Volume

Dagstuhl Seminar Proceedings, Volume 4351

Publication Details

  • published at: 2005-04-19
  • Publisher: Schloss Dagstuhl – Leibniz-Zentrum für Informatik

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DOI: 10.4230/DagSemProc.04351.1
04351 Abstracts Collection – Spatial Representation: Discrete vs. Continuous Computational Models

Authors: Ralph Kopperman, Prakash Panangaden, Michael B. Smyth, Dieter Spreen, and Julian Webster

Abstract
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.
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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}
}
Document
DOI: 10.4230/DagSemProc.04351.2
04351 Summary – Spatial Representation: Discrete vs. Continuous Computational Models

Authors: Ralph Kopperman, Prakash Panangaden, Michael B. Smyth, Dieter Spreen, and Julian Webster

Abstract
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.
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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}
}
Document
DOI: 10.4230/DagSemProc.04351.3
A Cartesian Closed Extension of the Category of Locales

Authors: Reinhold Heckmann

Abstract
We present a Cartesian closed category ELOC of equilocales, which contains the category LOC of locales as a reflective full subcategory. The embedding of LOC into ELOC preserves products and all exponentials of exponentiable locales.
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Reinhold Heckmann. A Cartesian Closed Extension of the Category of Locales. In Spatial Representation: Discrete vs. Continuous Computational Models. Dagstuhl Seminar Proceedings, Volume 4351, pp. 1-20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{heckmann:DagSemProc.04351.3,
  author =	{Heckmann, Reinhold},
  title =	{{A Cartesian Closed Extension of the Category of Locales}},
  booktitle =	{Spatial Representation: Discrete vs. Continuous Computational Models},
  pages =	{1--20},
  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.04351.3},
  URN =		{urn:nbn:de:0030-drops-1339},
  doi =		{10.4230/DagSemProc.04351.3},
  annote =	{Keywords: Locale , Cartesian closed category}
}
Document
DOI: 10.4230/DagSemProc.04351.4
A Category of Discrete Closure Spaces

Authors: John L. Pfaltz

Abstract
Discrete systems such as sets, monoids, groups are familiar categories. The internal strucutre of the latter two is defined by an algebraic operator. In this paper we describe the internal structure of the base set by a closure operator. We illustrate the role of such closure in convex geometries and partially ordered sets and thus suggestthe wide applicability of closure systems. Next we develop the ideas of closed and complete functions over closure spaces. These can be used to establish criteria for asserting that "the closure of a functional image under $f$ is equal to the functional image of the closure". Functions with these properties can be treated as categorical morphisms. Finally, the category "CSystem" of closure systems is shown to be cartesian closed.
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John L. Pfaltz. A Category of Discrete Closure Spaces. In Spatial Representation: Discrete vs. Continuous Computational Models. Dagstuhl Seminar Proceedings, Volume 4351, pp. 1-16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{pfaltz:DagSemProc.04351.4,
  author =	{Pfaltz, John L.},
  title =	{{A Category of Discrete Closure Spaces}},
  booktitle =	{Spatial Representation: Discrete vs. Continuous Computational Models},
  pages =	{1--16},
  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.04351.4},
  URN =		{urn:nbn:de:0030-drops-1253},
  doi =		{10.4230/DagSemProc.04351.4},
  annote =	{Keywords: Category , closure , antimatroid , function}
}
Document
DOI: 10.4230/DagSemProc.04351.5
A domain of spacetime intervals in general relativity

