A Logic for Spatial Reasoning in the Framework of Rough Mereology

  • Lech PolkowskiEmail author
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10810)


Spatial reasoning concerns a language in which spatial objects are described and argued about. Within the plethora of approaches, we single out the one set in the framework of mereology - the theory of concepts employing the notion of a part as the primitive one. Within mereology, we can choose between the approach based on part as the basic notion or the approach based on the notion of a connection from which the notion of a part is defined. In this work, we choose the former approach modified to the rough mereology version in which the notion of a part becomes ‘fuzzified’ to the notion of a part to a degree. The prevalence of this approach lies in the fact that it does allow for quantitative assessment of relations among spatial objects in distinction to only qualitative evaluation of those relations in case of other mereology based approaches.

In this work, we introduce sections on mereology based reasoning, covering part and connection based variants as well as rough mereology in order to provide the Reader with the conceptual environment we work in. We recapitulate shortly those approaches along with based on them methods for spatial reasoning. We then introduce the mereological approach in the topological context used in spatial reasoning, i.e., in collections of regular open or regular closed sets known to form complete Boolean algebras. In this environment, we create a logic for reasoning about parts and degrees of inclusion based on an abstract notion of a mass which generalizes geometric measure of area or volume and extends in the abstract manner the Lukasiewicz logical rendering of probability calculus. We give some applications, notably, we extend the relation of betweenness applied by us earlier in robot navigation and we give it the abstract characterization.


