Philippe Balbiani. Region-based theories of space

We will present formal languages interpreted over classes of structures featuring regions and relations between them. These languages stem from Whitehead’s system in which the “being in contact” relation was assumed as primitive and Grzegorczyk’s system in which the “being included” and “being separated” relations were assumed as primitives (Biacino and Gerla, 1996). We will provide a context for studying the confluence of four research areas: first-order mereotopologies, algebras of regions, region-based propositional modal logics of space and qualitative spatial reasoning.

First-order mereotopologies are languages whose variables range over  regions and whose non-logical constants denote relations between them (Pratt and Lemon, 1997; Roeper, 1997). We will interpret such languages over the Euclidean plane, analyze their expressive power (Pratt and Schoop, 2000) and study their axiomatizability (Pratt and Schoop, 1998).

The origins of Boolean contact algebras go back to the works of Lesniewski on mereology and Leonard and Goodman on the calculus of individuals. As abstractions of Boolean algebras of regular closed sets, we will study their axiomatizability (Stell, 2000) and give
representation theorems (Düntsch and Winter, 2005; Vakarelov, Düntsch and Bennett, 2001).

Turning to region-based propositional modal logics of space, we will tell the story of two different directions in spatial logics: Tarski’s interpretation of modal logic based on the interior and closure operators of topology (van Benthem and Bezhanishvili, 2007) and von Wright’s interpretation of modal logic based on the “being included” and “being in contact” relations of mereotopology (Balbiani, Tinchev and Vakarelov, 2007; Wolter and Zakharyaschev, 2000).

As for qualitative spatial reasoning, we will introduce a number of different constraint calculi covering topology, orientation, etc (Cohn and Hazarika, 2001). We will identify tractable subsets of these calculi (Renz, 2002) and give examples of how calculi requiring
different aspects of space can be combined (Gerevini and Renz, 2002).

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