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Granular Geometry

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Towards the Future of Fuzzy Logic

Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 325))

Abstract

Many approaches have been proposed in the fuzzy logic research community to fuzzifying classical geometries. From the field of geographic information science (GIScience) arises the need for yet another approach, where geometric points and lines have granularity: Instead of being “infinitely precise”, points and lines can have size. With the introduction of size as an additional parameter, the classical bivalent geometric predicates such as equality, incidence, parallelity or duality become graduated, i.e., fuzzy. The chapter introduces the Granular Geometry Framework (GGF) as an approach to establishing axiomatic theories of geometries that allow for sound, i.e., reliable, geometric reasoning with points and lines that have size. Following Lakoff’s and Núñez’ cognitive science of mathematics, the proposed framework is built upon the central assumption that classical geometry is an idealized abstraction of geometric relations between granular entities in the real world. In a granular world, an ideal classical geometric statement is sometimes wrong, but can be “more or less true”, depending on the relative sizes and distances of the involved granular points and lines. The GGF augments every classical geometric axiom with a degree of similarity to the truth that indicates its reliability in the presence of granularity. The resulting fuzzy set of axioms is called a granular geometry, if all truthlikeness degrees are greater than zero. As a background logic, Łukasiewicz Fuzzy Logic with Evaluated Syntax is used, and its deduction apparatus allows for deducing the reliability of derived statements. The GGF assigns truthlikeness degrees to axioms in order to embed information about the intended granular model of the world in the syntax of the logical theory. As a result, a granular geometry in the sense of the framework is sound by design. The GGF allows for interpreting positional granules by different modalities of uncertainty (e.g. possibilistic or veristic). We elaborate the framework for possibilistic positional granules and exemplify it’s application using the equality axioms and Euclid’s First Postulate.

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Notes

  1. 1.

    An well-known example for the use of crowed-sourced geographic data in participatory mapping is OpenStreetMap (www.openstreetmap.org).

  2. 2.

    cf., e.g., [48].

  3. 3.

    www.openstreetmap.org.

  4. 4.

    www.apple.com/ios/siri.

  5. 5.

    cf., e.g., [11, 36, 60–63].

  6. 6.

    cf., e.g., [10].

  7. 7.

    cf. [58].

  8. 8.

    cf, e.g., [27].

  9. 9.

    Axioms are often universally quantified. In Łukasiewicz predicate logic, the universal quantifier \(\forall \) is interpreted as the infimum operator. As a result, the truthlikeness degree of a universally quantified axiom \(\forall x. [Statement(x)]\) is the infimum over all truthlikeness degrees of all instances \(Statement(x)\), i.e., the worst case truthlikeness degree. Depending on the intended interpretation, this may result to zero.

  10. 10.

    For details on signed formulas in \(FL_{ev}\)(Ł\(\forall \)) see [41].

  11. 11.

    cf., e.g., [30].

  12. 12.

    More specifically, \(\phi \) is a crisp relation. Yet, in principle, the Granular Geometry Framework can be extended to consider not only crisp position granules \(P\), but also fuzzy position granules. In the GIS community, fuzzy position granules are discussed under the name “vague regions”, cf., e.g. [10].

References

  1. Amelunxen, C.: An approach to geocoding based on volunteered spatial data. In: Zipf, A., Behnke, K., Hillen, F., Schäfermeyer, J. (eds.) Geoinformatik 2010. Die Welt im Netz, pp. 7–20 (2010)

    Google Scholar 

  2. Bennett, B.: A categorical axiomatisation of region-based geometry. Fundamenta Informaticae 46, 145–158 (2001)

    MATH  MathSciNet  Google Scholar 

  3. Bennett, B., Cohn, A.G., Torrini, P., Hazarika, S.M.: Describing rigid body motions in a qualitative theory of spatial regions. In: Kautz, H.A., Porter, B. (eds.) Proceedings of AAAI-2000, pp. 503–509 (2000)

    Google Scholar 

  4. Bennett, B., Cohn, A.G., Torrini, P., Hazarika, S.M.: A foundation for region-based qualitative geometry. In: Horn, W. (ed.) Proceedings of ECAI-2000, pp. 204–208 (2000)

    Google Scholar 

  5. Bennett, B., Cohn, A.G., Torrini, P., Hazarika, S.M.: Region-based qualitative geometry. Research report series, University of Leeds, School of Computer Studies (2000)

