arXiv subjects

Daniel Luckhardt

Publications and source records attributed to Daniel Luckhardt.

5 recordsLinked to original sources

arXiv2025

Expressivity of bisimulation pseudometrics over analytic state spaces

A Markov decision process (MDP) is a state-based dynamical system capable of describing probabilistic behaviour with rewards. In this paper, we view MDPs as coalgebras living in the category of analytic spaces, a very general class of measurable spaces. Note that analytic spaces were already studied in the literature on labelled Markov processes and bisimulation relations. Our results are twofold. First, we define bisimulation pseudometrics over such coalgebras using the framework of fibrations. Second, we develop a quantitative modal logic for such coalgebras and prove a quantitative form of Hennessy-Milner theorem in this new setting stating that the bisimulation pseudometric corresponds to the logical distance induced by modal formulae.

cs.LO↗

arXiv2024

On Kleisli liftings and decorated trace semantics

It is well known that Kleisli categories provide a natural language to model side effects. For instance, in the theory of coalgebras, behavioural equivalence coincides with language equivalence (instead of bisimilarity) when nondeterministic automata are modelled as coalgebras living in the Kleisli category of the powerset monad. In this paper, our aim is to establish decorated trace semantics based on language and ready equivalences for conditional transition systems (CTSs) with/without upgrades. To this end, we model CTSs as coalgebras living in the Kleisli category of a relative monad. Our results are twofold. First, we reduce the problem of defining a Kleisli lifting for the machine endofunctor in the context of a relative monad to the classical notion of Kleisli lifting. Second, we provide a recipe based on indexed categories to construct a Kleisli lifting for general endofunctors.

cs.LO↗

arXiv2021

Norms on Categories and Analogs of the Schr\"oder-Bernstein Theorem

We generalize the concept of a norm on a vector space to one of a norm on a category. This provides a unified perspective on many specific matters in many different areas of mathematics like set theory, functional analysis, measure theory, topology, and metric space theory. We will especially address the two last areas in which the monotone-light factorization and, respectively, the Gromov-Hausdorff distance will naturally appear. In our formalization a Schr\"oder-Bernstein property becomes an axiom of a norm which constitutes interesting properties of the categories in question. The proposed concept provides a convenient framework for metrizations.

math.CT↗

arXiv2020

A volume comparison theorem for characteristic numbers

We show that assuming lower bounds on the Ricci curvature and the injectivity radius the absolute value of certain characteristic numbers of a Riemannian manifold, including all Pontryagin and Chern numbers, is bounded proportionally to the volume. The proof relies on Chern-Weil theory applied to a connection constructed from Euclidean connections on charts in which the metric tensor is harmonic and has bounded H\"older norm. We generalize this theorem to a Gromov-Hausdorff closed class of rough Riemannian manifolds defined in terms of H\"older regularity. Assuming an additional upper Ricci curvature bound, we show that also the Euler characteristic is bounded proportionally to the volume. Additionally, we remark on a volume comparison theorem for Betti numbers of manifolds with an additional upper bound on sectional curvature. It is a consequence of a result by Bowen.

math.DG↗

arXiv2020

Curvature bounds for regularized riemannian metrics

We investigate regularization of riemannian metrics by mollification. Assuming both-sided bounds on the Ricci tensor and a lower injectivity radius bound we obtain a uniform estimate on the change of the sectional curvature. Actually, our result holds for any metric with a uniform bound on the $W^{2,p}$-harmonic radius.

math.DG↗