  1. 1.
    Agah, A.: Robot teams, human workgroups and animal sociobiology. A review of research on natural and artificial multi-agent autonomous systems. Adv. Robot. 10, 523–545 (1997)CrossRefGoogle Scholar
  2. 2.
    van Benthem, J.: The Logic of Time. Reidel. Dordrecht (1983)Google Scholar
  3. 3.
    Cao, Y.U., Fukunaga, A.S., Kahng, A.B.: Cooperative mobile robotics: antecedents and directions. Auton. Robot. 4, 7–27 (1997)CrossRefGoogle Scholar
  4. 4.
    Casati, R., Varzi, A.C.: Parts and Places. The Structures of Spatial Representation. MIT Press, Cambridge (1999)Google Scholar
  5. 5.
    Clarke, B.L.: A calculus of individuals based on connection. Notre Dame J. Form. Log. 22(2), 204–218 (1981)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Cohn, A.G.: Calculi for qualitative spatial reasoning. In: Calmet, J., Campbell, J.A., Pfalzgraf, J. (eds.) AISMC 1996. LNCS, vol. 1138, pp. 124–143. Springer, Heidelberg (1996). Scholar
  7. 7.
    Cohn, A.G., Gooday, J.M., Bennett, B., Gotts, N.M.: A logical approach to representing and reasoning about space. In: Calmet, J., Campbell, J.A., Pfalzgraf, J. (eds.) Artificial Intelligence and Symbolic Mathematical Computation. Lecture Notes in Computer Science, vol. 1138, pp. 124–143. Springer, Heidelberg (1996). Scholar
  8. 8.
    Cohn, A.G., Gotts, N.M.: Representing spatial vagueness: a mereological approach. In: Proceedings of the 5th International Conference on Principles of Knowledge Representation and Reasoning, KR 1996, pp. 230–241. Morgan Kaufmann, San Francisco (1996)Google Scholar
  9. 9.
    Cohn, A.G., Randell, D., Cui, Z., Bennett, B.: Qualitative spatial reasoning and representation. In: Carrete, N., Singh, M. (eds.) Qualitative Reasoning and Decision Technologies, Barcelona, pp. 513–522 (1993)Google Scholar
  10. 10.
    Cohn, A.G., Varzi, A.C.: Connections relations in mereotopology. In: Prade H. (ed.) Proceedings of ECAI 1998 13th European Conference on Artificial Intelligence, pp. 150–154. Wiley, Chichester (1998)Google Scholar
  11. 11.
    Egenhofer, M.J.: Reasoning about binary topological relations. In: Gunther, O., Schek, H.(eds.) Proceedings of Advances in Spatial Databases, SSD 1991, Berlin, pp. 143–160 (1991)Google Scholar
  12. 12.
    Gotts, N.M., Gooday, J.M., Cohn, A.G.: A connection based approach to commonsense topological description and reasoning. Monist 79(1), 51–75 (1996)CrossRefGoogle Scholar
  13. 13.
    Gotts, N.M., Cohn, A.G.: A mereological approach to representing spatial vagueness. In: Working papers. The Ninth International Workshop on Qualitative Reasoning, QR 1995 (1995)Google Scholar
  14. 14.
    H\(\acute{a}\)jek, P.: Metamathematics of Fuzzy Logic. Kluwer, Dordrecht (1998)Google Scholar
  15. 15.
    de Laguna, T.: Point, line and surface as sets of solids. J. Philos. 19, 449–461 (1922)CrossRefGoogle Scholar
  16. 16.
    Leśniewski, S.: Foundations of the General Theory of Sets (in Polish). Moscow (1916)Google Scholar
  17. 17.
    Ling, C.-H.: Representation of associative functions. Publ. Math. Debr. 12, 189–212 (1965)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Łukasiewicz, J.: Die Logischen Grundlagen der Wahrscheinlichkeitsrechnung. Kraków, 1913. Cf. Borkowski, L. (ed.) Selected Works. North Holland-PWN, Amsterdam-Warszawa, pp. 16–63 (1970)Google Scholar
  19. 19.
    Matarić M.: Interaction and intelligent behavior. Ph.D. dissertation. MIT EECS Department (1994)Google Scholar
  20. 20.
    Nicolas, D.: The logic of mass expressions. In: Stanford Enc. Phil.
  21. 21.
    Ośmiałowski, P.: On path planning for mobile robots: introducing the mereological potential field method in the framework of mereological spatial reasoning. J. Autom. Mob. Robot. Intell. Syst. (JAMRIS) 3(2), 24–33 (2009)Google Scholar
  22. 22.
    Osmialowski P.: Planning and navigation for mobile autonomous robots. Ph.D. dissertation. Polkowski, L. Supervisor, Polish-Japanese Academy IT. PJAIT Publishers, Warszawa (2011)Google Scholar
  23. 23.
    Pawlak, Z.: Rough Sets: Theoretical Aspects of Data Analysis. Kluwer, Dordrecht (1992)Google Scholar
  24. 24.
    O’smiaıowski, P., Polkowski, L.: Spatial reasoning based on rough mereology: a notion of a robot formation and path planning problem for formations of mobile autonomous robots. In: Peters, J.F., Skowron, A., Słowiński, R., Lingras, P., Miao, D., Tsumoto, S. (eds.) Transactions on Rough Sets XII. LNCS, vol. 6190, pp. 143–169. Springer, Heidelberg (2010). Scholar
  25. 25.
    Polkowski, L.: Rough Sets. Mathematical Foundations. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  26. 26.
