Abstract
In the psychology of reasoning, spatial reasoning capacities are often explained by postulating models in the mind. According to the Space To Reason theory, these models only consist of the spatial qualities of the considered situation, such as the topology or the relative orientation, without containing any quantitative measures. It turns out that a field of computer science, called Qualitative Spatial Reasoning, is entirely dedicated to formalizing such qualitative representations. Although the formalism of qualitative spatial reasoning has already been used in the space to reason theory, it has not yet entirely been exploited. Indeed, it can also be used to formally characterize spatial models and account for our reasoning on them. To exemplify this claim, two typical problems of spatial reasoning are exhaustively analyzed through the framework of qualitative constraint networks (QCN). It is shown that for both problems every aspect can be formally captured, as for example the integration of premises into one single model, or the prediction of alternative models. Therefore, this framework represents an opportunity to completely formalize the space to reason theory and, what is more, diversify the type of spatial reasoning accounted by it. The most substantial element of this formal translation is that a spatial model and a satisfiable atomic QCN - a scenario - turn out to have exactly the same conditions of possibility.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
It is worth noticing that such alternative spatial models are different from the alternative models of the classical Mental Model theory [8]. Alternative models in the classical mental model theory are obtained by negating some clauses of the problem, whereas alternative spatial models are all contained in the same valuation of the problem, namely, the one where all the premises are true. In terms of propositional logic, alternative spatial models are all contained in the same first row of the classical truth-tables. This distinction in the nature of alternative models between the two theories sometimes does not seem to be fully taken into account in the study of spatial reasoning [15].
- 2.
The constraint C(u, v) is indistinctly noted by its infix notation (u C v).
- 3.
The graph of a qualitative constraint network is often denoted \(G(\mathcal {N})\), but since this additional notation does not bring any relevant distinction, the graph of a QCN will be indistinctly designated by its corresponding QCN symbol \(\mathcal {N}\).
- 4.
SparQ contains many common calculi of qualitative spatial reasoning, and the procedures of this article can be easily checked with it. The software is downloadable on the website https://www.uni-bamberg.de/en/sme/research/sparq/.
- 5.
Note that it exists an infinity of solutions for a scenario since it exists an infinity of possible instantiations satisfying the constraints.
References
Allen, J.: Maintaining knowledge about temporal intervals (1983)
Craik, K.W.: The Nature of Explanation. Cambridge University Press, Cambridge (1943)
Dechter, R., et al.: Constraint Processing. Morgan Kaufmann, Burlington (2003)
Dylla, F., et al.: A survey of qualitative spatial and temporal calculi: algebraic and computational properties. ACM Comput. Surv. (CSUR) 50(1), 1–39 (2017)
Dylla, F., Mossakowski, T., Schneider, T., Wolter, D.: Algebraic properties of qualitative spatio-temporal calculi. In: Tenbrink, T., Stell, J., Galton, A., Wood, Z. (eds.) COSIT 2013. LNCS, vol. 8116, pp. 516–536. Springer, Cham (2013). https://doi.org/10.1007/978-3-319-01790-7_28
Johnson-Laird, P.: Imagery, visualization, and thinking. Perception and cognition at century’s end, pp. 441–467 (1998)
Johnson-Laird, P.N.: Mental Models. Harvard University Press (1983)
Khemlani, S.S., Byrne, R.M., Johnson-Laird, P.N.: Facts and possibilities: a model-based theory of sentential reasoning. Cogn. Sci. 42(6), 1887–1924 (2018)
Knauff, M.: Space to Reason: A Spatial Theory of Human Thought. MIT Press, Cambridge (2013)
Knauff, M.: Visualization, reasoning, and rationality. In: Endres, D., Alam, M., Şotropa, D. (eds.) ICCS 2019. LNCS (LNAI), vol. 11530, pp. 3–10. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-23182-8_1
Ligozat, G.: Qualitative Spatial and Temporal Reasoning. Wiley, Hoboken (2013)
Ligozat, G., Renz, J.: What Is a qualitative calculus? a general framework. In: Zhang, C., W. Guesgen, H., Yeap, W.-K. (eds.) PRICAI 2004. LNCS (LNAI), vol. 3157, pp. 53–64. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-28633-2_8
Ligozat, G.É.: Reasoning about cardinal directions. J. Vis. Lang. Comput. 9(1), 23–44 (1998)
Ragni, M., Knauff, M.: A theory and a computational model of spatial reasoning with preferred mental models. Psychol. Rev. 120(3), 561 (2013)
Ragni, M., Sonntag, T., Johnson-Laird, P.N.: Spatial conditionals and illusory inferences. J. Cogn. Psychol. 28(3), 348–365 (2016)
Randell, D.A., Cui, Z., Cohn, A.G.: A spatial logic based on regions and connection. KR 92, 165–176 (1992)
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_40
Sioutis, M.: Algorithmic contributions to qualitative constraint-based spatial and temporal reasoning. Ph.D. thesis, Artois (2017)
Sioutis, M., Long, Z., Li, S.: Efficiently reasoning about qualitative constraints through variable eliminiation. In: Proceedings of the 9th Hellenic Conference on Artificial Intelligenc, pp. 1–10 (2016)
Tarski, A.: On the calculus of relations. J. Symbolic Logic 6(3), 73–89 (1941)
Vilain, M.B., Kautz, H.A.: Constraint propagation algorithms for temporal reasoning. AAAI 86, 377–382 (1986)
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 2006. LNCS (LNAI), vol. 4387, pp. 39–58. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-75666-8_3
Acknowledgments
I would like to thank Romain Bourdoncle, Benjamin Icard and Paul Egré for the interesting discussions. I also thank Ellen Boes and Anne Clarenne for their support. Finally, I thank PSL University (ANR-10-IDEX-0001-02 PSL) and the programs FrontCog (ANR-17-EURE-0017).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
Olivier, F. (2020). A Logical Framework for Spatial Mental Models. In: Šķilters, J., Newcombe, N., Uttal, D. (eds) Spatial Cognition XII. Spatial Cognition 2020. Lecture Notes in Computer Science(), vol 12162. Springer, Cham. https://doi.org/10.1007/978-3-030-57983-8_20
Download citation
DOI: https://doi.org/10.1007/978-3-030-57983-8_20
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-57982-1
Online ISBN: 978-3-030-57983-8
eBook Packages: Computer ScienceComputer Science (R0)