30 Search Results for "Kopperman, Ralph"

Document
DOI: 10.4230/DagSemProc.06341.1
06341 Abstracts Collection – Computational Structures for Modelling Space, Time and Causality

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

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

Abstract
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.
Cite as

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-dev.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}
}
Document
DOI: 10.4230/DagSemProc.06341.2
A convenient category of domains

Authors: Ingo Battenfeld, Matthias Schröder, and Alex Simpson

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

Abstract
We motivate and define a category of "topological domains", whose objects are certain topological spaces, generalising the usual $omega$-continuous dcppos of domain theory. Our category supports all the standard constructions of domain theory, including the solution of recursive domain equations. It also supports the construction of free algebras for (in)equational theories, provides a model of parametric polymorphism, and can be used as the basis for a theory of computability. This answers a question of Gordon Plotkin, who asked whether it was possible to construct a category of domains combining such properties.
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Ingo Battenfeld, Matthias Schröder, and Alex Simpson. A convenient category of domains. In Computational Structures for Modelling Space, Time and Causality. Dagstuhl Seminar Proceedings, Volume 6341, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2007)

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@InProceedings{battenfeld_et_al:DagSemProc.06341.2,
  author =	{Battenfeld, Ingo and Schr\"{o}der, Matthias and Simpson, Alex},
  title =	{{A convenient category of domains}},
  booktitle =	{Computational Structures for Modelling Space, Time and Causality},
  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-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.06341.2},
  URN =		{urn:nbn:de:0030-drops-8945},
  doi =		{10.4230/DagSemProc.06341.2},
  annote =	{Keywords: Domain theory, topology of datatypes}
}
Document
DOI: 10.4230/DagSemProc.06341.3
Closure and Causality

Authors: John L. Pfaltz

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

Abstract
We present a model of causality which is defined by the intersection of two distinct closure systems, ${cal I}$ and ${cal T}$. Next we present empirical evidence to demonstrate that this model has practical validity by examining computer trace data to reveal causal dependencies between individual code modules. From over 498,000 events in the transaction manager of an open source system we tease out 66 apparent causal dependencies. Finally, we explore how to mathematically model the transformation of a causal topology resulting from unforlding events.
Cite as

John L. Pfaltz. Closure and Causality. In Computational Structures for Modelling Space, Time and Causality. Dagstuhl Seminar Proceedings, Volume 6341, pp. 1-13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2007)

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@InProceedings{pfaltz:DagSemProc.06341.3,
  author =	{Pfaltz, John L.},
  title =	{{Closure and Causality}},
  booktitle =	{Computational Structures for Modelling Space, Time and Causality},
  pages =	{1--13},
  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-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.06341.3},
  URN =		{urn:nbn:de:0030-drops-8978},
  doi =		{10.4230/DagSemProc.06341.3},
  annote =	{Keywords: Closure, causality, antimatroid, temporal, software engineering}
}
Document
DOI: 10.4230/DagSemProc.06341.4
Elementary Differential Calculus on Discrete, Continuous and Hybrid Spaces

Authors: Howard Blair

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

Abstract
We unify a variety of continuous and discrete types of change of state phenomena using a scheme whose instances are differential calculi on structures that embrace both topological spaces and graphs as well as hybrid ramifications of such structures. These calculi include the elementary differential calculus on real and complex vector spaces. One class of spaces that has been increasingly receiving attention in recent years is the class of convergence spaces [cf. Heckmann, R., TCS v.305, (159--186)(2003)]. The class of convergence spaces together with the continuous functions among convergence spaces forms a Cartesian-closed category CONV that contains as full subcategories both the category TOP of topological spaces and an embedding of the category DIGRAPH of reflexive directed graphs. (More can importantly be said about these embeddings.) These properties of CONV serve to assure that we can construct continuous products of continuous functions, and that there is always at least one convergence structure available in function spaces with respect to which the operations of function application and composition are continuous. The containment of TOP and DIGRAPH in CONV allows to combine arbitrary topological spaces with discrete structures (as represented by digraphs) to obtain hybrid structures, which generally are not topological spaces. We give a differential calculus scheme in CONV that addresses three issues in particular. 1. For convergence spaces $X$ and $Y$ and function $f: X longrightarrow Y$, the scheme gives necessary and sufficient conditions for a candidate differential $df: X longrightarrow Y$ to be a (not necessarily "the", depending on the spaces involved) differential of $f$ at $x_0$. 2. The chain rule holds and the differential relation between functions distributes over Cartesian products: e.g. if $Df$, $Dg$ and $Dh$ are, respectively, differentials of $f$ at $(g(x_0),h(x_0))$ and $g$ and $h$ at $x_0$, then $Df circ (Dg times Dh)$ is a differential of $f circ (g times h)$ at $x_0$. 3. When specialized to real and complex vector spaces, the scheme is in agreement with ordinary elementary differential calculus on these spaces. Moreover, with two additional constraints having to do with self-differentiation of differentials and translation invariance (for example, a linear operator on, say, $C^2$, is its own differential everywhere) there is a (unique) maximum differential calculus in CONV.
Cite as