Authors: Keye Martin and Prakash Panangaden

Abstract
We prove that a globally hyperbolic spacetime with its causality relation is a bicontinuous poset whose interval topology is the manifold topology. This implies that from only a countable dense set of events and the causality relation, it is possible to reconstruct a globally hyperbolic spacetime in a purely order theoretic manner. The ultimate reason for this is that globally hyperbolic spacetimes belong to a category that is equivalent to a special category of domains called interval domains. We obtain a mathematical setting in which one can study causality independently of geometry and differentiable structure, and which also suggests that spacetime emanates from something discrete.
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Keye Martin and Prakash Panangaden. A domain of spacetime intervals in general relativity. In Spatial Representation: Discrete vs. Continuous Computational Models. Dagstuhl Seminar Proceedings, Volume 4351, pp. 1-28, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{martin_et_al:DagSemProc.04351.5,
  author =	{Martin, Keye and Panangaden, Prakash},
  title =	{{A domain of spacetime intervals in general relativity}},
  booktitle =	{Spatial Representation: Discrete vs. Continuous Computational Models},
  pages =	{1--28},
  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.04351.5},
  URN =		{urn:nbn:de:0030-drops-1350},
  doi =		{10.4230/DagSemProc.04351.5},
  annote =	{Keywords: Causality , spacetime , global hyperbolicity , interval domains , bicontinuous posets , spacetime topology}
}
Document
DOI: 10.4230/DagSemProc.04351.6
A geometry of information, I: Nerves, posets and differential forms

Authors: Jonathan Gratus and Timothy Porter

Abstract
The main theme of this workshop is 'Spatial Representation: Continuous vs. Discrete'. Spatial representation has two contrasting but interacting aspects (i) representation \emph{of} spaces' and (ii) representation \emph{by} spaces. In this paper we will examine two aspects that are common to both interpretations of the theme, namely nerve constructions and refinement. Representations change, data changes, spaces change. We will examine the possibility of a 'differential geometry' of spatial representations of both types, and in the sequel give an algebra of differential forms that has the potential to handle the dynamical aspect of such a geometry. We will discuss briefly a conjectured class of spaces, generalising the Cantor set which would seem ideal as a test-bed for the set of tools we are developing.
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Jonathan Gratus and Timothy Porter. A geometry of information, I: Nerves, posets and differential forms. In Spatial Representation: Discrete vs. Continuous Computational Models. Dagstuhl Seminar Proceedings, Volume 4351, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{gratus_et_al:DagSemProc.04351.6,
  author =	{Gratus, Jonathan and Porter, Timothy},
  title =	{{A geometry of information, I: Nerves, posets and differential forms}},
  booktitle =	{Spatial Representation: Discrete vs. Continuous Computational Models},
  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.04351.6},
  URN =		{urn:nbn:de:0030-drops-1268},
  doi =		{10.4230/DagSemProc.04351.6},
  annote =	{Keywords: Chu spaces , nerves , differential forms}
}
Document
DOI: 10.4230/DagSemProc.04351.7
A geometry of information, II: Sorkin models, and biextensional collapses

Authors: Jonathan Gratus and Timothy Porter

Abstract
In this second part of our contribution to the workshop, we look in more detail at the Sorkin model, its relationship to constructions in Chu space theory, and then compare it with the Nerve constructions given in the first part.
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Jonathan Gratus and Timothy Porter. A geometry of information, II: Sorkin models, and biextensional collapses. In Spatial Representation: Discrete vs. Continuous Computational Models. Dagstuhl Seminar Proceedings, Volume 4351, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{gratus_et_al:DagSemProc.04351.7,
  author =	{Gratus, Jonathan and Porter, Timothy},
  title =	{{A geometry of information, II: Sorkin models, and biextensional collapses}},
  booktitle =	{Spatial Representation: Discrete vs. Continuous Computational Models},
  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.04351.7},
  URN =		{urn:nbn:de:0030-drops-1271},
  doi =		{10.4230/DagSemProc.04351.7},
  annote =	{Keywords: Chu space , Sorkin model , Nerve}
}
Document
DOI: 10.4230/DagSemProc.04351.8
Auxiliary relations and sandwich theorems

Authors: Chris God, Achim Jung, Robin Knight, and Ralph Kopperman

Abstract
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.
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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}
}
Document
DOI: 10.4230/DagSemProc.04351.9
Compactness in apartness spaces?

