An Incremental SAT-Based Approach to Reason Efficiently on Qualitative Constraint Networks
Abstract
The \({\mathcal {RCC}}8 \) language is a widely-studied formalism for describing topological arrangements of spatial regions. Two fundamental reasoning problems that are associated with \({\mathcal {RCC}}8 \) are the problems of satisfiability and realization. Given a qualitative constraint network (QCN) of \({\mathcal {RCC}}8 \), the satisfiability problem is deciding whether it is possible to assign regions to the spatial variables of the QCN in such a way that all of its constraints are satisfied (solution). The realization problem is producing an actual spatial model that can serve as a solution. Researchers in \({\mathcal {RCC}}8 \) focus either on symbolically checking the satisfiability of a QCN or on presenting a method to realize (valuate) a satisfiable QCN. To the best of our knowledge, a combination of those two lines of research has not been considered in the literature in a unified and homogeneous approach, as the first line deals with native constraint-based methods, and the second one with rich mathematical structures that are difficult to implement. In this article, we combine the two aforementioned lines of research and explore the opportunities that surface by interrelating the corresponding reasoning problems, viz., the problems of satisfiability and realization. We restrict ourselves to QCNs that, when satisfiable, are realizable with rectangles. In particular, we propose an incremental \(\mathrm {SAT}\)-based approach for providing a framework that reasons about the \({\mathcal {RCC}}8 \) language in a counterexample-guided manner. The incrementality of our approach also avoids the usual blow-up and the lack of scalability in \(\mathrm {SAT}\)-based encodings. Specifically, our \(\mathrm {SAT}\)-translation is parsimonious, i.e, constraints are added incrementally in a manner that guides the embedded \(\mathrm {SAT}\)-solver and forbids it to find the same counter-example twice. We experimentally evaluated our approach and studied its scalability against state-of-the-art solvers for reasoning about \({\mathcal {RCC}}8 \) relations using a varied dataset of instances. The approach scales up and is competitive with the state of the art for the considered benchmarks.
Keywords
\({\mathcal {RCC}} \) Qualitative Spatial and Temporal Reasoning \(\mathrm {SAT}\) \({\text {CEGAR}}\)Notes
Acknowledgments
The authors would like to thank the anonymous reviewers for their insightful comments. Part of this work was supported by the French Ministry for Higher Education and Research, the Haut-de-France Regional Council through the “Contrat de Plan État Région (CPER) DATA” and an EC FEDER grant.
References
- 1.Sioutis, M., Alirezaie, M., Renoux, J., Loutfi, A.: Towards a synergy of qualitative spatio-temporal reasoning and smart environments for assisting the elderly at home. In: IJCAI Workshop on Qualitative Reasoning (2017)Google Scholar
- 2.Bhatt, M., Guesgen, H., Wölfl, S., Hazarika, S.: Qualitative spatial and temporal reasoning: emerging applications, trends, and directions. Spat. Cogn. Comput. 11, 1–14 (2011)CrossRefGoogle Scholar
- 3.Dubba, K.S.R., Cohn, A.G., Hogg, D.C., Bhatt, M., Dylla, F.: Learning relational event models from video. J. Artif. Intell. Res. 53, 41–90 (2015)MathSciNetCrossRefGoogle Scholar
- 4.Story, P.A., Worboys, M.F.: A design support environment for spatio-temporal database applications. In: Frank, A.U., Kuhn, W. (eds.) COSIT 1995. LNCS, vol. 988, pp. 413–430. Springer, Heidelberg (1995). https://doi.org/10.1007/3-540-60392-1_27CrossRefGoogle Scholar
- 5.Randell, D.A., Cui, Z., Cohn, A.: A spatial logic based on regions and connection. In: KR (1992)Google Scholar
- 6.Bouzy, B.: Les concepts spatiaux dans la programmation du go. Revue d’Intelligence Artificielle 15, 143–172 (2001)CrossRefGoogle Scholar
