knowledge, also the notion of quotient of a qualitative calculus (defined using methods from universal algebra) appears for the first time in this paper.

We have demonstrated a prototypical application of reasoning about qualitative position information in relative reference systems. In this scenario about cognitive spatial agents and qualitative map building, coarse locally perceived street configuration information has to be integrated by constraint propagation in order to get survey knowledge. The well-known path-consistency method which is implemented with standard QSR tools can make use of our new dipole calculus composition table and compute the desired result in polynomial time.

Such concrete but generalizable application scenarios for relative position calculi are the more important since a recent result by Wolter and Lee [27] showed that relative position calculi are intractable even in base relations. For this reason, it is necessary to gain experience in which application contexts the unavoidable approximate reasoning is effective and produces relevant inference results. With our street network example, we have a test case which puts emphasis on deriving implicit knowledge as the output of qualitative spatial reasoning based on observed data. This is a prototypical application scenario which in the future can also be applied to other relative position calculi.

Since the observed data in the case of error-free perception leads to consistent input constraints, the general consistency problem can be avoided - we instead rely on logical consequence. Now both problems are intractable and need to be approximated using algebraic closure; however, in our setting, approximate losses are less harmful, since we do not risk working with inconsistent scenarios.

Our future work will address the question of how in general the quality of approximations for relative position reasoning can also be assessed with quantitative measures. Another part of our future QSR research will apply our new condensed semantics method to other calculi.

## Acknowledgment

The authors would like to thank Diedrich Wolter, Jochen Renz, Frank Dylla, Christian Freksa, Franz Kalhoff, Stefan W¨olfl, Lutz Schr¨oder, and Brandon Bennett for interesting and helpful discussions related to the topic of the paper. Our work was supported by the DFG Transregional Collaborative Research Center SFB/TR 8 "Spatial Cognition".

## References

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[2] C. Freksa, Using Orientation Information for Qualitative Spatial Reasoning, in: A. U. Frank, I. Campari, U. Formentini (Eds.), Theories and Meth-