Dagstuhl Seminar Proceedings, Volume 6341
Dagstuhl Seminar Proceedings
DagSemProc
https://www.dagstuhl.de/dagpub/1862-4405
https://dblp.org/db/series/dagstuhl
1862-4405
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
6341
2007
https://drops.dagstuhl.de/entities/volume/DagSemProc-volume-6341
06341 Abstracts Collection – Computational Structures for Modelling Space, Time and Causality
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.
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\-topology
partial metrics
1-23
Regular Paper
Ralph
Kopperman
Ralph Kopperman
Prakash
Panangaden
Prakash Panangaden
Michael B.
Smyth
Michael B. Smyth
Dieter
Spreen
Dieter Spreen
10.4230/DagSemProc.06341.1
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A convenient category of domains
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.
Domain theory
topology of datatypes
1-0
Regular Paper
Ingo
Battenfeld
Ingo Battenfeld
Matthias
Schröder
Matthias Schröder
Alex
Simpson
Alex Simpson
10.4230/DagSemProc.06341.2
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Closure and Causality
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.
Closure
causality
antimatroid
temporal
software engineering
1-13
Regular Paper
John L.
Pfaltz
John L. Pfaltz
10.4230/DagSemProc.06341.3
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Elementary Differential Calculus on Discrete, Continuous and Hybrid Spaces
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.
Hybrid space
convergence space
differential
calculus
chain rule
hybrid dynamical system
discrete structure
topological space
1-2
Regular Paper
Howard
Blair
Howard Blair
10.4230/DagSemProc.06341.4
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Enriched categories and models for spaces of dipaths
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.
Enriched category
1-0
Regular Paper
Timothy
Porter
Timothy Porter
10.4230/DagSemProc.06341.5
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Instant topological relationships hidden in the reality
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".
Pointless topology
reality
1-2
Regular Paper
Martin Maria
Kovár
Martin Maria Kovár
10.4230/DagSemProc.06341.6
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