Abstract
Declarative spatial reasoning denotes the ability to (declaratively) specify and solve real-world problems related to geometric and qualitative spatial representation and reasoning within standard knowledge representation and reasoning (KR) based methods (e.g., logic programming and derivatives). One approach for encoding the semantics of spatial relations within a declarative programming framework is by systems of polynomial constraints. However, solving such constraints is computationally intractable in general (i.e. the theory of real-closed fields).
We present a new algorithm, implemented within the declarative spatial reasoning system CLP(QS), that drastically improves the performance of deciding the consistency of spatial constraint graphs over conventional polynomial encodings. We develop pruning strategies founded on spatial symmetries that form equivalence classes (based on affine transformations) at the qualitative spatial level. Moreover, pruning strategies are themselves formalised as knowledge about the properties of space and spatial symmetries. We evaluate our algorithm using a range of benchmarks in the class of contact problems, and proofs in mereology and geometry. The empirical results show that CLP(QS) with knowledge-based spatial pruning outperforms conventional polynomial encodings by orders of magnitude, and can thus be applied to problems that are otherwise unsolvable in practice.
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Notes
- 1.
Incompleteness refers to the inability of a spatial reasoning method to determine whether a given network of qualitative spatial constraints is consistent or inconsistent in general. Relation-algebraic spatial reasoning (i.e. using algebraic closure based on weak composition) has been shown to be incomplete for a number of spatial languages and cannot guarantee consistency in general, e.g. relative directions [23] and containment relations between linearly ordered intervals [22], Theorem 5.9.
- 2.
We emphasise that this analytic geometry approach that we also adopt is not qualitative spatial reasoning in the relation algebraic sense; the foundations are similar (i.e. employing a finite language of spatial relations that are interpreted as infinite sets of configurations, determining consistency in the complete absence of numeric information, and so on) but the methods for determining consistency etc. come from different branches of spatial reasoning.
- 3.
Standard geometric constraint languages of approaches including [12, 17, 27] consist of points, lines, circles, ellipses, and coincidence, tangency, perpendicularity, parallelism, and numerical dimension constraints; note the absence of e.g. mereotopology and “common-sense” relative orientation relations [35].
- 4.
Spatial Reasoning (CLP(QS)). www.spatial-reasoning.com.
- 5.
Important factors in determining the applicability of various analytic approaches are the degree of the polynomials (particularly the distinction between linear and nonlinear) and whether both equality and inequalities are permitted in the constraints.
- 6.
That is, constructive methods may fail in building a consistent solution, and iterative root finding methods may fail to converge.
- 7.
The properties of affine transformations and the geometric objects that they preserve are well understood; further information is readily available in introductory texts such as [26]. Our key contribution is formalising and exploiting this spatial knowledge as modular and extensible common-sense rules in intelligent knowledge-based spatial assistance systems.
- 8.
All cases have been verified using Reduce as presented in Theorem 2.