Howard Blair. Elementary Differential Calculus on Discrete, Continuous and Hybrid Spaces. In Computational Structures for Modelling Space, Time and Causality. Dagstuhl Seminar Proceedings, Volume 6341, pp. 1-2, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2007)

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@InProceedings{blair:DagSemProc.06341.4,
  author =	{Blair, Howard},
  title =	{{Elementary Differential Calculus on Discrete, Continuous and Hybrid Spaces}},
  booktitle =	{Computational Structures for Modelling Space, Time and Causality},
  pages =	{1--2},
  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-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.06341.4},
  URN =		{urn:nbn:de:0030-drops-8956},
  doi =		{10.4230/DagSemProc.06341.4},
  annote =	{Keywords: Hybrid space, convergence space, differential, calculus, chain rule, hybrid dynamical system, discrete structure, topological space}
}
Document
DOI: 10.4230/DagSemProc.06341.5
Enriched categories and models for spaces of dipaths

Authors: Timothy Porter

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

Abstract
Partially ordered sets, causets, partially ordered spaces and their local counterparts are now often used to model systems in computer science and theoretical physics. The order models `time' which is often not globally given. In this setting directed paths are important objects of study as they correspond to an evolving state or particle traversing the system. Many physical problems rely on the analysis of models of the path space of space-time manifold. Many problems in concurrent systems use `spaces' of paths in a system. Here we review some ideas from algebraic topology that suggest how to model the dipath space of a pospace by a simplicially enriched category.
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Timothy Porter. Enriched categories and models for spaces of dipaths. In Computational Structures for Modelling Space, Time and Causality. Dagstuhl Seminar Proceedings, Volume 6341, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2007)

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@InProceedings{porter:DagSemProc.06341.5,
  author =	{Porter, Timothy},
  title =	{{Enriched categories and models for spaces of dipaths}},
  booktitle =	{Computational Structures for Modelling Space, Time and Causality},
  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-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.06341.5},
  URN =		{urn:nbn:de:0030-drops-8989},
  doi =		{10.4230/DagSemProc.06341.5},
  annote =	{Keywords: Enriched category}
}
Document
DOI: 10.4230/DagSemProc.06341.6
Instant topological relationships hidden in the reality

Authors: Martin Maria Kovár

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

Abstract
In most applications of general topology, topology usually is not the first, primary structure, but the information which finally leads to the construction of the certain, for some purpose required topology, is filtered by more or less thick filter of the other mathematical structures. This fact has two main consequences: (1) Most important applied constructions may be done in the primary structure, bypassing the topology. (2) Some topologically important information from the reality may be lost (filtered out by the other, front-end mathematical structures). Thus some natural and direct connection between topology and the reality could be useful. In this contribution we will discuss a pointless topological structure which directly reflects relationship between various locations which are glued together by possible presence of a physical object or a virtual ``observer".
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Martin Maria Kovár. Instant topological relationships hidden in the reality. In Computational Structures for Modelling Space, Time and Causality. Dagstuhl Seminar Proceedings, Volume 6341, pp. 1-2, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2007)

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@InProceedings{kovar:DagSemProc.06341.6,
  author =	{Kov\'{a}r, Martin Maria},
  title =	{{Instant topological relationships hidden in the reality}},
  booktitle =	{Computational Structures for Modelling Space, Time and Causality},
  pages =	{1--2},
  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-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.06341.6},
  URN =		{urn:nbn:de:0030-drops-8962},
  doi =		{10.4230/DagSemProc.06341.6},
  annote =	{Keywords: Pointless topology, reality}
}
Document
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

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

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.
Cite as

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-dev.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.04351.3
A Cartesian Closed Extension of the Category of Locales

Authors: Reinhold Heckmann

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

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-dev.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

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

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-dev.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

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

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.
Cite as

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-dev.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

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

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.
Cite as

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-dev.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

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

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.
Cite as

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-dev.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

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

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.
Cite as

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-dev.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.04351.10
Continued Radicals

Authors: Jamie Johnson and Tom Richmond

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

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.
Cite as

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-dev.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

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

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.
Cite as

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-dev.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}
}
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