    Google Scholar 

  6. Biacino, L., Gerla, G.: Connection structures: Grzegorczyk’s and Whitehead’s definitions of points. Notre Dame J. Form. Log. 37, 431–439 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  7. Biacino, L., Gerla, G.: Logics with approximate premises. Int. J. Intell. Syst. 13, 1–10 (1998)

    Article  MATH  Google Scholar 

  8. Buckley, J., Eslami, E.: Fuzzy plane geometry I: points and lines. Fuzzy Sets Syst. 86, 179–187 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  9. Buckley, J., Eslami, E.: Fuzzy plane geometry II: circles and polygons. Fuzzy Sets Styst. 87, 79–85 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  10. Burrough, P.A., Frank, A.U. (eds.): Geographic Objects with Indeterminate Boundaries. GISDATA. Taylor & Francis, London (1996)

    Google Scholar 

  11. Buyong, T.B., Frank, A.U., Kuhn, W.: A conceptual model of measurement-based multipurpose cadastral systems. J. Urban Reg. Inf. Syst. Assoc. URISA 3(2), 35–49 (1991)

    Google Scholar 

  12. Cheng, S.-C., Mordeson, J.N.: Fuzzy spheres. In: Annual Meeting of the North American Fuzzy information Processing Society (NAFIPS’97). Fuzzy Information Processing Society (1997)

    Google Scholar 

  13. Clementini, E.: A model for uncertain lines. J. Vis. Lang. Comput. 16(4), 271–288 (2005)

    Article  Google Scholar 

  14. Dilo, A.: Representation of and reasoning with vagueness in spatial information—a system for handling vague objects. Ph.D. thesis, C.T. de Wit Graduate School for Production Ecology and Resource Conservation (PE&RC) in Wageningen University, The Netherlands (2006)

    Google Scholar 

  15. Douglas, D., Peucker, T.: Algorithms for the reduction of the number of points required to represent a digitized line or its caricature. Can. Cartogr. 10, 112–122 (1973)

    Article  Google Scholar 

  16. Dutta, S.: Qualitative Spatial Reasoning: A Semi-quantitative Approach Using Fuzzy Logic, pp. 345–364. Springer, Heidelberg (1990)

    Google Scholar 

  17. Gerla, G.: Pointless metric spaces. J. Symb. Log. 55(1), 207–219 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  18. Gerla, G.: Pointless geometries. In: Buekenhout, F. (ed.) Handbook of Incidence Geometry. Buildings and Foundations, Chapter 18, 1st edn., pp. 1015–1031. Elsevier, North-Holland (1995)

    Google Scholar 

  19. Gerla, G.: Fuzzy Logic: Mathematical Tools for Approximate Reasoning. Trends in Logic, vol. 11. Kluwer Academic Publishers, Dordrecht (2001)

    Google Scholar 

  20. Gerla, G.: Approximate similarities and Poincaré paradox. Notre Dame J. Form. Log. 49(2), 203–226 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  21. Gerla, G., Miranda, A.: Graded inclusion and point-free geometry. Int. J. Appl. Math. 11(1), 63–81 (2004)

    MATH  MathSciNet  Google Scholar 

  22. Gerla, G., Volpe, R.: Geometry without points. Am. Math. Mon. 92, 707–711 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  23. Ghilani, C.D., Wolf, P.R.: Adjustment Computation—Spatial Data Analysis, 4th edn. Wiley, Hoboken (2006)

    Book  Google Scholar 

  24. Godo, L., Rodríguez, R.O.: Logical approaches to fuzzy similarity-based reasoning: an overview. In: Riccia, G.D., Dubois, D., Kruse, R., Lenz, H.-J. (eds.) Preferences and Similarities, chapter 2, vol. 504, pp. 75–128. Springer, New York (2008)

    Chapter  Google Scholar 

  25. Goodchild, M.: Citizens as sensors: the world of volunteered geography. GeoJournal 69(4), 211–221 (2007)

    Article  Google Scholar 

  26. Gupta, K.C., Ray, S.: Fuzzy plane projective geometry. Fuzzy Sets Syst. 54, 191–206 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  27. Hájek, P.: Metamathematics of Fuzzy Logic. Trends in Logic, vol. 4. Kluwer Academic Publishers, Dordrecht (1998)

    MATH  Google Scholar 

  28. Haklay, M.: How good is volunteered geographical information? A comparative study of OpenStreetMap and ordnance survey datasets. Environ. Plan. B: Plan. Des. 37(4), 682–703 (2010)