- 7.Lattner, A.D., Timm, I.J., Lorenz, M., Herzog, O.: Knowledge-based risk assessment for intelligent vehicles. In: KIMAS (2005)Google Scholar
- 8.Heintz, F., de Leng, D.: Spatio-temporal stream reasoning with incomplete spatial information. In: ECAI (2014)Google Scholar
- 9.Randell, D.A., Galton, A., Fouad, S., Mehanna, H., Landini, G.: Mereotopological correction of segmentation errors in histological imaging. J. Imaging 3(4), 63 (2017)CrossRefGoogle Scholar
- 10.Renz, J., Nebel, B.: On the complexity of qualitative spatial reasoning: a maximal tractable fragment of the region connection calculus. Artif. Intell. 108(1–2), 69–123 (1999)MathSciNetCrossRefGoogle Scholar
- 11.Li, S.: On topological consistency and realization. Constraints 11, 31–51 (2006)MathSciNetCrossRefGoogle Scholar
- 12.Renz, J., Nebel, B.: Qualitative spatial reasoning using constraint calculi. In: Aiello, M., Pratt-Hartmann, I., Van Benthem, J. (eds.) Handbook of Spatial Logics, pp. 161–215. Springer, Dordrecht (2007). https://doi.org/10.1007/978-1-4020-5587-4_4CrossRefGoogle Scholar
- 13.Golumbic, M.C., Shamir, R.: Complexity and algorithms for reasoning about time: a graph-theoretic approach. J. ACM 40, 1108–1133 (1993)MathSciNetCrossRefGoogle Scholar
- 14.Clarke, E.M., Grumberg, O., Jha, S., Lu, Y., Veith, H.: Counterexample-guided abstraction refinement for symbolic model checking. J. ACM 50(5), 752–794 (2003)MathSciNetCrossRefGoogle Scholar
- 15.Huang, J., Li, J.J., Renz, J.: Decomposition and tractability in qualitative spatial and temporal reasoning. Artif. Intell. 195, 140–164 (2013)MathSciNetCrossRefGoogle Scholar
- 16.Brummayer, R., Biere, A.: Effective bit-width and under-approximation. In: Moreno-Díaz, R., Pichler, F., Quesada-Arencibia, A. (eds.) EUROCAST 2009. LNCS, vol. 5717, pp. 304–311. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-04772-5_40CrossRefGoogle Scholar
- 17.Seipp, J., Helmert, M.: Counterexample-guided cartesian abstraction refinement. In: Borrajo, D., et al. (eds.) Proceedings of ICAPS 2013. AAAI (2013)Google Scholar
- 18.Soh, T., Le Berre, D., Roussel, S., Banbara, M., Tamura, N.: Incremental SAT-based method with native boolean cardinality handling for the hamiltonian cycle problem. In: Fermé, E., Leite, J. (eds.) JELIA 2014. LNCS (LNAI), vol. 8761, pp. 684–693. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-11558-0_52CrossRefGoogle Scholar
- 19.Janota, M., Klieber, W., Marques-Silva, J., Clarke, E.M.: Solving QBF with counterexample guided refinement. Artif. Intell. 234, 1–24 (2016)MathSciNetCrossRefGoogle Scholar
- 20.Pulina, L.: The ninth QBF solvers evaluation - preliminary report. In: Lonsing, F., Seidl, M. (eds.) Proceedings of QBF@SAT 2016, CEUR Workshop Proceedings, vol. 1719. CEUR-WS.org (2016)Google Scholar
- 21.Hooker, J.N.: Logic-based methods for optimization. In: Borning, A. (ed.) PPCP 1994. LNCS, vol. 874, pp. 336–349. Springer, Heidelberg (1994). https://doi.org/10.1007/3-540-58601-6_111CrossRefGoogle Scholar
- 22.Chu, Y., Xia, Q.: A hybrid algorithm for a class of resource constrained scheduling problems. In: Barták, R., Milano, M. (eds.) CPAIOR 2005. LNCS, vol. 3524, pp. 110–124. Springer, Heidelberg (2005). https://doi.org/10.1007/11493853_10CrossRefGoogle Scholar
- 23.Hooker, J.N.: A hybrid method for the planning and scheduling. Constraints 10(4), 385–401 (2005)MathSciNetCrossRefGoogle Scholar
- 24.Tran, T.T., Beck, J.C.: Logic-based benders decomposition for alternative resource scheduling with sequence dependent setups. In: Proceedings of ECAI 2012 (2012)Google Scholar
- 25.de Moura, L., Rueß, H., Sorea, M.: Lazy theorem proving for bounded model checking over infinite domains. In: Voronkov, A. (ed.) CADE 2002. LNCS (LNAI), vol. 2392, pp. 438–455. Springer, Heidelberg (2002). https://doi.org/10.1007/3-540-45620-1_35CrossRefGoogle Scholar
