Details

Autor(en) / Beteiligte

• Kontchakov, Roman
• Pratt-Hartmann, Ian
• Zakharyaschev, Michael

Titel

Spatial reasoning with and connectedness constraints in Euclidean spaces

Ist Teil von

• Artificial intelligence, 2014-12, Vol.217, p.43-75

Erscheinungsjahr

2014

Quelle

ScienceDirect

Beschreibungen/Notizen

• The language is a widely-studied formalism for describing topological arrangements of spatial regions. The variables of this language range over the collection of non-empty, regular closed sets of n-dimensional Euclidean space, here denoted , and its non-logical 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 -constraints in m variables, determine whether there exists an m-tuple of elements of satisfying them. These problems are known to coincide for all , so that -satisfiability is independent of dimension. This common satisfiability problem is NLogSpace-complete. Unfortunately, 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 : , in which we can state that a region is connected, and , 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 . Furthermore, in the case of , we show that there exist finite sets of constraints that are satisfiable over , but only by 'wild' regions having no possible physical meaning. This prompts us to consider interpretations over the more restrictive domain of non-empty, regular closed, polyhedral sets, . We show that (a) the satisfiability problems for (equivalently, ) over and are distinct and both NP-complete; (b) the satisfiability problems for over and are identical and NP-complete; (c) the satisfiability problems for over and are distinct, and the latter is NP-complete. Decidability of the satisfiability problem for over is open. For , and are not interestingly different from . We finish by answering the following question: given that a set of - or -constraints is satisfiable over or , how complex is the simplest satisfying assignment? In particular, we exhibit, for both languages, a sequence of constraints , satisfiable over , such that the size of grows polynomially in n, while the smallest configuration of polygons satisfying cuts the plane into a number of pieces that grows exponentially. We further show that, over , again requires exponentially large satisfying diagrams, while can force regions in satisfying configurations to have infinitely many components.