People often use qualitative relations to describe locations or directional information, especially in written communication, such as ‘the restaurant is located at the southeast corner of the square’. However, when a large number of spatial entities are involved, qualitative relations alone are not intuitive enough for people to understand a spatial configuration. In fact, many applications, e.g. pertaining to sharing travel experiences, use sketch maps, i.e. maps focusing on the main features of an area whilst abstracting exact scale measurements, to help demonstrate abstract qualitative relations with more intuitive geometric points. Current approaches for generating point sketch maps from qualitative spatial relations require a high level of expertise, face inherent difficulties with efficiently processing large-scale data in bulk, and are vulnerable to inaccurate or conflicting information contained in qualitative data. To address these limitations, by incorporating machine learning techniques, we propose to translate the problem into an optimization problem of data reconstruction, enabling a novel end-to-end approach for generating point sketch maps from qualitative directional relations in bulk. Experiments on real-world datasets show that the proposed approach has very high accuracy and is robust even with a large portion of inaccurate or incomplete information.

There are different models to characterize AGM belief revision framework. When the background language is finite propositional language, Katsuno and Mendelzon (KM) proposed in 1991 a representation model using total preorder on worlds. This ‘preorder’ model is very influential and has been extended to characterize epistemic state in iterated belief revision. KM showed an approach how to construct the preorder via a belief set and an AGM belif revision operator, however, this approach does not work well when the language is infinite. In this paper, we argue when the language is infinite propositional language, how to construct a preorder on world to model AGM belief revision framework, and then we generalize the representation theorem of KM over an infinite language.