Rational Region-Based Affine Logic of the Real Plane

Rational Region-Based Affine Logic of the Real Plane ADAM TRYBUS, The University of Manchester The region-based spatial logics, where variables are set to range over certain subsets of geometric space, are the focal point of the qualitative spatial reasoning, a subfield of the KR&R research area. A lot of attention has been devoted to developing the topological spatial logics, leaving other systems relatively underexplored. We are concerned with a specific example of a region-based affine spatial logic. Building on the previous results on spatial logics with convexity, we axiomatise the theory of M = ROQ(R2 ), conv M , M , where ROQ(R2 ) is the set of regular open rational polygons of the real plane; conv M is the convexity property and M is the inclusion relation. The axiomatisation uses two infinitary rules of inference and a number of axiom schemas. Categories and Subject Descriptors: F.4.1 [Mathematical Logic and Formal Languages]: Mathematical Logic--Model theory; F.4.1 [Mathematical Logic and Formal Languages]: Mathematical Logic--Proof theory; I.2.4 [Artificial Intelligence]: Knowledge Representation Formalisms and Methods--Predicate logic General Terms: Theory of Computation, Logic Additional Key Words and Phrases: Spatial logic, convexity, axiomatisation ACM Reference Format: Adam Trybus. 2016. Rational region-based affine logic of the real plane. ACM Trans. Comput. Logic 17, 3, Article 21 (April 2016), 18 pages. DOI: http://dx.doi.org/10.1145/2897190 1. INTRODUCTION How to represent everyday objects in a mathematically satisfactory way that would allow reasoning about their properties? One answer involves formalisation in (firstorder) logic, focusing on subsets of a given geometric space (representing the objects) and their properties. This--region-based--approach has become a standard in terms of representation of and reasoning about geometric knowledge. There are many such formalisms, probed to various degrees. For convenience, these spatial...