Qualitative Spatio-Temporal Representation

Qualitative Spatio-Temporal Representation & Reasoning for Vague Regions

by Zina M. Ibrahim

A Thesis Submitted to the Faculty of Graduate Studies and Research through the School of Computer Science in Partial Fulfillment of the Requirements for the Degree of Master of Science at the University of Windsor

Windsor, Ontario, Canada 2004

c Zina M. Ibrahim, 2004 ° All Rights Reserved

Qualitative Spatio-Temporal Representation & Reasoning for Vague Regions by Zina M. Ibrahim

APPROVED BY:

E. Abdel-Raheem Department of Electrical Engineering D. Wu Department of Computer Science

A. Tawfik, Advisor Department of Computer Science R. El-Marakby, Chair of Defense Department of Computer Science

December 2, 2004

Abstract The ability to perform commonsense reasoning about complex concepts such as change is a powerful trait. When acquired by an agent, this ability facilitates performing useful tasks that are hard to carry out otherwise, and makes possible dealing with heterogeneous situations in different environments. It is especially useful when reasoning about motion and spatio-temporal change as it grants the power of commonsense reasoning to mobile agents as they perform their navigation tasks. Motion is changing one’s location over time. Therefore, in order to be able to perform commonsense reasoning tasks about motion, the concepts of space, time and space-time must be represented in a fashion that enables commonsense reasoning. In other words, they must be represented in a qualitative fashion. Making realistic applications that use a qualitative representation of a spatiotemporal concept such as motion requires having realistic assumptions about the world. This introduces many elements into the application, one of which is the issue of vagueness, as spatial objects do not usually have well-defined boundaries. This thesis presents an extension of an existing spatio-temporal theory to incorporate the possibility that regions under study have vague boundaries. We formulate an ontology, along with reasoning techniques to build a qualitative reasoning engine which reasons about vague spatio-temporal regions in an urban rescue application.

iv

Dedication

To my parents, For safely placing me in the boat And teaching me how to row In the ocean of life...

To the land where it all began, Where my heart lives And dies, every day...

v

Contents Abstract

iv

Dedication

v

Acknowledgements

vi

Contents

viii

List of Tables

xii

List of Figures

xiii

1 Introduction

1

1.1

Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.2

Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.3

Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.4

Thesis Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

2 Qualitative Space, Time and Space-time

7

2.1

Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

2.2

Importance of Qualitative Representation and Reasoning . . . . . . .

8

2.3

Applications of Qualitative Representations . . . . . . . . . . . . . .

9

2.3.1

Temporal Applications . . . . . . . . . . . . . . . . . . . . . .

9

2.3.2

Spatial Applications . . . . . . . . . . . . . . . . . . . . . . .

11

2.3.3

Spatio-temporal Applications . . . . . . . . . . . . . . . . . .

13

Representing Qualitative Spatial & Temporal Knowledge . . . . . . .

15

2.4.1

Ontology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

2.4.2

Formulating the Relations . . . . . . . . . . . . . . . . . . . .

17

Qualitative Calculi for Space and Time . . . . . . . . . . . . . . . . .

19

2.4

2.5

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Chapter 0

2.6

Spatio-temporal Reasoning for Vague Regions

2.5.1

Qualitative Temporal Calculi . . . . . . . . . . . . . . . . . .

20

2.5.2

Logical Approaches to Representing Time . . . . . . . . . . .

24

2.5.3

Qualitative Spatial Calculi . . . . . . . . . . . . . . . . . . . .

26

Qualitative Spatio-temporal Representation & Reasoning . . . . . . .

31

2.6.1

The Conceptual Neighborhood . . . . . . . . . . . . . . . . . .

31

2.6.2

Formulating a Spatio-Temporal Theory . . . . . . . . . . . . .

33

2.6.3

Theories of Space-Time . . . . . . . . . . . . . . . . . . . . . .

35

3 Spatial Vagueness

41

3.1

Definitions and Motivation . . . . . . . . . . . . . . . . . . . . . . . .

41

3.2

Philosophical Views on Vagueness . . . . . . . . . . . . . . . . . . . .

