Dependence logics are a modern family of logics of independence and dependence which mimic notions of database theory. In this paper, we aim to initiate the study of enumeration complexity in the field of dependence logics and thereby get a new point of view on enumerating answers of database queries. Consequently, as a first step, we investigate the problem of enumerating all satisfying teams of formulas from a given fragment of propositional dependence logic. We distinguish between restricting the team size by arbitrary functions and the parametrised version where the parameter is the team size. We show that a polynomial delay can be reached for polynomials and otherwise in the parametrised setting we reach FPT delay. However, the constructed enumeration algorithm with polynomial delay requires exponential space. We show that an incremental polynomial delay algorithm exists which uses polynomial space only. Negatively, we show that for the general problem without restricting the team size, an enumeration algorithm running in polynomial space cannot exist.T |= x :⇔ s(x) = 1 ∀s ∈ T, T |= ¬x :⇔ s(x) = 0 ∀s ∈ T, T |= 1 :⇔ true,We say that T satisfies ϕ iff T |= ϕ holds.Note that we have T |= (x ∧ ¬x) iff T = ∅. This observation motivates the definition for T |= 0. Observe that the evaluation in classical propositional logic occurs as the special case of evaluating singletons in team-based propositional logic.Definition 2 (Downward closure) A team-based propositional formula ϕ is called downward closed, if for every team T we have that T |= ϕ ⇒ ∀S ⊆ T : S |= ϕ. An operator • of arity k is called downward closed, if •(ϕ 1 , . . . , ϕ k ) is downward closed for all downward closed formulas ϕ i , i = 1, . . . , k. A class φ of team-based propositional formulas is called downward closed, if all formulas in φ are downward closed.
We use k‐order Voronoi diagrams to assess the stability of k‐neighbourhoods in ensembles of 2D point sets, and apply it to analyse the robustness of a dimensionality reduction technique to variations in its input configurations. To measure the stability of k‐neighbourhoods over the ensemble, we use cells in the k‐order Voronoi diagrams, and consider the smallest coverings of corresponding points in all point sets to identify coherent point subsets with similar neighbourhood relations. We further introduce a pairwise similarity measure for point sets, which is used to select a subset of representative ensemble members via the PageRank algorithm as an indicator of an individual member's value. The stability information is embedded into the k‐order Voronoi diagrams of the representative ensemble members to emphasize coherent point subsets and simultaneously indicate how stable they lie together in all point sets. We use the proposed technique for visualizing the robustness of t‐distributed stochastic neighbour embedding and multi‐dimensional scaling applied to high‐dimensional data in neural network layers and multi‐parameter cloud simulations.
Küken Karlchen ist zu spät geschlüpft und sucht seine Geschwister. Auf dem Weg durchs Kükent(h)al muss es kniffl ige Rätsel lösen. Die richtigen sechs Buchstaben ergeben zusammen das Lösungswort. Ein langer Weg (Multiple Choice) © TAMAGO POTATO / Fotolia 1. Wie heißen die Gebilde, mit denen der Eidotter in der Mitte des Eis gehalten wird?
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