    Article  Google Scholar 

  29. Haklay, M.M., Basiouka, S., Antoniou, V., Ather, A.: How many volunteers does it take to map an area well? The validity of Linus law to volunteered geographic information. Cartogr. J. 47, 315–322 (2010)

    Article  Google Scholar 

  30. Hartshorne, R.: Geometry: Euclid and Beyond. Undergraduate Texts in Mathematics. Springer, New York (2000)

    Book  Google Scholar 

  31. Hobbs, J.R.: Granularity. In: Proceedings of Ninth International Joint Conference on Artificial Intelligence, pp. 432–435. Morgan Kaufmann, Los Angeles (1985)

    Google Scholar 

  32. Katz, M.: Inexact geometry. Notre Dame J. Form. Log. 21, 521–535 (1980)

    Article  MATH  Google Scholar 

  33. Kuijken, L.: Fuzzy projective geometries. In: Mayor, G., Jaume, J.S. (eds.) Proceedings of the 1999 EUSFLAT-ESTYLF Joint Conference, Palma de Mallorca, Spain (1999)

    Google Scholar 

  34. Kuijken, L., Maldeghem, H.V.: Fibered geometries. Discret. Math. 255(1–3), 259–274 (2002)

    Article  MATH  Google Scholar 

  35. Lakoff, G., Núñez, R.E.: Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being. Basic Books, New York (2000)

    Google Scholar 

  36. Leung, Y., Ma, J.-H., Goodchild, M.F.: A general framework for error analysis in measurement-based GIS, parts 1–4. J. Geogr. Syst. 6(4), 323–428 (2004)

    Article  Google Scholar 

  37. Liu, H., Coghill, G.M.: Can we do trigonometry qualitatively? In: Proceedings of the 19th International Workshop on Qualitative Reasoning QR-05 (2005)

    Google Scholar 

  38. Lobačevskij, N.I.: Novye načala geometry s polnoj teoriej parallel’nyh (New principles of geometry with complete theory of parallels). Polnoe sobranie socinenij 2 (1835)

    Google Scholar 

  39. Menger, K.: Propabilistic geometry. In: PNAS Proceedings of the National Academy of the United States of America, vol. 37, pp. 226–229 (1951)

    Google Scholar 

  40. Menger, K.: Selected Papers in Logic and Foundations, Didactics, Economics. Vienna Circle Collection, vol. 10. D. Reidel Publishing Company, Dordrecht (1979)

    Book  MATH  Google Scholar 

  41. Novák, V., Perfilieva, I., Močkoř, J.: Mathematical Principles of Fuzzy Logic. Kluwer Academic Publishers, Boston (1999)

    Book  MATH  Google Scholar 

  42. Pavelka, J.: On fuzzy logic I, II, III. Zeitschrift für Mathematik, Logik und Grundlagen der Mathematik 25, 45–52, 119–134, 447–464 (1979)

    Google Scholar 

  43. Perkal, J.: On epsilon length. Bulletin de l’Academie Polonaise des Sciences 4, 399–403 (1956)

    MATH  MathSciNet  Google Scholar 

  44. Perkal, J.: On the length of empirical curves. discussion paper no. 10. Michigan Inter-University Community of Mathematical Geographers, Ann Arbor (1966)

    Google Scholar 

  45. Peucker, T.K.: Chapter A theory of the cartographic line. International Yearbook of Cartography, vol. 16, pp. 134–143 (1975)

    Google Scholar 

  46. Poincaré, H.: Science and Hypothesis. Walter Scott Publishing, New York (1905). First published by Flammarion, Paris (1902)

    Google Scholar 

  47. Poston. T.: Fuzzy Geometry. PhD thesis, University of Warwick (1971)

    Google Scholar 

  48. Pullar, D.V.: Consequences of using a tolerance paradigm in spatial overlay. Proc. Auto-Carto 11, 288–296 (1993)

    Google Scholar 

  49. Ramer, U.: An iterative procedure for the polygonal approximation of plane curves. Comput. Graph. Image Process. 1(3), 244–256 (1972)

