Dieter Spreen. 06341 Abstracts Collection – Computational Structures for Modelling Space, Time and Causality. In Computational Structures for Modelling Space, Time and Causality. Dagstuhl Seminar Proceedings, Volume 6341, pp. 1-23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2007)

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@InProceedings{kopperman_et_al:DagSemProc.06341.1, author = {Kopperman, Ralph and Panangaden, Prakash and Smyth, Michael B. and Spreen, Dieter}, title = {{06341 Abstracts Collection – Computational Structures for Modelling Space, Time and Causality}}, booktitle = {Computational Structures for Modelling Space, Time and Causality}, pages = {1--23}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2007}, volume = {6341}, editor = {Ralph Kopperman and Prakash Panangaden and Michael B. Smyth and Dieter Spreen}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06341.1}, URN = {urn:nbn:de:0030-drops-9000}, doi = {10.4230/DagSemProc.06341.1}, annote = {Keywords: Borel hierarchy, causets, Chu spaces, computations in higher types, computable analysis, constructive topology, differential calculus, digital topology, dihomotopy, domain theory, domain representation, formal topology, higher dimensional automata, mereo\backslash-topology, partial metrics} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 4351, Spatial Representation: Discrete vs. Continuous Computational Models (2005)

From 22.08.04 to 27.08.04, the Dagstuhl Seminar 04351
``Spatial Representation: Discrete vs. Continuous Computational Models''
was held in the International Conference and Research Center (IBFI),
Schloss Dagstuhl.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available.

Ralph Kopperman, Prakash Panangaden, Michael B. Smyth, Dieter Spreen, and Julian Webster. 04351 Abstracts Collection – Spatial Representation: Discrete vs. Continuous Computational Models. In Spatial Representation: Discrete vs. Continuous Computational Models. Dagstuhl Seminar Proceedings, Volume 4351, pp. 1-24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{kopperman_et_al:DagSemProc.04351.1, author = {Kopperman, Ralph and Panangaden, Prakash and Smyth, Michael B. and Spreen, Dieter and Webster, Julian}, title = {{04351 Abstracts Collection – Spatial Representation: Discrete vs. Continuous Computational Models}}, booktitle = {Spatial Representation: Discrete vs. Continuous Computational Models}, pages = {1--24}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2005}, volume = {4351}, editor = {Ralph Kopperman and Michael B. Smyth and Dieter Spreen and Julian Webster}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.04351.1}, URN = {urn:nbn:de:0030-drops-1742}, doi = {10.4230/DagSemProc.04351.1}, annote = {Keywords: Domain theory , formal topology , constructive topology , domain representation, space-time , quantum gravity , inverse limit construction, matroid geometry , descriptive set theory , Borel hierarchy , Hausdorff difference hierarchy , Wadge degree partial metric , fractafold , region geometry , oriented projective geometry} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 4351, Spatial Representation: Discrete vs. Continuous Computational Models (2005)

A well-known topological theorem due to Kat\v etov states:
Suppose $(X,\tau)$ is a normal topological space, and let $f:X\to[0,1]$ be upper semicontinuous, $g:X\to[0,1]$ be lower semicontinuous, and $f\leq g$. Then there is a continuous $h:X\to[0,1]$ such that $f\leq h\leq g$.
We show a version of this theorem for many posets with auxiliary relations. In particular, if $P$ is a Scott domain and $f,g:P\to[0,1]$ are such that $f\leq g$, and $f$ is lower continuous and $g$ Scott continuous, then for some $h$, $f\leq h\leq g$ and $h$ is both Scott and lower continuous.
As a result, each Scott continuous function from $P$ to $[0,1]$, is the sup of the functions below it which are both Scott and lower continuous.

Chris God, Achim Jung, Robin Knight, and Ralph Kopperman. Auxiliary relations and sandwich theorems. In Spatial Representation: Discrete vs. Continuous Computational Models. Dagstuhl Seminar Proceedings, Volume 4351, pp. 1-4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{god_et_al:DagSemProc.04351.8, author = {God, Chris and Jung, Achim and Knight, Robin and Kopperman, Ralph}, title = {{Auxiliary relations and sandwich theorems}}, booktitle = {Spatial Representation: Discrete vs. Continuous Computational Models}, pages = {1--4}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2005}, volume = {4351}, editor = {Ralph Kopperman and Michael B. Smyth and Dieter Spreen and Julian Webster}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.04351.8}, URN = {urn:nbn:de:0030-drops-1348}, doi = {10.4230/DagSemProc.04351.8}, annote = {Keywords: Adjoint , auxiliary relation , continuous poset , pairwise completely regular (and pairwise normal) bitopological space , upper (lower) semicontinuous Urysohn relation} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 4351, Spatial Representation: Discrete vs. Continuous Computational Models (2005)

Partial metrics were introduced in 1992
as a metric to allow the distance of a point from
itself to be non zero. This notion of self distance, designed to extend
metrical concepts to Scott topologies as used
in computing, has little intuition for the mainstream Hausdorff topologist.
The talk will show that a partial metric over a set can be represented by a metric over that set with a so-called 'base point'.
Thus we establish that a partial metric is essentially a structure combining both a metric space and a skewed view of that space from the base point. From this we can deduce what it is that partial metrics are really all about.