42

3.3

Representing and Reasoning about Vague Regions . . . . . . . . . . .

43

3.3.1

The egg-yolk Calculus . . . . . . . . . . . . . . . . . . . . . .

44

3.3.2

The Broad Boundary Theory . . . . . . . . . . . . . . . . . .

45

3.3.3

Using Rough Sets . . . . . . . . . . . . . . . . . . . . . . . . .

46

3.3.4

The Scrambled-egg Theory . . . . . . . . . . . . . . . . . . . .

47

3.3.5

A Note About the Calculi . . . . . . . . . . . . . . . . . . . .

48

4 Extending a Spatio-temporal Representation to Vague Regions

49

Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

4.1.1

Limitations of Current Approaches . . . . . . . . . . . . . . .

50

4.1.2

Potential Benefits . . . . . . . . . . . . . . . . . . . . . . . . .

50

4.2

Aim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

4.3

New Ontology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

4.3.1

New Ontological Primitive . . . . . . . . . . . . . . . . . . . .

51

4.3.2

New Primitive Relations . . . . . . . . . . . . . . . . . . . . .

51

Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

4.4.1

From RCC-8 to RCC-5 . . . . . . . . . . . . . . . . . . . . . .

52

4.4.2

From RCC-5 to the Egg-Yolk Relations . . . . . . . . . . . . .

54

4.4.3

Abstracting the Clusters . . . . . . . . . . . . . . . . . . . . .

57

Motion and Vague Regions . . . . . . . . . . . . . . . . . . . . . . . .

59

4.5.1

Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . .

60

4.5.2

Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

4.5.3

Certain Motion Classes . . . . . . . . . . . . . . . . . . . . . .

61

4.1

4.4

4.5

University of Windsor, 2004

ix

Chapter 0


4.5.4 4.6

4.7

4.8

4.9

Possible Motion Classes . . . . . . . . . . . . . . . . . . . . .

68

Continuity of the Motion Patterns . . . . . . . . . . . . . . . . . . . .

72

4.6.1

Defining the Continuity Constraints . . . . . . . . . . . . . . .

72

4.6.2

Continuity at the First Certainty Level . . . . . . . . . . . . .

73

4.6.3

Continuity At the Second Certainty Level . . . . . . . . . . .

73

4.6.4

A Note About the Continuous Extension . . . . . . . . . . . .

74

Composite Motion Classes . . . . . . . . . . . . . . . . . . . . . . . .

74

4.7.1

The CROSS Motion Class . . . . . . . . . . . . . . . . . . . .

75

4.7.2

The Complete REACH Motion Class . . . . . . . . . . . . . .

78

4.7.3

The Complete LEAVE Motion Class . . . . . . . . . . . . . .

79

Reasoning about Motion . . . . . . . . . . . . . . . . . . . . . . . . .

79

4.8.1

Reasoning about Temporal Information . . . . . . . . . . . . .

80

4.8.2

Reasoning about Spatial Knowledge . . . . . . . . . . . . . . .

81

Scrambled Eggs and Egg Splitting . . . . . . . . . . . . . . . . . . . .

83

5 The Experiments 5.1

5.2

5.3

86

The Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

86

5.1.1

The GIS, Constructing the City . . . . . . . . . . . . . . . . .

87

5.1.2

The Sub-simulators . . . . . . . . . . . . . . . . . . . . . . . .

88

5.1.3

The Rescue Agents . . . . . . . . . . . . . . . . . . . . . . . .

90

A Qualitative Commentator . . . . . . . . . . . . . . . . . . . . . . .

90

5.2.1

Ontological Primitives . . . . . . . . . . . . . . . . . . . . . .

90

5.2.2

Motion Patterns . . . . . . . . . . . . . . . . . . . . . . . . . .

91

5.2.3

The Commentator . . . . . . . . . . . . . . . . . . . . . . . .

91

Experiments and Results . . . . . . . . . . . . . . . . . . . . . . . . .

92

5.3.1

Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . .

92

5.3.2

Merit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

5.3.3

Expressiveness . . . . . . . . . . . . . . . . . . . . . . . . . . .