    Article  Google Scholar 

  50. Roberts, F.S.: Representations of indifference relations. Ph.D. thesis, Stanford University (1968)

    Google Scholar 

  51. Roberts, F.S.: Tolerance geometry. Notre Dame J. Form. Log. 14(1), 68–76 (1973)

    Article  MATH  Google Scholar 

  52. Roberts, F.S., Suppes, P.: Some problems in the geometry of visual perception. Synthese 17, 173–201 (1967)

    Article  MATH  Google Scholar 

  53. Rosenfeld, A.: The diameter of a fuzzy set. Fuzzy Sets Syst. 13(3), 241–246 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  54. Rosenfeld, A.: Fuzzy rectangles. Pattern Recognit. Lett. 11(10), 677–679 (1990)

    Article  MATH  Google Scholar 

  55. Rosenfeld, A.: Fuzzy plane geometry: triangles. Fuzzy Syst. IEEE World Congr. Comput. Intell. 2, 891–893 (1994)

    Google Scholar 

  56. Rosenfeld, A., Haber, S.: The perimeter of a fuzzy set. Pattern Recognit. 18(2), 125–130 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  57. Ruspini, E.: On the semantics of fuzzy logic. Int. J. Approx. Reason. 5, 45–88 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  58. Salesin, D., Stolfi, J., Guibas, L.: Epsilon geometry: building robust algorithms from imprecise computations. In: SCG’89: Proceedings of the Fifth Annual Symposium on Computational Geometry, pp. 208-217. ACM, New York (1989)

    Google Scholar 

  59. Schmidtke, H.: A geometry for places: Representing extension and extended objects. In: Kuhn, W., Worboys, M., Timpf, S. (eds.) Spatial Information Theory: Foundations of Geographic Information Science, pp. 221–238. Springer, Berlin (2003)

    Chapter  Google Scholar 

  60. Shi, W.: A generic statistical approach for modelling error of geometric features in GIS. Int. J. Geogr. Inf. Sci. 12(2), 131–143 (1998)

    Article  Google Scholar 

  61. Shi, W.: Principles of Modeling Uncertainties in Spatial Data and Spatial Analyses. CRC Press Inc., Boca Raton (2009)

    Book  Google Scholar 

  62. Shi, W., Cheung, C.K., Zhu, C.: Modelling error propagation in vector-based buffer analysis. Int. J. Geogr. Inf. Sci. 17(3), 251–271 (2003)

    Article  Google Scholar 

  63. Shi, W., Liu, W.: A stochastic process-based model for the positional error of line segments in gis. Int. J. Geogr. Inf. Sci. 14(1), 51–66 (2000)

    Article  MathSciNet  Google Scholar 

  64. Suppes, P., Krantz, D.H., Luce, D.R., Tversky, A.: Foundations of Measurement Volume I–III. Dover Publications, New York (2007). Originally published by Academic Press (1989)

    Google Scholar 

  65. Suppes, P., Krantz, D.H., Luce, D.R., Tversky, A.: Foundations of Measurement Volume II: Geometrical, Threshold, and Probabilistic Representations. Dover Publications, New York (2007). Originally published by Academic Press (1989)

    Google Scholar 

  66. Tarski, A.: Methodology of deductive sciences. Logics, Semantics, Metamathematics. Oxford University Press, Oxford (1956)

    Google Scholar 

  67. Tarski, A.: What is elementary geometry? In: Henkin, L., Suppes, P., Tarski, A. (eds.) The Axiomatic Method. With Special Reference to Geometry and Physics. Proceedings of an International Symposium Held at the University of California, Berkeley, Studies in Logic and the Foundations of Mathematics, pp. 16–29. North-Holland, Amsterdam (1958)

    Google Scholar 

  68. Topaloglou, T.: First order theories of approximate space. In: AAAI-94 Workshop on Spatial and Temporal Reasoning, pp. 47–53 (1994)

    Google Scholar 

  69. Trillas, E., Valverde, L.: An inquiry into indistinguishability operators. In: Skala, H., Termini, S., Trillas, E. (eds.) Aspects of Vagueness. Theory and Decision Library, vol. 39, pp. 231–256. Springer, Netherlands (1984)

    Chapter  Google Scholar 

  70. Vakarelov, D.: Region-based theory of space: Algebras of regions, representation theory, and logics. In: Gabbay, D., Goncharov, S., Zakharyaschev, M. (eds.) Mathematical Problems from Applied Logic II. Logics for the XXIst Century. International Mathematical Series, vol. 5. Springer, New York (2007)