Authors: Douglas Bridges, Hajime Ishihara, Peter Schuster, and Luminita S. Vita

Abstract
A major problem in the constructive theory of apartness spaces is that of finding a good notion of compactness. Such a notion should (i) reduce to ``complete plus totally bounded'' for uniform spaces and (ii) classically be equivalent to the usual Heine-Borel-Lebesgue property for the apartness topology. The constructive counterpart of the smallest uniform structure compatible with a given apartness, while not constructively a uniform structure, offers a possible solution to the compactness-definition problem. That counterpart turns out to be interesting in its own right, and reveals some additional properties of an apartness that may have uses elsewhere in the theory.
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Douglas Bridges, Hajime Ishihara, Peter Schuster, and Luminita S. Vita. Compactness in apartness spaces?. In Spatial Representation: Discrete vs. Continuous Computational Models. Dagstuhl Seminar Proceedings, Volume 4351, pp. 1-7, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{bridges_et_al:DagSemProc.04351.9,
  author =	{Bridges, Douglas and Ishihara, Hajime and Schuster, Peter and Vita, Luminita S.},
  title =	{{Compactness in apartness spaces?}},
  booktitle =	{Spatial Representation: Discrete vs. Continuous Computational Models},
  pages =	{1--7},
  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.04351.9},
  URN =		{urn:nbn:de:0030-drops-1175},
  doi =		{10.4230/DagSemProc.04351.9},
  annote =	{Keywords: Apartness , constructive , compact uniform space}
}
Document
DOI: 10.4230/DagSemProc.04351.10
Continued Radicals

Authors: Jamie Johnson and Tom Richmond

Abstract
A nested radical with terms $a_1, a_2, \ldots , a_N$ is an expression of form $\sqrt{a_N + \cdots + \sqrt{a_2 + \sqrt{a_1}}}$. The limit as $N$ approaches infinity of such an expression, if it exists, is called a continued radical. We consider the set of real numbers $S(M)$ representable as a continued radical whose terms $a_1, a_2, \ldots$ are all from a finite set $M$ of nonnegative real numbers. We give conditions on the set $M$ for $S(M)$ to be (a) an interval, and (b) homeomorphic to the Cantor set.
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Jamie Johnson and Tom Richmond. Continued Radicals. 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{johnson_et_al:DagSemProc.04351.10,
  author =	{Johnson, Jamie and Richmond, Tom},
  title =	{{Continued Radicals}},
  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.04351.10},
  URN =		{urn:nbn:de:0030-drops-1286},
  doi =		{10.4230/DagSemProc.04351.10},
  annote =	{Keywords: Continued radical}
}
Document
DOI: 10.4230/DagSemProc.04351.11
Continuous Semantics for Termination Proofs

Authors: Ulrich Berger

Abstract
We prove a general strong normalization theorem for higher type rewrite systems based on Tait's strong computability predicates and a strictly continuous domain-theoretic semantics. The theorem applies to extensions of Goedel's system $T$, but also to various forms of bar recursion for which termination was hitherto unknown.
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Ulrich Berger. Continuous Semantics for Termination Proofs. In Spatial Representation: Discrete vs. Continuous Computational Models. Dagstuhl Seminar Proceedings, Volume 4351, pp. 1-19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{berger:DagSemProc.04351.11,
  author =	{Berger, Ulrich},
  title =	{{Continuous Semantics for Termination Proofs}},
  booktitle =	{Spatial Representation: Discrete vs. Continuous Computational Models},
  pages =	{1--19},
  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.04351.11},
  URN =		{urn:nbn:de:0030-drops-1300},
  doi =		{10.4230/DagSemProc.04351.11},
  annote =	{Keywords: Higher-order term rewriting , termination , domain theory}
}
Document
DOI: 10.4230/DagSemProc.04351.12
Deadlocks and Dihomotopy in Mutual Exclusion Models

Authors: Martin Raussen

Abstract
Parallel processes in concurrency theory can be modelled in a geometric framework. A convenient model are the Higher Dimensional Automata of V. Pratt and E. Goubault with cubical complexes as their mathematical description. More abstract models are given by (locally) partially ordered topological spaces, the directed ($d$-spaces) of M.Grandis and the flows of P. Gaucher. All models invite to use or modify ideas from algebraic topology, notably homotopy. In specific semaphore models for mutual exclusion, we have developed methods and algorithms that can detect deadlocks and unsafe regions and give information about essentially different schedules using higher dimensional ``geometric'' representations of the state space and executions (directed paths) along it.
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Martin Raussen. Deadlocks and Dihomotopy in Mutual Exclusion Models. In Spatial Representation: Discrete vs. Continuous Computational Models. Dagstuhl Seminar Proceedings, Volume 4351, pp. 1-8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{raussen:DagSemProc.04351.12,
  author =	{Raussen, Martin},
  title =	{{Deadlocks and Dihomotopy in Mutual Exclusion Models}},
  booktitle =	{Spatial Representation: Discrete vs. Continuous Computational Models},
  pages =	{1--8},
  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.04351.12},
  URN =		{urn:nbn:de:0030-drops-1364},
  doi =		{10.4230/DagSemProc.04351.12},
  annote =	{Keywords: Mutual exclusion , deadlock detection , dihomotopy}
}
Document
DOI: 10.4230/DagSemProc.04351.13
Dihomotopy Classes of Dipaths in the Geometric Realization of a Cubical Set: from Discrete to Continuous and back again

