Measure Logics for Spatial Reasoning

Abstract

Although it is quite common in spatial reasoning to utilize topology for representing spatial information in a qualitative manner, in this paper an alternative method is investigated which has no connection to topology but to measure theory. I propose two logics to speak about measure theoretic information. First I investigate a highly expressive, first-order measure logic and besides providing models which with respect to this logic is sound and complete, I also show that it is actually undecidable. In the second half of the paper, a propositional measure logic is constructed which is much less expressive but computationally much more attractive than its first-order counterpart. Most importantly, in this propositional measure logic we can express spatial relations which are very similar to well-known topological relations of RCC-8 although the most efficient known logic system to express these topological relations is propositional intuitionistic logic which is undoubtedly harder than propositional measure logic.