References
Aiello, M., Pratt-Hartmann, I.E., van Benthem, J.F.: Handbook of Spatial Logics. Springer-Verlag New York Inc., Secaucus (2007). ISBN 978-1-4020-5586-7
Arnon, D.S., Collins, G.E., McCallum, S.: Cylindrical algebraic decomposition I: the basic algorithm. SIAM J. Comput. 13(4), 865–877 (1984)
Bhatt, M., Wallgrün, J.O.: Geospatial narratives and their spatio-temporal dynamics: Commonsense reasoning for high-level analyses in geographic information systems. ISPRS Int. J. Geo-Information 3(1), 166–205 (2014). doi:10.3390/ijgi3010166. http://dx.doi.org/10.3390/ijgi3010166
Bhatt, M., Lee, J.H., Schultz, C.: CLP(QS): a declarative spatial reasoning framework. In: Egenhofer, M., Giudice, N., Moratz, R., Worboys, M. (eds.) COSIT 2011. LNCS, vol. 6899, pp. 210–230. Springer, Heidelberg (2011)
Bhatt, M., Schultz, C., Freksa, C.: The ‘Space’ in spatial assistance systems: conception, formalisation and computation. In: Tenbrink, T., Wiener, J., Claramunt, C. (eds.) Representing space in cognition: Interrelations of behavior, language, and formal models. Series: Explorations in Language and Space. Oxford University Press (2013). 978-0-19-967991-1
Bhatt, M., Suchan, J., Schultz, C.: Cognitive interpretation of everyday activities - toward perceptual narrative based visuo-spatial scene interpretation. In: Finlayson, M., Fisseni, B., Loewe, B., Meister, J.C. (eds.) Computational Models of Narrative (CMN) 2013, a satellite workshop of CogSci 2013: The 35th meeting of the Cognitive Science Society., Dagstuhl, Germany, OpenAccess Series in Informatics (OASIcs) (2013)
Bhatt, M., Schultz, C.P.L., Thosar, M.: Computing narratives of cognitive user experience for building design analysis: KR for industry scale computer-aided architecture design. In: Baral, C., Giacomo, G.D., Eiter, T. (eds.) Principles of Knowledge Representation and Reasoning: Proceedings of the Fourteenth International Conference, KR 2014, Vienna, Austria, 20–24 July, 2014. AAAI Press (2014). ISBN 978-1-57735-657-8
Borgo, S.: Spheres, cubes and simplexes in mereogeometry. Logic Logical Philos. 22(3), 255–293 (2013)
Bouhineau, D.: Solving geometrical constraint systems using CLP based on linear constraint solver. In: Pfalzgraf, J., Calmet, J., Campbell, J. (eds.) AISMC 1996. LNCS, vol. 1138, pp. 274–288. Springer, Heidelberg (1996)
Bouhineau, D., Trilling, L., Cohen, J.: An application of CLP: Checking the correctness of theorems in geometry. Constraints 4(4), 383–405 (1999)
Buchberger, B.: Bruno Buchberger’s PhD thesis 1965: an algorithm for finding the basis elements of the residue class ring of a zero dimensional polynomial ideal (English translation). J. Symbolic Comput. 41(3), 475–511 (2006)
Chou, S.-C.: Mechanical Geometry Theorem Proving, vol. 41. Springer Science and Business Media, Dordrecht (1988)
Collins, G.E.: Quantifier elimination for real closed fields by cylindrical algebraic decompostion. In: Brakhage, H. (ed.) GI-Fachtagung 1975. LNCS, vol. 33, pp. 134–183. Springer, Heidelberg (1975)
Collins, G.E., Hong, H.: Partial cylindrical algebraic decomposition for quantifier elimination. J. Symbolic Comput. 12(3), 299–328 (1991). ISSN 0747–7171
Dolzmann, A., Seidl, A., Sturm, T.: REDLOG User Manual, Edition 3.0, Apr 2004
Gantner, Z., Westphal, M., Wölfl, S.: GQR-A fast reasoner for binary qualitative constraint calculi. In: Proceedings of AAAI, vol. 8 (2008)
Hadas, N., Hershkowitz, R., Schwarz, B.B.: The role of contradiction and uncertainty in promoting the need to prove in dynamic geometry environments. Educ. Stud. Mathe. 44(1–2), 127–150 (2000)
Haunold, P., Grumbach, S., Kuper, G., Lacroix, Z.: Linear constraints: Geometric objects represented by inequalitiesl. In: Frank, A.U. (ed.) COSIT 1997. LNCS, vol. 1329, pp. 429–440. Springer, Heidelberg (1997)
Jaffar, J., Maher, M.J.: Constraint logic programming: A survey. J. Logic Prog. 19, 503–581 (1994)