- 26.Ji, X., Ma, F.: An efficient lazy SMT solver for nonlinear numerical constraints. In: Proceedings of WETICE 2012 (2012)Google Scholar
- 27.Renz, J.: A canonical model of the region connection calculus. JANCL 12, 469–494 (2002)MathSciNetzbMATHGoogle Scholar
- 28.Renz, J., Ligozat, G.: Weak composition for qualitative spatial and temporal reasoning. In: van Beek, P. (ed.) CP 2005. LNCS, vol. 3709, pp. 534–548. Springer, Heidelberg (2005). https://doi.org/10.1007/11564751_40CrossRefzbMATHGoogle Scholar
- 29.Li, S., Ying, M.: Region connection calculus: its models and composition table. Artif. Intell. 145, 121–146 (2003)MathSciNetCrossRefGoogle Scholar
- 30.Long, Z., Schockaert, S., Li, S.: Encoding large RCC8 scenarios using rectangular pseudo-solutions. In: Proceedings of KR 2016 (2016)Google Scholar
- 31.Tseitin, G.S.: On the complexity of derivation in propositional calculus. In: Siekmann, J.H., Wrightson, G. (eds.) Automation of Reasoning, pp. 466–483. Springer, Heidelberg (1983). https://doi.org/10.1007/978-3-642-81955-1_28CrossRefGoogle Scholar
- 32.Lagniez, J.M., Le Berre, D., de Lima, T., Montmirail, V.: A recursive shortcut for CEGAR: application to the modal logic K satisfiability problem. In: Proceedings of IJCAI 2017 (2017)Google Scholar
- 33.Long, Z.: Qualitative spatial and temporal representation and reasoning: efficiency in time and space. Ph.D. thesis, Faculty of Engineering and Information Technology, University of Technology Sydney (UTS), January 2017Google Scholar
- 34.Savicky, P., Vomlel, J.: Triangulation heuristics for BN2O networks. In: Sossai, C., Chemello, G. (eds.) ECSQARU 2009. LNCS (LNAI), vol. 5590, pp. 566–577. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-02906-6_49CrossRefGoogle Scholar
- 35.Sioutis, M., Koubarakis, M.: Consistency of chordal RCC-8 networks. In: Proceedings of ICTAI 2012. IEEE Computer Society (2012)Google Scholar
- 36.Dechter, R., Meiri, I., Pearl, J.: Temporal constraint networks. Artif. Intell. 49(1–3), 61–95 (1991)MathSciNetCrossRefGoogle Scholar
- 37.Long, Z., Sioutis, M., Li, S.: Efficient path consistency algorithm for large qualitative constraint networks. In: Proceedings of IJCAI 2016 (2016)Google Scholar
- 38.Sioutis, M., Long, Z., Li, S.: Leveraging variable elimination for efficiently reasoning about qualitative constraints. Int. J. Artif. Intell. Tools (2018, in press)Google Scholar
- 39.Audemard, G., Lagniez, J.-M., Simon, L.: Improving glucose for incremental SAT solving with assumptions: application to MUS extraction. In: Järvisalo, M., Van Gelder, A. (eds.) SAT 2013. LNCS, vol. 7962, pp. 309–317. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-39071-5_23CrossRefzbMATHGoogle Scholar
- 40.Eén, N., Sörensson, N.: An extensible SAT-solver. In: Giunchiglia, E., Tacchella, A. (eds.) SAT 2003. LNCS, vol. 2919, pp. 502–518. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-24605-3_37CrossRefGoogle Scholar
- 41.Westphal, M., Wölfl, S., Gantner, Z.: GQR: a fast solver for binary qualitative constraint networks. In: Proceedings of the AAAI Spring Symposium. AAAI (2009)Google Scholar
- 42.Renz, J., Nebel, B.: Efficient methods for qualitative spatial reasoning. J. Artif. Intell. Res. 15, 289–318 (2001)MathSciNetCrossRefGoogle Scholar
- 43.Sioutis, M., Condotta, J.-F.: Tackling large qualitative spatial networks of scale-free-like structure. In: Likas, A., Blekas, K., Kalles, D. (eds.) SETN 2014. LNCS (LNAI), vol. 8445, pp. 178–191. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-07064-3_15CrossRefGoogle Scholar
- 44.Sioutis, M., Condotta, J., Koubarakis, M.: An efficient approach for tackling large real world qualitative spatial networks. Int. J. Artif. Intell. Tools 25, 1–33 (2016)CrossRefGoogle Scholar
- 45.Barabási, A.L., Albert, R.: Emergence of scaling in random networks. Science 286(5439), 509–512 (1999)MathSciNetCrossRefGoogle Scholar