On the Decidability of Connectedness Constraints in 2D and 3D Euclidean Spaces

We investigate (quantifier-free) spatial constraint languages with equality, contact and connectedness predicates as well as Boolean operations on regions, interpreted over low-dimensional Euclidean spaces. We show that the complexity of reasoning varies dramatically depending on the dimension of the space and on the type of regions considered. For example, the logic with the interior-connectedness predicate (and without contact) is undecidable over polygons or regular closed sets in the Euclidean plane, NP-complete over regular closed sets in three-dimensional Euclidean space, and ExpTime-complete over polyhedra in three-dimensional Euclidean space.Comment: Accepted for publication in the IJCAI 2011 proceeding

On Distributive Subalgebras of Qualitative Spatial and Temporal Calculi

Qualitative calculi play a central role in representing and reasoning about qualitative spatial and temporal knowledge. This paper studies distributive subalgebras of qualitative calculi, which are subalgebras in which (weak) composition distributives over nonempty intersections. It has been proven for RCC5 and RCC8 that path consistent constraint network over a distributive subalgebra is always minimal and globally consistent (in the sense of strong $n$-consistency) in a qualitative sense. The well-known subclass of convex interval relations provides one such an example of distributive subalgebras. This paper first gives a characterisation of distributive subalgebras, which states that the intersection of a set of $n\geq 3$ relations in the subalgebra is nonempty if and only if the intersection of every two of these relations is nonempty. We further compute and generate all maximal distributive subalgebras for Point Algebra, Interval Algebra, RCC5 and RCC8, Cardinal Relation Algebra, and Rectangle Algebra. Lastly, we establish two nice properties which will play an important role in efficient reasoning with constraint networks involving a large number of variables.Comment: Adding proof of Theorem 2 to appendi

Modal logic of planar polygons

We study the modal logic of the closure algebra $P_2$, generated by the set of all polygons in the Euclidean plane $\mathbb{R}^2$. We show that this logic is finitely axiomatizable, is complete with respect to the class of frames we call "crown" frames, is not first order definable, does not have the Craig interpolation property, and its validity problem is PSPACE-complete

A semantic web rule language for geospatial domains

Retrieval of geographically-referenced information on the Internet is now a common activity. The web is increasingly being seen as a medium for the storage and exchange of geographic data sets in the form of maps. The geospatial-semantic web (GeoWeb) is being developed to address the need for access to current and accurate geo-information. The potential applications of the GeoWeb are numerous, ranging from specialised application domains for storing and analysing geo-information to more common applications by casual users for querying and visualising geo-data, e.g. finding locations of services, descriptions of routes, etc. Ontologies are at the heart of W3C's semantic web initiative to provide the necessary machine understanding to the sheer volumes of information contained on the internet. For the GeoWeb to succeed the development of ontologies for the geographic domain are crucial. Semantic web technologies to represent ontologies have been developed and standardised. OWL, the Web Ontology Language, is the most expressive of these enabling a rich form of reasoning, thanks to its formal description logic underpinnings. Building geo-ontologies involves a continuous process of update to the originally modelled data to reflect change over time as well as to allow for ontology expansion by integrating new data sets, possibly from different sources. One of the main challenges in this process is finding means of ensuring the integrity of the geo-ontology and maintaining its consistency upon further evolution. Representing and reasoning with geographic ontologies in OWL is limited. Firstly, OWL is not an integrity checking language due to it's non-unique name and open world assumptions. Secondly, it can not represent spatial datatypes, can not compute information using spatial operators and does not have any form of spatial index. Finally, OWL does not support complex property composition needed to represent qualitative spatial reasoning over spatial concepts. To address OWL's representational inefficiencies, new ontology languages have been proposed based on the intersection or union of OWL (in particular the DL family corresponding to OWL) with logic programs (rule languages). In this work, a new Semantic Web Spatial Rule Language (SWSRL) is proposed, based on the syntactic core of the Description Logic Programs paradigm (DLP), and the semantics of a Logic Program. The language is built to support the expression of geospatial ontological axioms and geospatial integrity and deduction rules. A hybrid framework to integrate both qualitative symbolic information in SWSRL with quantitative, geometric information using spatial datatypes in a spatial database is proposed. Two notable features of SWSRL are 1) the language is based on a prioritised de fault logic that allows the expression of default integrity rules and their exceptions and 2) the implementation of the language uses an interleaved mode of inference for on the fly computation (either qualitative or quantitative) deduction of spatial relations. SWSRL supports an OGC complaint spatial syntax, and a standardised definition of rule meta data. Both features aid the construction, description, identification and categorisation of designed and implemented rules within large rule sets. The language and the developed engine are evaluated using synthetic as well as real data sets in the context of developing geographic ontologies for geographic information retrieval on the Semantic Web. Empirical experiments are also presented to test the scalability and applicability of the developed framework