spatial reasoning - university of .spatial reasoning theory and practice academisch proefschrift

Download Spatial Reasoning - University of .Spatial Reasoning Theory and Practice ACADEMISCH PROEFSCHRIFT

Post on 05-Oct-2018

218 views

Category:

Documents

0 download

Embed Size (px)

TRANSCRIPT

  • Spatial Reasoning

    Theory and Practice

    Marco Aiello

  • Spatial Reasoning

    Theory and Practice

  • ILLC Dissertation Series 2002-2

    For further information about ILLC-publications, please contact

    Institute for Logic, Language and ComputationUniversiteit van AmsterdamPlantage Muidergracht 24

    1018 TV Amsterdamphone: +31-20-525 6051

    fax: +31-20-525 5206e-mail: illc@science.uva.nl

    homepage:http://www.illc.uva.nl/

  • Spatial Reasoning

    Theory and Practice

    ACADEMISCH PROEFSCHRIFT

    ter verkrijging van de graad van doctor aan deUniversiteit van Amsterdam

    op gezag van de Rector Magnificusprof.mr. P.F. van der Heijden

    ten overstaan van een door het college voor promoties ingesteldecommissie, in het openbaar te verdedigen in de

    Aula der Universiteitop vrijdag 22 februari 2002, te 12.00 uur

    door

    Marco Aiello

    geboren te Fabriano, Italie.

  • Promotie commissie:

    Promotores:prof.dr J.F.A.K. van Benthemprof.dr ir A.M.W. Smeulders

    Overige leden:prof.dr L. Farinas del Cerro, Universite Paul Sabatier, Frankrijkprof.dr ir F. Giunchiglia, Universita di Trento, Italieprof.dr ir R. Scha, Universiteit van Amsterdam, Nederlanddr M. de Rijke, Universiteit van Amsterdam, Nederlanddr Y. Venema, Universiteit van Amsterdam, Nederland

    Faculteit der Natuurwetenschappen, Wiskunde en InformaticaUniversiteit van AmsterdamNederland

    The research was supported by the Institute for Logic, Language and Computation andby the Informatics Institute of the University of Amsterdam.

    Copyright c 2002 by Marco Aiellohttp://www.aiellom.it

    Cover design and photography by the author.Typeset in pdfLATEX.Printed and bound by Print Partners Ipskamp, Enschede.

    ISBN: 9057760797

  • A Mario e Gigina.

    v

  • CONTENTS

    Acknowledgments xi

    1 Introduction 11.1 Reasoning about space . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Theory and practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

    2 The topo approach: expressiveness 72.1 Basic modal logic of space . . . . . . . . . . . . . . . . . . . . . . . 8

    2.1.1 Topological bisimulation . . . . . . . . . . . . . . . . . . . . 112.1.2 Connections with topology . . . . . . . . . . . . . . . . . . . 132.1.3 Topo-bisimilar reductions . . . . . . . . . . . . . . . . . . . 14

    2.2 Games that compare visual scenes . . . . . . . . . . . . . . . . . . . 142.2.1 Strategies and modal formulas . . . . . . . . . . . . . . . . . 18

    2.3 Logical variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

    3 The topo approach: axiomatics 233.1 Topological spaces and Kripke models . . . . . . . . . . . . . . . . . 23

    3.1.1 The basic connection . . . . . . . . . . . . . . . . . . . . . . 233.1.2 Analogies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

    3.2 General completeness . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2.1 The main argument . . . . . . . . . . . . . . . . . . . . . . . 263.2.2 Topological comments . . . . . . . . . . . . . . . . . . . . . 283.2.3 Finite spaces suffice . . . . . . . . . . . . . . . . . . . . . . 29

    3.3 Completeness on the reals . . . . . . . . . . . . . . . . . . . . . . . 313.3.1 Cantorization . . . . . . . . . . . . . . . . . . . . . . . . . . 323.3.2 Counterexamples on the reals . . . . . . . . . . . . . . . . . 353.3.3 Logical non-finiteness on the reals . . . . . . . . . . . . . . . 39

    3.4 Axiomatizing special kinds of regions . . . . . . . . . . . . . . . . . 423.4.1 Serial sets on the real line . . . . . . . . . . . . . . . . . . . 42

    vii

  • 3.4.2 Formulas in one variable over the serial sets . . . . . . . . . . 453.4.3 Countable unions of convex sets on the real line . . . . . . . . 483.4.4 Generalization toIR2 . . . . . . . . . . . . . . . . . . . . . . 49

    3.5 A general picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.5.1 The deductive landscape . . . . . . . . . . . . . . . . . . . . 51

    4 Logical extensions 534.1 Universal reference . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.2 Alternative extensions . . . . . . . . . . . . . . . . . . . . . . . . . . 60