Kanellakis, P.C., Kuper, G.M., Revesz, P.Z.: Constraint query languages. In: Rosenkrantz, D.J., Sagiv, Y. (eds.) Proceedings of the Ninth ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems, Nashville, Tennessee, USA, 2–4 April, 1990, pp. 299–313. ACM Press (1990). ISBN 0-89791-352-3
Kapur, D., Mundy, J.L. (eds.): Geometric Reasoning. MIT Press, Cambridge (1988). ISBN 0-262-61058-2
Ladkin, P.B., Maddux, R.D.: On binary constraint problems. J. ACM (JACM) 41(3), 435–469 (1994)
Lee, J.H.: The complexity of reasoning with relative directions. In: 21st European Conference on Artificial Intelligence (ECAI 2014) (2014)
Ligozat, G.: Qualitative Spatial and Temporal Reasoning. Wiley-ISTE, Hoboken (2011)
Ligozat, G.F.: Qualitative triangulation for spatial reasoning. In: Campari, I., Frank, A.U. (eds.) COSIT 1993. LNCS, vol. 716, pp. 54–68. Springer, Heidelberg (1993)
Martin, G.E.: Transformation geometry: An introduction to symmetry. Springer, New York (1982)
Owen, J.C.: Algebraic solution for geometry from dimensional constraints. In: Proceedings of the First ACM Symposium on Solid Modeling Foundations and CAD/CAM Applications, pp. 397–407. ACM (1991)
Pesant, G., Boyer, M.: QUAD-CLP (R): Adding the power of quadratic constraints. In: Borning, A. (ed.) PPCP 1994. LNCS, vol. 874, pp. 95–108. Springer, Heidelberg (1994)
Pesant, G., Boyer, M.: Reasoning about solids using constraint logic programming. J. Automated Reasoning 22(3), 241–262 (1999)
Randell, D.A., Cohn, A.G., Cui Z.: Computing transitivity tables: A challenge for automated theorem provers. In 11th International Conference on Automated Deduction (CADE-11), pp. 786–790 (1992)
Ratschan, S.: Approximate quantified constraint solving by cylindrical box decomposition. Reliable Comput. 8(1), 21–42 (2002)
Ratschan, S.: Efficient solving of quantified inequality constraints over the real numbers. ACM Trans. Comput. Logic (TOCL) 7(4), 723–748 (2006)
Schultz, C., Bhatt, M.: Towards a declarative spatial reasoning system. In: 20th European Conference on Artificial Intelligence (ECAI 2012) (2012)
Schultz, C., Bhatt, M.: Declarative spatial reasoning with boolean combinations of axis-aligned rectangular polytopes. In: ECAI 2014–21st European Conference on Artificial Intelligence, pp. 795–800 (2014)
Schultz, C., Bhatt, M., Borrmann, A.: Bridging qualitative spatial constraints and parametric design - a use case with visibility constraints. In: EG-ICE: 21st International Workshop - Intelligent Computing in Engineering 2014 (2014)
Tarski, A.: A general theorem concerning primitive notions of Euclidean geometry. Indagationes Mathematicae 18(468), 74 (1956)
Varzi, A.C.: Parts, wholes, and part-whole relations: The prospects of mereotopology. Data Knowl. Eng. 20(3), 259–286 (1996)
Walega, P., Bhatt, M., Schultz, C.: ASPMT(QS): non-monotonic spatial reasoning with answer set programming modulo theories. In: LPNMR: Logic Programming and Nonmonotonic Reasoning - 13th International Conference (2015). http://lpnmr2015.mat.unical.it
Wallgrün, J.O., Frommberger, L., Wolter, D., Dylla, F., Freksa, C.: Qualitative spatial representation and reasoning in the SparQ-Toolbox. In: Barkowsky, T., Knauff, M., Ligozat, G., Montello, D.R. (eds.) Spatial Cognition 2007. LNCS (LNAI), vol. 4387, pp. 39–58. Springer, Heidelberg (2007)
Wenjun, W.: Basic principles of mechanical theorem proving in elementary geometries. J. Syst. Sci. Math. Sci. 4(3), 207–235 (1984)
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Schultz, C., Bhatt, M. (2015). Spatial Symmetry Driven Pruning Strategies for Efficient Declarative Spatial Reasoning. In: Fabrikant, S., Raubal, M., Bertolotto, M., Davies, C., Freundschuh, S., Bell, S. (eds) Spatial Information Theory. COSIT 2015. Lecture Notes in Computer Science(), vol 9368. Springer, Cham. https://doi.org/10.1007/978-3-319-23374-1_16
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