Inproceedings,
A much better polynomial time approximation of consistency in the LR calculus
Proceedings of the 5th Starting AI Researcher Symposium (STAIRS 2010), volume 222 of Frontiers in Artificial Intelligence and Applications, page 175-185. IOS Press; Amsterdam; http://www.iospress.nl, (2010)
Abstract
In the area of qualitative spatial reasoning, the LR calculus is a
quite simple constraint calculus that forms the core of several
orientation calculi like the dipole calculi and the OPRA-1
calculus.
For many qualitative spatial calculi, algebraic closure is applied
as standard polynomial time decision procedure. For a long time it
was believed that this can decide the consistency of scenarios of
the quite simple and basic LR calculus (a refinement of Ligozat's
flip-flop calculus). However, Lücke et al. showed that algebraic
closure is a quite bad approximation of consistency of LR scenarios:
scenarios in the base relations "Left" and "Right" are always
algebraically closed. So algebraic closure is completely useless
here. Furthermore, Wolter and Lee have proved that the consistency
problem for any calculus with relative orientation containing the
relations "Left" and "Right" is NP-hard.
In this paper we propose a new polynomial time approximation
procedure for this NP-hard problem. It is based on the angles of
triangles in the Euclidean plane. LR scenarios are translated to
sets of linear inequations over the real numbers. We evaluate the
quality of this procedure by comparing it both to the old
approximation using algebraic closure and to the (exact but
exponential time) Buchberger algorithm for Gröbner bases.