96

5.3.4

Summary of Results . . . . . . . . . . . . . . . . . . . . . . .

98

6 Conclusions and Future Research

99

6.1

Limitations and Loose Ends . . . . . . . . . . . . . . . . . . . . . . . 100

6.2

Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 6.2.1

Developing an Application Programming Interface(API) . . . 101


x

Chapter 0

6.3


Other Aspects of Theory . . . . . . . . . . . . . . . . . . . . . . . . . 101

Bibliography

102

Vita Auctoris

115


xi

List of Tables 3.1

Defining the RCC-5 Relations . . . . . . . . . . . . . . . . . . . . . .

47

4.1

The First Set of Egg-yolk Clusters . . . . . . . . . . . . . . . . . . . .

55

4.2

The Second Set of Egg-yolk Clusters . . . . . . . . . . . . . . . . . .

57

4.3

The RCC vw(wE) with its RCCv Makeup . . . . . . . . . . . . . . . . .

59

4.4

The RCC vw(yE) with its RCCv Makeup . . . . . . . . . . . . . . . . .

59

4.5

The Topology at Beginning and End of z for the 1st Certainty Level Motion Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

4.6

Pairs of Continuous Extensions at the First Certainty Level. . . . . .

75

4.7

The Topology at Beginning and End of z for the 2nd Certainty Level Motion Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76

4.8

Pairs of Continuous Extensions at the Second Certainty Level. . . . .

76

4.9

Composing Motion with Temporal Information . . . . . . . . . . . . .

81

4.10 Composing Motion with Spatial Information . . . . . . . . . . . . . .

83

xii

List of Figures 2.1

Allen’s Relations , pp. 835 of [1]

. . . . . . . . . . . . . . . . . . . .

20

2.2

Reasoning about Allen’s Relations, pp. 836 of [1] . . . . . . . . . . .

22

2.3

Cyclic Nature of Time, pp. 6 of [78]

. . . . . . . . . . . . . . . . . .

24

2.4

Cyclic Intervals Relations, pp. 7 of [78] . . . . . . . . . . . . . . . . .

24

2.5

RCC-5 & RCC-8 Relations pp. 2 of [41], and pp. 9 of [35] . . . . . .

27

2.6

Composition Table for RCC-8, pp. 20 of [35] . . . . . . . . . . . . . .

28

2.7

The 8 Relations of 4-Intersection Model, pp. 3 of [51]

. . . . . . . .

29

2.8

Lines of Site (LOS-14), pp. 2 of [65] . . . . . . . . . . . . . . . . . .

30

2.9

Occlusion Topological Relations, pp. 1 of [83] . . . . . . . . . . . . .

31

2.10 Meets and Overlaps are Conceptual Neighbors . . . . . . . . . . . . .

32

2.11 Before and Equals are Not Conceptual Neighbors . . . . . . . . . . .

32

2.12 The RCC-5 Conceptual Neighborhood, pp. 275 of [88]

33

. . . . . . . .

2.13 The RCC-8 Conceptual Neighborhood, pp. 6 of [38], and pp. 23 or [35] 33 2.14 The OCC Conceptual Neighborhood, pp. 6 of [83]

. . . . . . . . . .

34

2.15 The LOS-14 Conceptual Neighborhood, pp. 5 of [65] . . . . . . . . .

35

2.16 Muller’s Classes of Motion, pp. 1 of [98] . . . . . . . . . . . . . . . .

40

3.1

The Egg-yolk Representation, pp. 2 of [88] . . . . . . . . . . . . . . .

44

3.2

The 46 Possible egg-yolk Combinations, pp. 10 of [67]

. . . . . . . .

45

3.3

Approximating a Spatial Region . . . . . . . . . . . . . . . . . . . . .

46

4.1

The First Form of egg-yolk Clusters . . . . . . . . . . . . . . . . . . .

55

4.2

The Second Form of egg-yolk Clusters . . . . . . . . . . . . . . . . . .

56

4.3

Abstraction the First Form of egg-yolk Clusters . . . . . . . . . . . .