Authors: Lisbeth Fajstrup

Abstract
The geometric models of concurrency - Dijkstra's PV-models and V. Pratt's Higher Dimensional Automata - rely on a translation of discrete or algebraic information to geometry. In both these cases, the translation is the geometric realisation of a semi cubical complex, which is then a locally partially ordered space, an lpo space. The aim is to use the algebraic topology machinery, suitably adapted to the fact that there is a preferred time direction. Then the results - for instance dihomotopy classes of dipaths, which model the number of inequivalent computations should be used on the discrete model and give the corresponding discrete objects. We prove that this is in fact the case for the models considered: Each dipath is dihomottopic to a combinatorial dipath and if two combinatorial dipaths are dihomotopic, then they are combinatorially equivalent. Moreover, the notions of dihomotopy (LF., E. Goubault, M. Raussen) and d-homotopy (M. Grandis) are proven to be equivalent for these models - hence the Van Kampen theorem is available for dihomotopy. Finally we give an idea of how many spaces have a local po-structure given by cubes. The answer is, that any cubicalized space has such a structure after at most one subdivision. In particular, all triangulable spaces have a cubical local po-structure.
Cite as

Lisbeth Fajstrup. Dihomotopy Classes of Dipaths in the Geometric Realization of a Cubical Set: from Discrete to Continuous and back again. In Spatial Representation: Discrete vs. Continuous Computational Models. Dagstuhl Seminar Proceedings, Volume 4351, pp. 1-3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{fajstrup:DagSemProc.04351.13,
  author =	{Fajstrup, Lisbeth},
  title =	{{Dihomotopy Classes of Dipaths in the Geometric Realization of a Cubical Set: from Discrete to Continuous and back again}},
  booktitle =	{Spatial Representation: Discrete vs. Continuous Computational Models},
  pages =	{1--3},
  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.04351.13},
  URN =		{urn:nbn:de:0030-drops-1328},
  doi =		{10.4230/DagSemProc.04351.13},
  annote =	{Keywords: Cubical Complex , Higher Dimensional Automaton , Ditopology}
}
Document
DOI: 10.4230/DagSemProc.04351.14
Discrete classical vs. continuous quantum data in abstract quantum mechanics

