Dual Logic Concepts based on Mathematical Morphology in Stratified Institutions: Applications to Spatial Reasoning

Several logical operators are defined as dual pairs, in different types of logics. Such dual pairs of operators also occur in other algebraic theories, such as mathematical morphology. Based on this observation, this paper proposes to define, at the abstract level of institutions, a pair of abstract dual and logical operators as morphological erosion and dilation. Standard quantifiers and modalities are then derived from these two abstract logical operators. These operators are studied both on sets of states and sets of models. To cope with the lack of explicit set of states in institutions, the proposed abstract logical dual operators are defined in an extension of institutions, the stratified institutions, which take into account the notion of open sentences, the satisfaction of which is parametrized by sets of states. A hint on the potential interest of the proposed framework for spatial reasoning is also provided.Comment: 36 page

The State-of-the-Art of Set Visualization

Sets comprise a generic data model that has been used in a variety of data analysis problems. Such problems involve analysing and visualizing set relations between multiple sets defined over the same collection of elements. However, visualizing sets is a non-trivial problem due to the large number of possible relations between them. We provide a systematic overview of state-of-the-art techniques for visualizing different kinds of set relations. We classify these techniques into six main categories according to the visual representations they use and the tasks they support. We compare the categories to provide guidance for choosing an appropriate technique for a given problem. Finally, we identify challenges in this area that need further research and propose possible directions to address these challenges. Further resources on set visualization are available at http://www.setviz.net

Membership Functions for Spatial Proximity

Formalising nearness has been the subject of extensive work, resulting in many membership functions based on absolute distance metrics, relative distance metrics, and combinations of those. The possible strengths and weaknesses of these functions have been discussed and argued at length, but strangely enough, no experiment seems to have been conducted to assess the merits and shortcomings of competing approaches. Conducting such experiments can be expected not only to provide an objective evaluation of the various measures that have been proposed, but also to suggest new measures that outperform all those being analysed. This paper fulfills these expectations, and gives further evidence that fuzzy logic provides fruitful and powerful methods to formalise qualitative reasoning and capture fundamental qualitative notions. The proposed fuzzy membership functions can be directly used in qualitative reasoning about spatial proximity in Geographic Information Systems, which are becoming more and more important in software development for diverse purposes such as Tourist Information Systems or property development

ISIPTA'07: Proceedings of the Fifth International Symposium on Imprecise Probability: Theories and Applications

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