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A Logic for Spatial Reasoning in the Framework of Rough Mereology

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Transactions on Rough Sets XXI

Part of the book series: Lecture Notes in Computer Science ((TRS,volume 10810))

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Abstract

Spatial reasoning concerns a language in which spatial objects are described and argued about. Within the plethora of approaches, we single out the one set in the framework of mereology - the theory of concepts employing the notion of a part as the primitive one. Within mereology, we can choose between the approach based on part as the basic notion or the approach based on the notion of a connection from which the notion of a part is defined. In this work, we choose the former approach modified to the rough mereology version in which the notion of a part becomes ‘fuzzified’ to the notion of a part to a degree. The prevalence of this approach lies in the fact that it does allow for quantitative assessment of relations among spatial objects in distinction to only qualitative evaluation of those relations in case of other mereology based approaches.

In this work, we introduce sections on mereology based reasoning, covering part and connection based variants as well as rough mereology in order to provide the Reader with the conceptual environment we work in. We recapitulate shortly those approaches along with based on them methods for spatial reasoning. We then introduce the mereological approach in the topological context used in spatial reasoning, i.e., in collections of regular open or regular closed sets known to form complete Boolean algebras. In this environment, we create a logic for reasoning about parts and degrees of inclusion based on an abstract notion of a mass which generalizes geometric measure of area or volume and extends in the abstract manner the Lukasiewicz logical rendering of probability calculus. We give some applications, notably, we extend the relation of betweenness applied by us earlier in robot navigation and we give it the abstract characterization.

To Professor Andrzej Skowron on the seventy fifth birthday.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Chair of Mathematical Methods in Computer Science, University of Warmia and Mazury in Olsztyn, Słoneczna str. 54, 10–710, Olsztyn, Poland

    Lech Polkowski

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  1. Lech Polkowski
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  1. University of Manitoba, Winnipeg, MB, Canada

    James F. Peters

  2. University of Warsaw, Warsaw, Poland

    Andrzej Skowron

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Polkowski, L. (2019). A Logic for Spatial Reasoning in the Framework of Rough Mereology. In: Peters, J., Skowron, A. (eds) Transactions on Rough Sets XXI. Lecture Notes in Computer Science(), vol 10810. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-58768-3_5

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  • DOI: https://doi.org/10.1007/978-3-662-58768-3_5

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