Authors: Samson Abramsky and Bob Coecke

Abstract
``Quantum'' stands for for the concepts (both operational and formal) which had to be added to classical physics in order to understand otherwise unexplainable observed phenomena such as the structure of the spectral lines in atomic spectra. While the basic part of classical mechanics deals with the (essentially) reversible dynamics, quantum required adding the notions of ``measurement'' and (possibly non-local) ``correlations'' to the discussion. Crucially, all this comes with a ``probabilistic calculus''. The corresponding mathematical formalism was considered to have reached maturity in [von Neumann 1932], but there are some manifest problems with that formalism: (i) While measurements are applied to physical systems, application of their formal counterpart (i.e. a self-adjoint linear operator) to the vector representing that state of the system in no way reflects how the state changes during the act of measurement. Analogously, the composite of two self-adjoint operators has no physical significance while in practice measurements can be effectuated sequentially. More generally, the formal types in von Neumann's formalism do not reflect the nature of the corresponding underlying concept at all! (ii) Part of the problem regarding the measurements discussed above is that in the von Neumann formalism there is no place for storage, manipulation and exchange of the classical data obtained from measurements. Protocols such as quantum teleportation involving these cannot be given a full formal description. (iii) The behavioral properties of quantum entanglement which for example enable continuous data exchange using only finitary communication are hidden in the formalism. In [Abramsky and Coecke 2004] we addressed all these problems, and in addition provided a purely categorical axiomatization of quantum mechanics. The concepts of the abstract quantum mechanics are formulated relative to a strongly compact closed category with biproducts (of which the category FdHilb of finite dimensional Hilbert spaces and linear maps is an example). Preparations, measurements, either destructive or not, classical data exchange are all morphisms in that category, and their types fully reflect their kinds. Correctness properties of standard quantum protocols can be abstractly proven. Surprisingly, in this seemingly purely qualitative setting even the quantitative Born rule arises, that is the rule which tells you how to calculate the probabilities. Indeed, each such category has as endomorphism Hom of the tensor unit an abelian semiring of `scalars', and a special subset of these scalars will play the role of weights: each scalar induces a natural transformation which propagates through physical processes, and when a `state' undergoes a `measurement', the composition of the corresponding morphisms gives rise to the weight. Hence the probabilistic weights live within the category of processes. J. von Neumann. Mathematische Grundlagen der Quantenmechanik. Springer-Verlag (1932). English translation in Mathematical Foundations of Quantum Mechanics. Princeton University Press (1955). S. Abramsky and B. Coecke. A categorical semantics of quantum protocols. In the proceedings of LiCS'04 (2004). An extended version is available at arXiv:quant-ph/0402130 A more reader friendly version entitled `Quantum information flow, concretely, abstractly' is at http://www.vub.ac.be/CLEA/Bob/Papers/QPL.pdf
Cite as

Samson Abramsky and Bob Coecke. Discrete classical vs. continuous quantum data in abstract quantum mechanics. In Spatial Representation: Discrete vs. Continuous Computational Models. Dagstuhl Seminar Proceedings, Volume 4351, pp. 1-21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{abramsky_et_al:DagSemProc.04351.14,
  author =	{Abramsky, Samson and Coecke, Bob},
  title =	{{Discrete classical vs. continuous quantum data in abstract quantum mechanics}},
  booktitle =	{Spatial Representation: Discrete vs. Continuous Computational Models},
  pages =	{1--21},
  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.04351.14},
  URN =		{urn:nbn:de:0030-drops-1316},
  doi =		{10.4230/DagSemProc.04351.14},
  annote =	{Keywords: Category theory , strong compact closure , quantum information-flow}
}
Document
DOI: 10.4230/DagSemProc.04351.15
Dyadic Subbases and Representations of Topological Spaces

Authors: Hideki Tsuiki

Abstract
We explain topological properties of the embedding-based approach to computability on topological spaces. With this approach, he considered a special kind of embedding of a topological space into Plotkin's $T^\omega$, which is the set of infinite sequences of $T = \{0,1,\bot \}$. We show that such an embedding can also be characterized by a dyadic subbase, which is a countable subbase $S = (S_0^0, S_0^1, S_1^0, S_1^1, \ldots)$ such that $S_n^j$ $(n = 0,1,2,\ldots; j = 0,1$ are regular open and $S_n^0$ and $S_n^1$ are exteriors of each other. We survey properties of dyadic subbases which are related to efficiency properties of the representation corresponding to the embedding.
Cite as

Hideki Tsuiki. Dyadic Subbases and Representations of Topological Spaces. In Spatial Representation: Discrete vs. Continuous Computational Models. Dagstuhl Seminar Proceedings, Volume 4351, pp. 1-8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{tsuiki:DagSemProc.04351.15,
  author =	{Tsuiki, Hideki},
  title =	{{Dyadic Subbases and Representations of Topological Spaces}},
  booktitle =	{Spatial Representation: Discrete vs. Continuous Computational Models},
  pages =	{1--8},
  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.04351.15},
  URN =		{urn:nbn:de:0030-drops-1376},
  doi =		{10.4230/DagSemProc.04351.15},
  annote =	{Keywords: Dyadic subbase , embedding , computation over topological spaces , Plotkin's \$T^\backslashomega\$}
}
Document
DOI: 10.4230/DagSemProc.04351.16
Integrating Topology and Geometry for Macro-Molecular Simulations

Authors: Edward L. F. Moore, Thomas J. Peters, David R. Ferguson, and Neil F. Stewart