58

4.4

Abstracting the Second Form of egg-yolk Clusters . . . . . . . . . . .

58

4.5

The RCCv Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

4.6

The Conceptual Neighborhood of the RCCv Set . . . . . . . . . . . .

59

xiii

4.7

Definite Motion Classes for the First Certainty Level . . . . . . . . .

62

4.8

Forms (5,1), (6,1), (7,1) and (8,1) of D LEAVE . . . . . . . . . . . .

63

4.9

Forms (2,9) and (5,9) of D REACH . . . . . . . . . . . . . . . . . . .

64

4.10 Partial Motion Classes for the First Certainty Level . . . . . . . . . .

65

4.11 Forms (9,5), (8,5) and (8,2) of P LEAVEv z x y . . . . . . . . . . . .

66

4.12 Forms (1,6) and (1,7) of P REACHv z x y . . . . . . . . . . . . . . .

67

4.13 The Six Scenarios of Possible LEAVE, (9,1), (9,2), (9,5), (8,1), (8,2), (8,5). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

4.14 The Scenarios of Possible REACH,(1,5), (1,6), (1,7), (1,8), (1,9) . . .

70

4.15 Scenarios (6,5), (5,5), (6,6), (5,9) and (9,9) of Possible INTERNAL .

71

4.16 (5,5), (1,5), (1,1) of Possible EXTERNAL . . . . . . . . . . . . . . .

72

4.17 D CROSSv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

4.18 P CROSSv z x y . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

4.19 Comlete REACHv z x y . . . . . . . . . . . . . . . . . . . . . . . . .

79

4.20 Complete LEAVEv z x y . . . . . . . . . . . . . . . . . . . . . . . . .

80

4.21 Query Example Demonstration . . . . . . . . . . . . . . . . . . . . .

80

4.22 Query Example Demonstration . . . . . . . . . . . . . . . . . . . . .

82

5.1

The Robocup Simulation Environment . . . . . . . . . . . . . . . . .

88

5.2

Compactness of the Qualitative Representation vs. the Quantitative

5.3

One. (a) The Size of the World (b) The Events Rate . . . . . . . . .

94

Expressiveness of Qualitative and Quantitative Representations. . . .

98

xiv

Chapter 1

Introduction A program with the ability to “automatically deduce for itself a sufficiently wide class of immediate consequences of anything it is told and what it already knows” [95] is a program with common-sense reasoning capabilities. This type of reasoning has been a research focus for decades and large efforts have been invested to represent knowledge in forms that facilitate common-sense reasoning by capturing the inherent properties of the objects in concern and the intuitive relations that can connect them to other objects. From this perspective, the field of qualitative representation [85] has arisen, aiming to formalize high-level knowledge about the world in order to allow reasoning through common-sense. This type of representation abstracts the underlying numerical implementation of an application to capture physical objects. In other words, a qualitative representation sees the world in terms of shapes and colors, not numbers and digits. A class of applications that has been specifically receiving a lot of attention during recent years focuses on the use of computing machinery to model movement. The need for such applications arises from the wide interest in Navigational and Geographical applications. Originally, numerical methods have been used to model and implement such applications and they served their purpose in many fields such as robotics [104, 103]. However, As the task becomes harder and the environment more complex and dynamic, these methods proved inefficient giving rise to the concept of qualitative motion [100]. However, before attempting to model motion qualitatively, its constituent parts must be studied. Motion is one of the oldest concepts which captured the attention of schol-