Ralph Kopperman, Steve Matthews, and Homeira Pajoohesh. What do partial metrics represent?. In Spatial Representation: Discrete vs. Continuous Computational Models. Dagstuhl Seminar Proceedings, Volume 4351, pp. 1-4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{kopperman_et_al:DagSemProc.04351.22, author = {Kopperman, Ralph and Matthews, Steve and Pajoohesh, Homeira}, title = {{What do partial metrics represent?}}, booktitle = {Spatial Representation: Discrete vs. Continuous Computational Models}, pages = {1--4}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2005}, volume = {4351}, editor = {Ralph Kopperman and Michael B. Smyth and Dieter Spreen and Julian Webster}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.04351.22}, URN = {urn:nbn:de:0030-drops-1239}, doi = {10.4230/DagSemProc.04351.22}, annote = {Keywords: Metric , partial metric , base point} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 4351, Spatial Representation: Discrete vs. Continuous Computational Models (2005)

Topological notions and methods are used in various areas of the physical sciences and engineering, and therefore computer processing of topological data is important. Separate from this, but closely related, are computer science uses of topology: applications to programming language semantics and computing with exact real numbers are important examples. The seminar concentrated on an important approach, which is basic to all these applications, i.e. spatial representation.

Ralph Kopperman, Prakash Panangaden, Michael B. Smyth, Dieter Spreen, and Julian Webster. 04351 Summary – Spatial Representation: Discrete vs. Continuous Computational Models. In Spatial Representation: Discrete vs. Continuous Computational Models. Dagstuhl Seminar Proceedings, Volume 4351, pp. 1-5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{kopperman_et_al:DagSemProc.04351.2, author = {Kopperman, Ralph and Panangaden, Prakash and Smyth, Michael B. and Spreen, Dieter and Webster, Julian}, title = {{04351 Summary – Spatial Representation: Discrete vs. Continuous Computational Models}}, booktitle = {Spatial Representation: Discrete vs. Continuous Computational Models}, pages = {1--5}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2005}, volume = {4351}, editor = {Ralph Kopperman and Michael B. Smyth and Dieter Spreen and Julian Webster}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.04351.2}, URN = {urn:nbn:de:0030-drops-1710}, doi = {10.4230/DagSemProc.04351.2}, annote = {Keywords: Domain theory , formal topology , constructive topology , domain representation , space-time , quantum gravity , inverse limit construction , matroid geometry , descriptive set theory , Borel hierarchy , Hausdorff difference hierarchy , Wadge degree , partial metric , fractafold , region geometry} }

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**Published in:** Dagstuhl Seminar Reports. Dagstuhl Seminar Reports, Volume 1 (2021)

Ralph Kopperman, Michael B. Smyth, and Dieter Spreen. Mathematical Structures for Computable Topology and Geometry (Dagstuhl Seminar 02221). Dagstuhl Seminar Report 342, pp. 1-38, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2002)

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@TechReport{kopperman_et_al:DagSemRep.342, author = {Kopperman, Ralph and Smyth, Michael B. and Spreen, Dieter}, title = {{Mathematical Structures for Computable Topology and Geometry (Dagstuhl Seminar 02221)}}, pages = {1--38}, ISSN = {1619-0203}, year = {2002}, type = {Dagstuhl Seminar Report}, number = {342}, institution = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemRep.342}, URN = {urn:nbn:de:0030-drops-152238}, doi = {10.4230/DagSemRep.342}, }

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**Published in:** Dagstuhl Seminar Reports. Dagstuhl Seminar Reports, Volume 1 (2021)

Ralph Kopperman, Mike Smyth, and Dieter Spreen. Topology in Computer Science: Constructivity; Asymmetry and Partiality; Digitization (Dagstuhl Seminar 00231). Dagstuhl Seminar Report 276, pp. 1-27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2001)

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@TechReport{kopperman_et_al:DagSemRep.276, author = {Kopperman, Ralph and Smyth, Mike and Spreen, Dieter}, title = {{Topology in Computer Science: Constructivity; Asymmetry and Partiality; Digitization (Dagstuhl Seminar 00231)}}, pages = {1--27}, ISSN = {1619-0203}, year = {2001}, type = {Dagstuhl Seminar Report}, number = {276}, institution = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemRep.276}, URN = {urn:nbn:de:0030-drops-151600}, doi = {10.4230/DagSemRep.276}, }

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