Abstract
Emerging macro-molecular simulations, such as supercoiling of DNA and protein unfolding, have an opportunity to profit from two decades of experience with geometric models within computer-aided geometric design (CAGD). For CAGD, static models are often sufficient, while form and function are inextricably related in biochemistry, resulting in greater attention to critical topological characteristics of these dynamic models. The greater emphasis upon dynamic change in macro-molecular simulations imposes increased demands for faithful integration of topology and geometry, as well as much stricter requirements for computational efficiency. This article presents transitions from the CAGD domain to meet the greater fidelity and performance demands for macro-molecular simulations.
Cite as

Edward L. F. Moore, Thomas J. Peters, David R. Ferguson, and Neil F. Stewart. Integrating Topology and Geometry for Macro-Molecular Simulations. In Spatial Representation: Discrete vs. Continuous Computational Models. Dagstuhl Seminar Proceedings, Volume 4351, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{moore_et_al:DagSemProc.04351.16,
  author =	{Moore, Edward L. F. and Peters, Thomas J. and Ferguson, David R. and Stewart, Neil F.},
  title =	{{Integrating Topology and Geometry for Macro-Molecular Simulations}},
  booktitle =	{Spatial Representation: Discrete vs. Continuous Computational Models},
  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.04351.16},
  URN =		{urn:nbn:de:0030-drops-1240},
  doi =		{10.4230/DagSemProc.04351.16},
  annote =	{Keywords: Computational topology , spline , approximation}
}
Document
DOI: 10.4230/DagSemProc.04351.17
On Maximality of Compact Topologies

Authors: Martin Kovar

Abstract
Using some advanced properties of the de Groot dual and some generalization of the Hofmann-Mislove theorem, we solve in the positive the question of D. E. Cameron: Is every compact topology contained in some maximal compact topology?
Cite as

Martin Kovar. On Maximality of Compact Topologies. In Spatial Representation: Discrete vs. Continuous Computational Models. Dagstuhl Seminar Proceedings, Volume 4351, pp. 1-10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{kovar:DagSemProc.04351.17,
  author =	{Kovar, Martin},
  title =	{{On Maximality of Compact Topologies}},
  booktitle =	{Spatial Representation: Discrete vs. Continuous Computational Models},
  pages =	{1--10},
  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.04351.17},
  URN =		{urn:nbn:de:0030-drops-1182},
  doi =		{10.4230/DagSemProc.04351.17},
  annote =	{Keywords: de Groot dual , compact saturated set , wide Scott open filter , maximal compact topology}
}
Document
DOI: 10.4230/DagSemProc.04351.18
The Construction of Finer Compact Topologies

Authors: Hans-Peter A. Künzi and Dominic van der Zypen

Abstract
It is well known that each locally compact strongly sober topology is contained in a compact Hausdorff topology; just take the supremum of its topology with its dual topology. On the other hand, examples of compact topologies are known that do not have a finer compact Hausdorff topology. This led to the question (first explicitly formulated by D.E. Cameron) whether each compact topology is contained in a compact topology with respect to which all compact sets are closed. (For the obvious reason these spaces are called maximal compact in the literature.) While this major problem remains open, we present several partial solutions to the question in our talk. For instance we show that each compact topology is contained in a compact topology with respect to which convergent sequences have unique limits. In fact each compact topology is contained in a compact topology with respect to which countable compact sets are closed. Furthermore we note that each compact sober T_1-topology is contained in a maximal compact topology and that each sober compact T_1-topology which is locally compact or sequential is the infimum of a family of maximal compact topologies.
Cite as

Hans-Peter A. Künzi and Dominic van der Zypen. The Construction of Finer Compact Topologies. 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{kunzi_et_al:DagSemProc.04351.18,
  author =	{K\"{u}nzi, Hans-Peter A. and Zypen, Dominic van der},
  title =	{{The Construction of Finer Compact Topologies}},
  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.04351.18},
  URN =		{urn:nbn:de:0030-drops-1224},
  doi =		{10.4230/DagSemProc.04351.18},
  annote =	{Keywords: Maximal compact , KC-space , sober , US-space , locally compact , sequential , sequentially compact}
}
Document
DOI: 10.4230/DagSemProc.04351.19
The de Groot dual for general collections of sets

