References
Balas, Egon (1985). “Disjunctive programming and a hierarchy of relaxations for discrete optimization problems”. In:SIAM J ALGEBRA DISCR6.3, pp. 466–486.
Berkelaar, Michel, Kjell Eikland, and Peter Notebaert (May 2010).lp solve 5.5, Open source (Mixed-Integer) Linear Programming system. Software. Available athttp://lpsolve.
sourceforge.net/5.5/. Last accessed April, 2013.
Bhatt, Mehul, Jae Hee Lee, and Carl Schultz (2011). “CLP(QS): a declarative spatial reasoning framework”. In:Proc. of Cosit, pp. 210–230.
Dylla, Frank, Till Mossakowski, Thomas Schneider, and Diedrich Wolter (2013). “Algebraic Properties of Qualitative Spatio-Temporal Calculi”. In:Proceedings of the Conference On Spatial Information Theory (COSIT-13). Vol. 8116. LNCS. North Yorkshire, UK: Springer, pp. 516–536.
Gottlob, Georg (2011). “On Minimal Constraint Networks”. In:Principles and Practice of Constraint Programming, CP 2011. Ed. by Jimmy Lee. Vol. 6876. LNCS. Springer Berlin Heidelberg, pp. 325–339.ISBN: 978-3-642-23785-0.
Hu´e, Julien, Matthias Westphal, and Stefan W¨olfl (2012). “An Automatic Decomposition Method for Qualitative Spatial and Temporal Reasoning”. In:Proc. ICTAI-2012. University Freiburg, SFB/TR 8, pp. 588–595.
Jonsson, Peter and Christer B¨ackstr¨om (1998). “A unifying approach to temporal constraint reasoning”. In:Artif. Intell.102.1, pp. 143–155.
Kontchakov, Roman, Ian E. Pratt-Hartmann, Frank Wolter, and Michael Zakharyaschev (2010).
“Spatial Logics with Connectedness Predicates”. In:LOG METH COMPUT SCI.
Lee, Jae Hee, Jochen Renz, and Diedrich Wolter (2013). “StarVars—Effective Reasoning about Relative Directions”. In:IJCAI. Ed. by Francesca Rossi. IJCAI.
Lee, Sangbum and Ignacio E. Grossmann (2000). “New algorithms for nonlinear general-ized disjunctive programming”. In:Computers and Chemical Engineering Journal24.9, pp. 2125–2141.
Li, Sanjiang, Weiming Liu, and Shengsheng Wang (2013). “Qualitative constraint satisfaction problems: An extended framework with landmarks”. In:Artificial Intelligence201, pp. 32–
58.
Ligozat, G´erard (2011). Qualitative Spatial and Temporal Reasoning. Wiley. ISBN: 978-1848212527.
Mossakowski, Till and Reinhard Moratz (2012). “Qualitative Reasoning about Relative Direc-tion of Oriented Points”. In:Artif. Intell.180–181, pp. 34–45.
Randell, David A., Zhan Cui, and Anthony G. Cohn (1992). “A Spatial Logic based on Regions and “Connection””. In:Proc, of KR’92. Ed. by Bernhard Nebel, Charles Rich, and William Swartout. San Mateo (CA), USA: Morgan Kaufmann, pp. 165–176.
Renz, Jochen and D. Mitra (2004). “Qualitative Direction Calculi with Arbitrary Granularity”.
In:Proc. of PRICAI-04. Vol. 3157. LNAI. Springer, pp. 65–74.
References
Renz, Jochen and Bernhard Nebel (2007). “Qualitative spatial reasoning using constraint calculi”. In:Handbook of Spatial Logics. Ed. by Marco Aiello, Ian E. Pratt-Hartmann, and Johan van Benthem. Springer, pp. 161–215.ISBN: 978-1-4020-5586-7.
T¨aubig, Holger, Udo Frese, Christoph Hertzberg, Christoph L¨uth, Stefan Mohr, Elena Vorobev, and Dennis Walter (2012). “Guaranteeing functional safety: design for provability and computer-aided verification”. In:Autonomous Robots32.3, pp. 303–331.
W¨olfl, Stefan and Matthias Westphal (2009). “On Combinations of Binary Qualitative Con-straint Calculi”. In:Proc. of IJCAI, pp. 967–973.
