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doi:10.1016/S0004-3702(00)00044-8 
Copyright © 2000 Elsevier Science B.V. All rights reserved.
A new approach to cyclic ordering of 2D orientations using ternary relation algebras*1
Amar Isli
, 
, a and Anthony G. Cohn, b
Received 24 September 1998;
Abstract
In Tarski's formalisation, the universe of a relation algebra (RA) consists of a set of binary relations. A first contribution of this work is the introduction of RAs whose universe is a set of ternary relations: these support rotation as an operation in addition to those present in Tarski's formalisation. Then we propose two particular RAs: a binary RA,
, whose universe is a set of (binary) relations on 2D orientations; and a ternary RA,
, whose universe is a set of (ternary) relations on 2D orientations. The RA
, more expressive than
, constitutes a new approach to cyclic ordering of 2D orientations. An atom of
expresses for triples of orientations whether each of the three orientations is equal to, to the left of, opposite to, or to the right of each of the other two orientations.
has 24 atoms and the elements of its universe consist of all possible 224 subsets of the set of all atoms. Amongst other results,
1. we provide for
a constraint propagation procedure computing the closure of a problem under the different operations, and show that the procedure is polynomial, and complete for a subset including all atoms;
2. we prove that another subset, expressing only information on parallel orientations, is NP-complete;
3. we show that provided that a subset
of
4. we derive from the previous result that for both RAs we “jump” from tractability to intractability if we add the universal relation to the set of all atoms.
A comparison to the most closely related work in the literature indicates that the approach is promising.
Author Keywords: Qualitative spatial reasoning; Relation algebra; Constraint satisfaction; Orientation; Computational complexity; Knowledge representation
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*1 A preliminary version of this paper, prepared while the first author was at the University of Leeds, has appeared in the Proceedings of the Fifteenth American Conference on Artificial Intelligence, pp. 643–649, AAAI Press, 1998. This work was supported by the Spatial Inference project of the DFG priority program on Spatial Cognition, under grant Fr 806/7-2, and by EPSRC under grants GR/K65041 and GR/M56807.
Corresponding author; email: isli@informatik.uni-hamburg.de