Authors: Martin Kovar

Abstract
A topology is de Groot dual of another topology, if it has a closed base consisting of all its compact saturated sets. Until 2001 it was an unsolved problem of J. Lawson and M. Mislove whether the sequence of iterated dualizations of a topological space is finite. In this paper we generalize the author's original construction to an arbitrary family instead of a topology. Among other results we prove that for any family $\C\subseteq 2^X$ it holds $\C^{dd}=\C^{dddd}$. We also show similar identities for some other similar and topology-related structures.
Cite as

Martin Kovar. The de Groot dual for general collections of sets. In Spatial Representation: Discrete vs. Continuous Computational Models. Dagstuhl Seminar Proceedings, Volume 4351, pp. 1-8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{kovar:DagSemProc.04351.19,
  author =	{Kovar, Martin},
  title =	{{The de Groot dual for general collections of sets}},
  booktitle =	{Spatial Representation: Discrete vs. Continuous Computational Models},
  pages =	{1--8},
  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.04351.19},
  URN =		{urn:nbn:de:0030-drops-1215},
  doi =		{10.4230/DagSemProc.04351.19},
  annote =	{Keywords: Saturated set , dual topology , compactness operator}
}
Document
DOI: 10.4230/DagSemProc.04351.20
The Hofmann-Mislove Theorem for general posets

Authors: Martin Kovar

Abstract
In this paper we attempt to find and investigate the most general class of posets which satisfy a properly generalized version of the Hofmann-Mislove theorem. For that purpose, we generalize and study some notions (like compactness, the Scott topology, Scott open filters, prime elements, the spectrum etc.), and adjust them for use in general posets. Then we characterize the posets satisfying the Hofmann-Mislove theorem by the relationship between the generalized Scott closed prime subsets and the generalized prime elements of the poset. The theory become classic for distributive lattices. Remark that the topologies induced on the generalized spectra in general need not be sober.
Cite as

Martin Kovar. The Hofmann-Mislove Theorem for general posets. In Spatial Representation: Discrete vs. Continuous Computational Models. Dagstuhl Seminar Proceedings, Volume 4351, pp. 1-16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{kovar:DagSemProc.04351.20,
  author =	{Kovar, Martin},
  title =	{{The Hofmann-Mislove Theorem for general posets}},
  booktitle =	{Spatial Representation: Discrete vs. Continuous Computational Models},
  pages =	{1--16},
  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.04351.20},
  URN =		{urn:nbn:de:0030-drops-1199},
  doi =		{10.4230/DagSemProc.04351.20},
  annote =	{Keywords: Posets , generalized Scott topology , Scott open filters , (filtered) compactness , saturated}
}
Document
DOI: 10.4230/DagSemProc.04351.21
The Hofmann-Mislove Theorem for general topological structures

Authors: Martin Kovar

Abstract
In this paper we prove a modification of Hofmann-Mislove theorem for a topological structure similar to the minusspaces of de Groot, in which the empty set "need not be open". This will extend, in a slightly relaxed form, the validity of the classical Hofmann-Mislove theorem also to some of those spaces, whose underlying topology need not be (quasi-) sober.
Cite as

Martin Kovar. The Hofmann-Mislove Theorem for general topological structures. In Spatial Representation: Discrete vs. Continuous Computational Models. Dagstuhl Seminar Proceedings, Volume 4351, pp. 1-9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{kovar:DagSemProc.04351.21,
  author =	{Kovar, Martin},
  title =	{{The Hofmann-Mislove Theorem for general topological structures}},
  booktitle =	{Spatial Representation: Discrete vs. Continuous Computational Models},
  pages =	{1--9},
  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.04351.21},
  URN =		{urn:nbn:de:0030-drops-1208},
  doi =		{10.4230/DagSemProc.04351.21},
  annote =	{Keywords: Compact saturated set , Scott open filter , (quasi-) sober space}
}
Document
DOI: 10.4230/DagSemProc.04351.22
What do partial metrics represent?

Authors: Ralph Kopperman, Steve Matthews, and Homeira Pajoohesh

Abstract
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.
Cite as

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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