eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2005-04-19
4351
1
24
10.4230/DagSemProc.04351.1
article
04351 Abstracts Collection – Spatial Representation: Discrete vs. Continuous Computational Models
Kopperman, Ralph
Panangaden, Prakash
Smyth, Michael B.
Spreen, Dieter
Webster, Julian
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.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol04351/DagSemProc.04351.1/DagSemProc.04351.1.pdf
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
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2005-04-19
4351
1
5
10.4230/DagSemProc.04351.2
article
04351 Summary – Spatial Representation: Discrete vs. Continuous Computational Models
Kopperman, Ralph
Panangaden, Prakash
Smyth, Michael B.
Spreen, Dieter
Webster, Julian
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.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol04351/DagSemProc.04351.2/DagSemProc.04351.2.pdf
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
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2005-04-19
4351
1
20
10.4230/DagSemProc.04351.3
article
A Cartesian Closed Extension of the Category of Locales
Heckmann, Reinhold
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.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol04351/DagSemProc.04351.3/DagSemProc.04351.3.pdf
Locale
Cartesian closed category
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2005-04-19
4351
1
16
10.4230/DagSemProc.04351.4
article
A Category of Discrete Closure Spaces
Pfaltz, John L.
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.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol04351/DagSemProc.04351.4/DagSemProc.04351.4.pdf
Category
closure
antimatroid
function
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2005-04-19
4351
1
28
10.4230/DagSemProc.04351.5
article
A domain of spacetime intervals in general relativity
Martin, Keye
Panangaden, Prakash
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.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol04351/DagSemProc.04351.5/DagSemProc.04351.5.pdf
Causality
spacetime
global hyperbolicity
interval domains
bicontinuous posets
spacetime topology
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2005-04-19
4351
1
0
10.4230/DagSemProc.04351.6
article
A geometry of information, I: Nerves, posets and differential forms
Gratus, Jonathan
Porter, Timothy
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.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol04351/DagSemProc.04351.6/DagSemProc.04351.6.pdf
Chu spaces
nerves
differential forms
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2005-04-19
4351
1
0
10.4230/DagSemProc.04351.7
article
A geometry of information, II: Sorkin models, and biextensional collapses
Gratus, Jonathan
Porter, Timothy
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.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol04351/DagSemProc.04351.7/DagSemProc.04351.7.pdf
Chu space
Sorkin model
Nerve
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2005-04-19
4351
1
4
10.4230/DagSemProc.04351.8
article
Auxiliary relations and sandwich theorems
God, Chris
Jung, Achim
Knight, Robin
Kopperman, Ralph
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.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol04351/DagSemProc.04351.8/DagSemProc.04351.8.pdf
Adjoint
auxiliary relation
continuous poset
pairwise completely regular (and pairwise normal) bitopological space
upper (lower) semicontinuous Urysohn relation
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2005-04-19
4351
1
7
10.4230/DagSemProc.04351.9
article
Compactness in apartness spaces?
Bridges, Douglas
Ishihara, Hajime
Schuster, Peter
Vita, Luminita S.
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.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol04351/DagSemProc.04351.9/DagSemProc.04351.9.pdf
Apartness
constructive
compact uniform space
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2005-04-19
4351
1
4
10.4230/DagSemProc.04351.10
article
Continued Radicals
Johnson, Jamie
Richmond, Tom
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.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol04351/DagSemProc.04351.10/DagSemProc.04351.10.pdf
Continued radical
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2005-04-19
4351
1
19
10.4230/DagSemProc.04351.11
article
Continuous Semantics for Termination Proofs
Berger, Ulrich
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.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol04351/DagSemProc.04351.11/DagSemProc.04351.11.pdf
Higher-order term rewriting
termination
domain theory
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2005-04-19
4351
1
8
10.4230/DagSemProc.04351.12
article
Deadlocks and Dihomotopy in Mutual Exclusion Models
Raussen, Martin
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.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol04351/DagSemProc.04351.12/DagSemProc.04351.12.pdf
Mutual exclusion
deadlock detection
dihomotopy
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2005-04-19
4351
1
3
10.4230/DagSemProc.04351.13
article
Dihomotopy Classes of Dipaths in the Geometric Realization of a Cubical Set: from Discrete to Continuous and back again
Fajstrup, Lisbeth
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.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol04351/DagSemProc.04351.13/DagSemProc.04351.13.pdf
Cubical Complex
Higher Dimensional Automaton
Ditopology
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2005-04-19
4351
1
21
10.4230/DagSemProc.04351.14
article
Discrete classical vs. continuous quantum data in abstract quantum mechanics
Abramsky, Samson
Coecke, Bob
``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
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol04351/DagSemProc.04351.14/DagSemProc.04351.14.pdf
Category theory
strong compact closure
quantum information-flow
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2005-04-19
4351
1
8
10.4230/DagSemProc.04351.15
article
Dyadic Subbases and Representations of Topological Spaces
Tsuiki, Hideki
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.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol04351/DagSemProc.04351.15/DagSemProc.04351.15.pdf
Dyadic subbase
embedding
computation over topological spaces
Plotkin's $T^\omega$
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2005-04-19
4351
1
0
10.4230/DagSemProc.04351.16
article
Integrating Topology and Geometry for Macro-Molecular Simulations
Moore, Edward L. F.
Peters, Thomas J.
Ferguson, David R.
Stewart, Neil F.
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.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol04351/DagSemProc.04351.16/DagSemProc.04351.16.pdf
Computational topology
spline
approximation
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2005-04-19
4351
1
10
10.4230/DagSemProc.04351.17
article
On Maximality of Compact Topologies
Kovar, Martin
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?
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol04351/DagSemProc.04351.17/DagSemProc.04351.17.pdf
de Groot dual
compact saturated set
wide Scott open filter
maximal compact topology
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2005-04-19
4351
1
5
10.4230/DagSemProc.04351.18
article
The Construction of Finer Compact Topologies
Künzi, Hans-Peter A.
Zypen, Dominic van der
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.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol04351/DagSemProc.04351.18/DagSemProc.04351.18.pdf
Maximal compact
KC-space
sober
US-space
locally compact
sequential
sequentially compact
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2005-04-19
4351
1
8
10.4230/DagSemProc.04351.19
article
The de Groot dual for general collections of sets
Kovar, Martin
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.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol04351/DagSemProc.04351.19/DagSemProc.04351.19.pdf
Saturated set
dual topology
compactness operator
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2005-04-19
4351
1
16
10.4230/DagSemProc.04351.20
article
The Hofmann-Mislove Theorem for general posets
Kovar, Martin
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.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol04351/DagSemProc.04351.20/DagSemProc.04351.20.pdf
Posets
generalized Scott topology
Scott open filters
(filtered) compactness
saturated
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2005-04-19
4351
1
9
10.4230/DagSemProc.04351.21
article
The Hofmann-Mislove Theorem for general topological structures
Kovar, Martin
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.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol04351/DagSemProc.04351.21/DagSemProc.04351.21.pdf
Compact saturated set
Scott open filter
(quasi-) sober space
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2005-04-19
4351
1
4
10.4230/DagSemProc.04351.22
article
What do partial metrics represent?
Kopperman, Ralph
Matthews, Steve
Pajoohesh, Homeira
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.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol04351/DagSemProc.04351.22/DagSemProc.04351.22.pdf
Metric
partial metric
base point