1

Chapter 1


ars. Motion was defined by the Greek philosopher Aristotle as “the actuality of a potentiality as such”. Then came the Muslim scholar Ibn Sina, otherwise known as Avicenna, who formulated a thorough study of motion that began by the rule that “No body begins to move, or comes to rest, of itself”. Ibn Sina formulated the first laws governing motion. Also, the French philosopher Ren´e Descartes studied motion through the criticism of the work of Aristotle. He eventually defined motion as “nothing more than the action by which any body passes from one place to another”. The latest formulation of motion is that of Isaac Newton, who built upon the laws of Ibn Sina and formulated three rules that govern every type of movement. All in all, these definitions imply that motion has to do with actions or change. More specifically, changing one’s location in a specific period of time. Hence, space and time are the two concepts essential to motion and in order to be able to model movement qualitatively, qualitative paradigms of these concepts must be constructed first. Qualitative representations of time have been under study for more than two decades. Formulating such representations and embedding reasoning techniques in them motivated the work of Allen [1, 2, 3, 4, 5] and continues to motivate research today. Similar representations for space [30] have been formulated and gave rise to several calculi [112, 46, 65]. Having this maturity, along with the need for a qualitative paradigm that combines both time and space has resulted in what is known as qualitative spatio-temporal representation and reasoning which aims at qualitatively modelling change, e.g. motion and deformation. Having a qualitative spatio-temporal theory is not enough to model a concept as complex as motion. In order for an agent to be able to navigate through the heterogeneous world it must be capable of dealing with the unpredictable situations that may arise. Also, due to the dynamic nature of the world, perfect information regarding both the spatial and temporal dimensions are rare, if not impossible to find. For example, a robot navigating through a room may rarely encounter the same layout the next time it comes back. Moving any piece of furniture changes the layout, which makes the model the robot has obsolete. From the above, it can be deduced that the ability to interact with insufficient information is needed as our perception of the world around us is often incorrect, incomplete and imprecise. Therefore, as part of their behavior, agents undergoing motion deal with these uncertainties and overcome them to achieve their goal given University of Windsor, 2004

2

Chapter 1


the task at hand, which may vary from navigating through a familiar room to finding a way out of a trap. There exist many types of uncertainty that may be present in the spatiotemporal domain. The uncertainty may concern the exact location of the spatial object undergoing motion, such as the location of a moving robot, or the boundaries of the objects, such as the boundaries of clouds or any type of geographical objects. In the latter type, the boundaries of the spatial objects are fuzzy or in other words, vague. Incorporating such mechanisms into a spatio-temporal theory only works towards making it more usable in the real world by giving the representation more power to deal with the unpredictable situations awaiting. Incorporating all types of uncertainty is a vast task that goes beyond this thesis. In this thesis, we focus on one type of uncertainty, which is spatial vagueness.

1.1

Objectives

Qualitative theories of space-time, as well as qualitative theories representing spatial vagueness are present in the literature. However, to our knowledge, there does not exist a qualitative theory for vague spatio-temporal (moving) objects, which is what has driven thesis. Incorporating the concept of vagueness into an existing spatio-temporal theory is a step towards making it more realistic. We aim to achieve the following: • Merge a qualitative theory of spatial vagueness with an existing qualitative spatio-temporal theory to form a general qualitative theory of moving objects. • Implement a reasoning engine which embeds the theory formulated in an application in a dynamic environment.

1.2

Methodology

To achieve the above objectives, we perform the following steps: • Create a formal notion of a vague spatio-temporal object by integrating features of vagueness with a formalization of spatio-temporal objects.


3

Chapter 1


• Formalize the means through which vague spatio-temporal objects undergo motion by making the concept of motion explicit and classify its various forms. • Establish reasoning techniques about the vague spatio-temporal objects defined as they undergo motion and use the techniques to predict the relative location as well as other information about the objects. • Use the formalization of vague spatio-temporal objects we have created to implement a qualitative abstraction of a quantitative simulated dynamic environment. • Implement a reasoning engine that uses the techniques we established to describe the motion of ‘agents’ traversing the objects of the qualitative simulated environment we created. • Use the engine to conduct experiments that assess the qualitative implementation of the simulated environment and compare it to the previous qualitative implementation.