    4.2.1 Hybrid reference . . . . . . . . . . . . . . . . . . . . . . . . 604.2.2 Until a boundary . . . . . . . . . . . . . . . . . . . . . . . . 62

    4.3 Standard logical analysis . . . . . . . . . . . . . . . . . . . . . . . . 65

    5 Geometrical extensions 675.1 Affine Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

    5.1.1 Basic geometry . . . . . . . . . . . . . . . . . . . . . . . . . 675.1.2 The general logic of betweenness . . . . . . . . . . . . . . . 695.1.3 Modal languages of betweenness . . . . . . . . . . . . . . . . 715.1.4 Modal logics of betweenness . . . . . . . . . . . . . . . . . . 745.1.5 Special logics . . . . . . . . . . . . . . . . . . . . . . . . . . 765.1.6 Logics of convexity . . . . . . . . . . . . . . . . . . . . . . . 765.1.7 First-order affine geometry . . . . . . . . . . . . . . . . . . . 81

    5.2 Metric geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825.2.1 The geometry of relative nearness . . . . . . . . . . . . . . . 825.2.2 Modal logic of nearness . . . . . . . . . . . . . . . . . . . . 865.2.3 First-order theory of nearness . . . . . . . . . . . . . . . . . 90

    5.3 Linear algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 915.3.1 Mathematical morphology and linear logic . . . . . . . . . . 935.3.2 Richer languages . . . . . . . . . . . . . . . . . . . . . . . . 96

    6 A game-based similarity for image retrieval 1016.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1016.2 A general framework for mereotopology . . . . . . . . . . . . . . . . 102

    6.2.1 Expressiveness . . . . . . . . . . . . . . . . . . . . . . . . . 1036.2.2 Comparison with RCC . . . . . . . . . . . . . . . . . . . . . 105

    6.3 Comparing spatial patterns . . . . . . . . . . . . . . . . . . . . . . . 1066.3.1 Model comparison games distance . . . . . . . . . . . . . . . 107

    6.4 Computing similarities . . . . . . . . . . . . . . . . . . . . . . . . . 1106.4.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . 1106.4.2 Polygons of the plane . . . . . . . . . . . . . . . . . . . . . . 1116.4.3 The topo-distance algorithm . . . . . . . . . . . . . . . . . . 115

    6.5 TheIRIS prototype . . . . . . . . . . . . . . . . . . . . . . . . . . 1196.5.1 Implementing the similarity measure . . . . . . . . . . . . . . 119

    viii

  • 6.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

    7 Thick 2D relations for document understanding 1257.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1257.2 A logical structure detection architecture . . . . . . . . . . . . . . . . 1287.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

    7.3.1 Document encoding rules . . . . . . . . . . . . . . . . . . . 1297.3.2 Relations adequate for documents . . . . . . . . . . . . . . . 1337.3.3 Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

    7.4 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1427.4.1 Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1437.4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1437.4.3 Discussion of the results . . . . . . . . . . . . . . . . . . . . 147

    7.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

    8 Conclusions 1518.1 Where we stand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1518.2 Final remarks on theory and practice . . . . . . . . . . . . . . . . . . 152

    A A bit of topology 155

    B Sorting transitive directed graphs 159

    C Implementations 163C.1 Topax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163C.2 IRIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169C.3 SpaRe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

    Bibliography 181

    Index 193

    Samenvatting 199

    ix

  • ACKNOWLEDGMENTS

    I believe that a PhD thesis is not the effort of an individual, but the final outcome ofa synergy. Since there is officially one author, I feel the urge to thank in these initialpages the people involved in a way or another with my PhD project.

    I arrived in Amsterdam four years ago as a lost soul. Unlike Voltaires Candide, Ifound generosity, humor, inspiration and, most importantly, solid scientific values tobelieve in. I realized my conversion to the Amsterdam school was complete when Ireceived an email in which I was addressed as a modal logician. What a joy. I didnot even know what a modal logic was till I moved my first steps in Amsterdam. Iftoday someone may recognize me as a credible scientist, I own it first and foremost toJohan van Benthem.

    One of the many qualities of Johan I had the privilege to appreciate, and I am sureI share this feeling with many others, is his natural disposition of putting everyoneat ease. I could always raise a question, no matter how silly, and get a simple yetilluminating answer. Every occasion to meet, discuss or even exchange emails withJohan have been pleasurable events which I have and I will be looking forward to. Inshort, Johan thank you.

    I am deeply in debt with Arnold Smeulders for his continuous interest in my work,for his warm supervision, and for an extreme availability. He provided me with vision-ary questions, while leaving me a considerable amount of freedom in my research. Ionly wish I could have answered more of his questions. Not to please him, but becauseif I did, I would be a famous scientis

Recommended

View more >