A much better polynomial time approximation of consistency in the LR calculus

• Authors: Dominik Lücke;Till Mossakowski
• Venue: Proceedings of the 2010 conference on STAIRS 2010: Proceedings of the Fifth Starting AI Researchers' Symposium
• Year: 2010

In the area of qualitative spatial reasoning, the LR calculus (a refinement of Ligozat's flip-flop calculus) is a quite simple constraint calculus that forms the core of several orientation calculi like the Dipole calculi and the OPRA1 calculus by introducing the left-right-dichotomy. For many qualitative spatial calculi, algebraic closure is applied as the standard polynomial time “decision” procedure. For a long time it was believed that this can decide the consistency of scenarios of the LR calculus. However, in [8] it was shown that algebraic closure is a bad approximation of consistency for LR scenarios: scenarios in the base relations “Left” and “Right” are always algebraically closed, no matter if those scenarios are consistent or not. So algebraic closure is completely useless here. Furthermore, in [15] it was 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 and better 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 inequalities over the real numbers. We evaluate the quality of this procedure by comparing it bot to the old approximation using algebraic closure and to the (exact but exponential time) Buchberger algorithm for Gröbner bases (used as a decision method).