    Google Scholar 

  71. Veelaert, P.: Geometric constructions in the digital plane. J. Math. Imaging Vis. 11, 99–118 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  72. Wilke, G.: Approximate geometric reasoning with extended geographic objects. In: Proceedings of the ISPRS COST-Workshop on Quality, Scale and Analysis Aspects of City Models, Lund, Sweden (2009)

    Google Scholar 

  73. Wilke, G.: Towards approximate tolerance geometry for GIS—a framework for formalizing sound geometric reasoning under positional tolerance. PhD thesis, Vienna University of Technology (2012)

    Google Scholar 

  74. Wilke, G.: Equality in approximate tolerance geometry. In: Kacprzyk, J. (ed.) Advances in Intelligent Systems and Computing. Springer, New York (2014)

    Google Scholar 

  75. Wilke, G.: Fuzzy logic with evaluated syntax for sound geometric reasoning with geographic information. In: Proceedings of the 2014 IEEE Conference on Norbert Wiener in the 21st Century, Boston MA, USA, 24–26 June 2014, incorporating the Proceedings of the Annual Conference of the North American Fuzzy Information Processing Society: NAFIPS 2014 (2014)

    Google Scholar 

  76. Wilke, G., Frank, A.: On equality of points and lines. In: GIScience 2010, Zürich, Switzerland (2010)

    Google Scholar 

  77. Wilke, G., Frank, A.U.: Tolerance geometry: Euclid’s first postulate for points and lines with extension. In: 18th ACM SIGSPATIAL International Symposium on Advances in Geographic Information Systems, ACM-GIS 2010, 3–5 November 2010, San Jose, CA, USA, Proceedings, pp. 162–171(2010)

    Google Scholar 

  78. Ying, M.: A logic for approximate reasoning. J. Symb. Log. 59(3), 830–837 (1994)

    Article  MATH  Google Scholar 

  79. Zadeh, L.A.: Fuzzy sets. Inf. Control 8(3), 338–353 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  80. Zadeh, L.A.: Similarity relations and fuzzy orderings. Inf. Sci. 3, 177–200 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  81. Zadeh, L.A.: Generalized theory of uncertainty (GTU)—principal concepts and ideas. Comput. Stat. Data Anal. 51, 15–46 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  82. Zadeh, L.A.: A note on z-numbers. Inf. Sci. 181(14), 2923–2932 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  83. Zadeh, L.A.: Toward a restriction-centered theory of truth and meaning (RCT). Inf. Sci. 248, 1–14 (2013)

    Article  MathSciNet  Google Scholar 

  84. Zandbergen, P.A.: Geocoding quality and implications for spatial analysis. Geogr. Compass 3(2), 647–680 (2009)

    Article  Google Scholar 

  85. Zeeman, E.: The topology of the brain and visual perception. Topology of 3-Manifolds and Related Topics (Proceedings of the University of Georgia Institute, 1961), pp. 240–256. Prentice-Hall, Upper Saddle River (1962)

    Google Scholar 

  86. Zulfiqar, N.: A study of the quality of OpenStreetMap.org maps: a comparison of OSM data and ordnance survey data. Master’s thesis, University College London (2008)

    Google Scholar 

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Acknowledgments

Most of the work is part of the author’s PhD thesis and was produced while having been employed as a research assistant at the Research Group Geoinformation at the Vienna University of Technology. It was supported by Scholarships of the Austrian Marshallplan Foundation and the Vienna University of Technology. The author would like to thank Prof. Andrew U. Frank, Dr. John Stell and Prof. Lotfi Zadeh for their continuous support and inspiration.

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  1. University of Applied Sciences and Arts Northwestern Switzerland (FHNW), Bachweg 32, 4600, Olten, Switzerland

    Gwendolin Wilke

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  1. Institut für Geschichte der Medizin, Naturwissenschaft und Technik, Friedrich-Schiller-Universität Jena, Berggasse, Jena, Germany

    Rudolf Seising

  2. Edificio de Investigación, 3ª Planta. C Gonzalo Gutiérrez Quirós S/N, European Centre for Soft Computing, Mieres, Asturias, Spain

    Enric Trillas

  3. Polish Academy of Sciences, Systems Research Institute, Warsaw, Poland

    Janusz Kacprzyk

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Wilke, G. (2015). Granular Geometry. In: Seising, R., Trillas, E., Kacprzyk, J. (eds) Towards the Future of Fuzzy Logic. Studies in Fuzziness and Soft Computing, vol 325. Springer, Cham. https://doi.org/10.1007/978-3-319-18750-1_6

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