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Bisimulation for Probabilistic Transition Systems: A Coalgebraic Approach
, 1998
"... . The notion of bisimulation as proposed by Larsen and Skou for discrete probabilistic transition systems is shown to coincide with a coalgebraic definition in the sense of Aczel and Mendler in terms of a set functor. This coalgebraic formulation makes it possible to generalize the concepts to a ..."
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Cited by 75 (15 self)
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. The notion of bisimulation as proposed by Larsen and Skou for discrete probabilistic transition systems is shown to coincide with a coalgebraic definition in the sense of Aczel and Mendler in terms of a set functor. This coalgebraic formulation makes it possible to generalize the concepts to a continuous setting involving Borel probability measures. Under reasonable conditions, generalized probabilistic bisimilarity can be characterized categorically. Application of the final coalgebra paradigm then yields an internally fully abstract semantical domain with respect to probabilistic bisimulation. Keywords. Bisimulation, probabilistic transition system, coalgebra, ultrametric space, Borel measure, final coalgebra. 1 Introduction For discrete probabilistic transition systems the notion of probabilistic bisimilarity of Larsen and Skou [LS91] is regarded as the basic process equivalence. The definition was given for reactive systems. However, Van Glabbeek, Smolka and Steffen s...
Recombination Spaces, Metrics, and Pretopologies
 Z. PHYS. CHEM
, 2002
"... The topological features of genotype spaces given a genetic operator have a substantial impact on the course of evolution. We explore the structure of the recombination spaces arising from four different unequal crossover models in the context of pretopological spaces. We show that all four models a ..."
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Cited by 7 (6 self)
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The topological features of genotype spaces given a genetic operator have a substantial impact on the course of evolution. We explore the structure of the recombination spaces arising from four different unequal crossover models in the context of pretopological spaces. We show that all four models are incompatible with metric distance measures due to a lack of symmetry.
On continuously Urysohn and strongly separating spaces
, 2000
"... A topological space X is continuously Urysohn if for each pair of distinct points x; y 2 X there is a continuous realvalued function f x;y 2 C(X) such that f x;y (x) 6= f x;y (y) and the correspondence (x; y) ! f x;y is a continuous function from X 2 n to C(X), where C(X) carries the topology of ..."
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Cited by 1 (0 self)
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A topological space X is continuously Urysohn if for each pair of distinct points x; y 2 X there is a continuous realvalued function f x;y 2 C(X) such that f x;y (x) 6= f x;y (y) and the correspondence (x; y) ! f x;y is a continuous function from X 2 n to C(X), where C(X) carries the topology of uniform convergence and = f(x; x) : x 2 Xg. Metric spaces are examples of continuously Urysohn spaces with the additional property that the functions f x;y depend on just one parameter. We show that spaces with this property are precisely the spaces that have a weaker metric topology. However, to nd an example of a continuously Urysohn space where the functions f x;y cannot be chosen independently of one of their parameters, it is easier to consider a much simpler property than \continuously Urysohn", given by the following denition: A topological space X is strongly separating if for each point x 2 X there is a continuous, realvalued function g x such that for any z 2 X , g x (x) = g...
Infinite Combinatorics and the theorems of Steinhaus and Ostrowski
, 2008
"... We define combinatorial principles which unify and extend the classical results of Steinhaus and Piccard on the existence of interior points in the distance set. Thus the measure and category versions are derived from one topological theorem on interior points applied to the usual topology and the d ..."
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We define combinatorial principles which unify and extend the classical results of Steinhaus and Piccard on the existence of interior points in the distance set. Thus the measure and category versions are derived from one topological theorem on interior points applied to the usual topology and the density topology on the line. Likewise we unify the subgroup theorem by reference to a Ramsey property. A combinatorial form of Ostrowski’s theorem (that a bounded additive function is linear) permits the deduction of both the measure and category automatic continuity theorem for additive functions.