    Polkowski, L.: A rough set paradigm for unifying rough set theory and fuzzy set theory. Fundam. Inform. 54, 67–88 (2003)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Polkowski, L.: Toward rough set foundations. Mereological approach. In: Tsumoto, S., Słowiński, R., Komorowski, J., Grzymała-Busse, J.W. (eds.) RSCTC 2004. LNCS (LNAI), vol. 3066, pp. 8–25. Springer, Heidelberg (2004). Scholar
  28. 28.
    Polkowski, L.: Formal granular calculi based on rough inclusions. In: Proceedings of IEEE 2005 Conference on Granular Computing GrC 2005, Beijing, China, pp. 57–62. IEEE Press (2005)Google Scholar
  29. 29.
    Polkowski, L.: Granulation of knowledge in decision systems: the approach based on rough inclusions. The method and its applications. In: Kryszkiewicz, M., Peters, J.F., Rybinski, H., Skowron, A. (eds.) RSEISP 2007. LNCS (LNAI), vol. 4585, pp. 69–79. Springer, Heidelberg (2007). Scholar
  30. 30.
    Polkowski, L.: A unified approach to granulation of knowledge and granular computing based on rough mereology: a survey. In: Pedrycz, W., Skowron, A., Kreinovich, V. (eds.) Handbook of Granular Computing, pp. 375–400. Wiley, Chichester (2008)Google Scholar
  31. 31.
    Polkowski, L.: Granulation of knowledge: similarity based approach in information and decision systems. In: Meyers, R.A. (ed.) Springer Encyclopedia of Complexity and System Sciences, pp. 1464–1487. Springer, Heidelberg (2009). Scholar
  32. 32.
    Polkowski, L.: Approaimate Reasoning by Parts. An Introduction to Rough Mereology. Springer, Heidelberg (2011). Scholar
  33. 33.
    Polkowski, L.: Mereology in engineering and computer science. In: Calosi, C., Graziani, P. (eds.) Mereology and the Sciences. SL, vol. 371, pp. 217–291. Springer, Cham (2014). Scholar
  34. 34.
    Polkowski, L.: From Leśniewski, Łukasiewicz, Tarski to Pawlak: enriching rough set based data analysis. A retrospective survey. Fundam. Inform. 154(1–4), 343–358 (2017)CrossRefGoogle Scholar
  35. 35.
    Polkowski, L.: The counterpart to the Bayes theorem in mass-based rough mereology. In: Proceedings CS&P 2018. Humboldt Universität zu Berlin, September 2018. Informatik-Berichte series. Informatik-Bericht 248, pp. 47–56 (2018).
  36. 36.
    Polkowski, L., Ośmiałowski, P.: Spatial reasoning with applications to mobile robotics. In: Aing-Jiang, J. (ed.): Mobile Robots Motion Planning. New Challenges. I-Tech, Vienna, pp. 433–453 (2008)zbMATHGoogle Scholar
  37. 37.
    Polkowski, L., Ośmiałowski, P.: Navigation for mobile autonomous robots and their formations: an application of spatial reasoning induced from rough mereological geometry. In: Barrera, A. (ed.) Mobile Robots Navigation, pp. 339–354. In Tech, Zagreb (2010)zbMATHGoogle Scholar
  38. 38.
    Reynolds, C.: Flocks, herds and schools. A distributed behavioral model. Comput. Graph. 21(4), 25–34 (1987)CrossRefGoogle Scholar
  39. 39.
    Polkowski, L., Skowron, A.: Rough mereology. In: Raś, Z.W., Zemankova, M. (eds.) ISMIS 1994. LNCS, vol. 869, pp. 85–94. Springer, Heidelberg (1994). Scholar
  40. 40.
    Polkowski, L., Skowron, A.: Rough mereology: a new paradigm for approaimate reasoning. Int. J. Approx. Reason. 15(4), 333–365 (1997)CrossRefGoogle Scholar
  41. 41.
    Randell D., Cui Z., Cohn A. G.: A spatial logic based on regions and connection. In: Proceedings of the 3rd International Conference on Principles of Knowledge Representation and Reasoning KR 1992. Morgan Kaufmann, San Mateo, pp. 165–176 (1992)Google Scholar
  42. 42.
    Tarski, A.: Zur Grundlegen der Booleschen Algebra I. Fund. Math. 24, 177–198 (1935)CrossRefGoogle Scholar
  43. 43.
    Tarski, A., Givant, S.: Symbolic logic. Bull 5(2), 175–214 (1959)Google Scholar
  44. 44.
    Whitehead, A.N.: La th\(\acute{e}\)orie relationniste de l’espace. Revue de M\(\acute{e}\)taphysique et de Morale 23, 423–454 (1916)Google Scholar
  45. 45.
    Whitehead, A.N.: An Enquiry Concerning the Principles of Natural Knowledge. Cambridge University Press, Cambridge (1919)zbMATHGoogle Scholar
  46. 46.
    Whitehead, A.N.: The Concept of Nature. Cambridge University Press, Cambridge (1920)zbMATHGoogle Scholar
  47. 47.
    Whitehead, A.N.: Process and Reality: An Essay in Cosmology. Macmillan, New York (1929)zbMATHGoogle Scholar
  48. 48.
    Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338–353 (1965)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Mathematics and Computer Science, Chair of Mathematical Methods in Computer ScienceUniversity of Warmia and Mazury in OlsztynOlsztynPoland

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