Wolter, Diedrich (2012). “Analyzing Qualitative Spatio-Temporal Calculi using Algebraic Geometry”. In:Spatial Cognition and Comp.12.1, pp. 23–52.
Wolter, Diedrich and Jae Hee Lee (2010). “Qualitative Reasoning with Directional Relations”.
In:Artif. Intell.174.18, pp. 1498–1507.
8 Conclusion
This chapter concludes my cumulative dissertation with a brief summary, a critical assessment of the approach and in particular the assumptions made. As every end is the start of something new, this chapter also includes an outlook towards possible future work, and closing with some final thoughts.
8.1 Summary
Four research questions were presented in detail in the introduction. In the following I will summarize my answers to each of these four research questions. To answer these research ques-tions, I have developed And/Or Linear Programming (And/Or LP; Chapter 7) and introduced two new logics: Conceptual Neighborhood Logic (CNL; Chapter 5) and QLTL (Chapter 6).
How can qualitative calculi be combined, i.e. how can one jointly reason with knowledge represented in distinct calculi?
And/Or Linear Programming (And/Or LP; Chapter 7) allows to map formulas of different (spatial) aspects into a single unified language. The basic assumption underlying And/Or Linear Programming is that non-linearities can be approximated by a set of systems of linear equalities and inequalities. If the set is finite, then all of these systems could (theoretically) be checked, one by one. However, the size of the set of systems of linear equalities and inequalities renders such an exhaustive search impractical. Consequently, a heuristic should be used that either directs towards a satisfiable set, or that can exclude subsets without checking all of its members.
The current heuristic used in And/Or LP is based on pruning the search space by coarsening qualitative relations (Chapter 5 and Chapter 7). Each coarsening adheres to the following property: If no solution can be found for a coarsened system of linear inequalities, then for the original system no solution can be found either. Therefore, the search space can be pruned, whenever no solution can be found. In case a solution is found, the relations are split into finer granular relations, again resulting in a set of systems of linear equalities and inequalities. This deepening is repeated until either no solution is found or the desired level of granularity is reached, in which case the solution is returned.
In the current implementation a depth-first approach is used, which fixes single relations one after the other, using the above described heuristic. In case no intermediate solution is found, backtracking is used.
How can qualitative representations incorporate grounded information, i.e. how can free-ranging and constrained variable domains (singleton, finite, numeric constraints) be mixed?
An assumption in qualitative spatial reasoning is that all variables range over the same domain.
However, in applications it is often required to restrict the domain of specific variables, such as restricting a solution to fit into a given floor plan. Such a floor plan can be represented as a polygon fixed in a (local) coordinate system. The free space can than be described by disjunctive sets of linear inequalities, representing a convex partition of the polygon. Robotic applications generally describe—or at least approximate—shapes with polygons. In contrast to a ground floor, other entities represented by a polygons are not necessarily fixed at a single location. To express that an entity is limited to a finite set of locations, the locations can be described as sets of equalities. As systems of equalities and systems of inequalities both are fundamental building blocks of And/Or LP, Therefore, they can be added to the And/Or LP tree resulting from a formula with no change of the reasoning method.
In Chapter 5 and Chapter 7 we demonstrate the encoding and incorporation of such kind of background knowledge. Further, we also describe, that a single spatial region, such as a
”breaking area”, can also have different shapes. While the breaking area depend on the current speed (category) of the vehicles, the formula, that no two breaking areas are allowed to overlap, holds regardlessly. This eases the burden on the modeling and ensures the transferability of the modeled knowledge. In summary, And/Or LP can directly reason with specific instances and qualitative descriptions in one unified manner.
How can a prototypical pictorial representation be derived from a (pure) qualitative description of a scene?
One result presented in Chapter 4 is the generation of high-level counter examples. Such counter-examples are hard to interpret when only presented as a logical formula. Provided that a qualitative description is realizable, then by calculating a solution to the And/Or LP tree all variables have a real valued assignment. Consequently, as every part is fully specified, i.e. has a fixed position, a known orientation, and so on, we can draw a pictorial representation of the qualitative description.
How can a spatial logic and Linear Temporal Logic be combined to yield a decidable formalism, that can be applied to various applications?
Regarding time, two aspects are prevailing, first, how to apply a temporal formula to control a robotic system, and second, how to think about time during modeling. Time can be viewed as branching (CTL, ), emphasizing on what could be, whereas time viewed as linear (LTL,