Spatial reasoning with RCC8 and connectedness constraints in Euclidean spaces
Kontchakov, Roman and PrattHartmann, I. and Zakharyaschev, Michael (2014) Spatial reasoning with RCC8 and connectedness constraints in Euclidean spaces. Artificial Intelligence 217 , pp. 4375. ISSN 00043702.

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Abstract
The language RCC8 is a widelystudied formalism for describing topological arrangements of spatial regions. The variables of this language range over the collection of nonempty, regular closed sets of ndimensional Euclidean space, here denoted RC+(R^n), and its nonlogical primitives allow us to specify how the interiors, exteriors and boundaries of these sets intersect. The key question is the satisfiability problem: given a finite set of atomic RCC8constraints in m variables, determine whether there exists an mtuple of elements of RC+(R^n) satisfying them. These problems are known to coincide for all n ≥ 1, so that RCC8satisfiability is independent of dimension. This common satisfiability problem is NLogSpacecomplete. Unfortunately, RCC8 lacks the means to say that a spatial region comprises a ‘single piece’, and the present article investigates what happens when this facility is added. We consider two extensions of RCC8: RCC8c, in which we can state that a region is connected, and RCC8c0, in which we can instead state that a region has a connected interior. The satisfiability problems for both these languages are easily seen to depend on the dimension n, for n ≤ 3. Furthermore, in the case of RCC8c0, we show that there exist finite sets of constraints that are satisfiable over RC+(R^2), but only by ‘wild’ regions having no possible physical meaning. This prompts us to consider interpretations over the more restrictive domain of nonempty, regular closed, polyhedral sets, RCP+(R^n). We show that (a) the satisfiability problems for RCC8c (equivalently, RCC8c0) over RC+(R) and RCP+(R) are distinct and both NPcomplete; (b) the satisfiability problems for RCC8c over RC+(R^2) and RCP+(R^2) are identical and NPcomplete; (c) the satisfiability problems for RCC8c0 over RC+(R^2) and RCP+(R^2) are distinct, and the latter is NPcomplete. Decidability of the satisfiability problem for RCC8c0 over RC+(R^2) is open. For n ≥ 3, RCC8c and RCC8c0 are not interestingly different from RCC8. We finish by answering the following question: given that a set of RCC8c or RCC8c0constraints is satisfiable over RC+(R^n) or RCP+(R^n), how complex is the simplest satisfying assignment? In particular, we exhibit, for both languages, a sequence of constraints Φ_n, satisfiable over RCP+(R^2), such that the size of Φ_n grows polynomially in n, while the smallest configuration of polygons satisfying Φ_n cuts the plane into a number of pieces that grows exponentially. We further show that, over RC+(R^2), RCC8c again requires exponentially large satisfying diagrams, while RCC8c0 can force regions in satisfying configurations to have infinitely many components.
Metadata
Item Type:  Article 

Additional Information:  “NOTICE: this is the author’s version of a work that was accepted for publication in Artificial Intelligence. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Artificial Intelligence, DOI:10.1016/j.artint.2014.07.012 
Keyword(s) / Subject(s):  qualitative spatial reasoning, spatial logic, Euclidean space, connectedness, satisfiability, complexity 
School:  Birkbeck Faculties and Schools > Faculty of Science > School of Computing and Mathematical Sciences 
Depositing User:  Roman Kontchakov 
Date Deposited:  11 Aug 2014 11:46 
Last Modified:  09 Aug 2023 12:35 
URI:  https://eprints.bbk.ac.uk/id/eprint/10303 
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