1.3

Contribution

As noted earlier, the contribution of this work lies in proposing a completely qualitative theory representing moving objects whose boundaries are vague. Because the concept of qualitative motion is a relatively new one, research in this field has so far made assumptions that are not workable. One of these is assuming that regions always have well-defined boundaries. Also, although qualitative representations of vague spatial objects exist, most of them have not gone beyond stationary objects. The ones that did, resorted to non-qualitative means such as probability theory when dealing with motion. Our approach is unique in that we tackle the two research directions and bring them together in a logical form. Moreover, our implementation has proven the ability of qualitative representations to tackle applications in dynamic environments and therefore should not be excluded in this area. Our implementation also outlines several advantages that our qualitative representation has over the existing quantitative one. Our idea has been published in [79] and appeared in the 17th Conference of the Canadian Society for Computational Studies held in Ottawa. Also, the results of the empirical analysis conducted using the implementation have been submitted for publication. University of Windsor, 2004

4

Chapter 1

1.4


Thesis Structure

In what is left of this chapter, we present the structure of the thesis. The first two chapters give the reader a clear background of the topics involved and motivates our work. Chapter 2 is a review of qualitative studies of space, time and space-time. The chapter introduces the concepts, discusses their importance and enumerates applications where they are useful. The chapter then moves on to representational issues by outlining the parameters that decide the ontology of a qualitative theory of space or time and discusses existing calculi. After presenting the fundamentals of qualitative spatial and temporal representations, the chapter discusses their merger and examines qualitative spatio-temporal representation and reasoning. The chapter ends with a preview of existing qualitative spatio-temporal calculi, hinting to the limitations that we wish to address. Chapter 3 is an overview of our second interest, which is the concept of spatial vagueness. The chapter is divided into two parts, the first aims at thoroughly defining the concept of spatial vagueness and making its importance clear to the reader. The second part reviews work found in the literature and identifies the its limitations. By the end of this chapter, the reader must have a clear idea regarding the deficiencies of existing representations, leading to our work. Chapter 4 is based on our earlier work [79] to construct the building blocks of a qualitative spatio-temporal theory of vague objects. The chapter goes through the process of formalizing the various aspects of the theory and constructing the reasoning techniques to be used to implement them. In chapter 5 we develop a set of experiments to test the adequacy of our qualitative spatio-temporal theory of vague regions. In the first part, the chapter describes the structure of a reasoning engine which abstracts an existing quantitative implementation of a simulated urban disaster space and uses this abstraction to describe and reason about the motion of various rescue agents traversing the simulated environment. The second part of the chapter uses the developed engine to conduct experiments which will evaluate our qualitative representation and compare it to its quantitative counterpart. We outline several parameters for the comparison and conduct the experiments, where each experiment tests one parameter. The chapter ends by summarizing the results which show the advantages of our qualitative


5

Chapter 1


representation. The final chapter completes the thesis by outlining the various conclusions we have drawn about the research conducted in this thesis. The chapter also discusses some loose ends and various future directions in both the representation as well as the implementation aspect.


6

Chapter 2

Qualitative Space, Time and Space-time Given a coordinate-based implementation of a ‘world’, a qualitative representation of the same world captures intuitive information about one (or more) of its aspects that we are interested in. This information can be used to fully describe the chosen aspect and make it explicit in a structured and a uniform manner to create a formal abstraction of the underlying numerical implementation. The techniques formulated to reason about this type of representation and are used to deduce new knowledge from existing one are called qualitative reasoning techniques. We are interested in qualitative studies of the concepts of space and time and the notion of space-time resulting from their co-existence in a single domain. The goal of the chapter is to appreciate the idea of motion from a qualitative perspective.

2.1

Definitions

Qualitative temporal representation is concerned with establishing a formal way to explicitly describe time so that algorithms can be used to reason about it. This is done by expressing the constituents of time, their properties and the relationships that may exist among them in a formal, high-level and abstract manner that does away with numerical details. The techniques used to reason about such representation and extract new knowledge from it are described as qualitative temporal reasoning techniques. A qualitative spatial representation of a spatial object must be capable of

7

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describing some or all of its spatial properties without having to resort to numerical methods [38] and use these properties to formulate ways to connect spatial objects to each other so that qualitative reasoning techniques can be applied to deduce new knowledge. The composite dimension of space-time arises when space and time are combined to formalize concepts that can only be defined by space and time joint together and not treated individually. A classical example is that of motion which requires handling the concepts of space and time as a composite entity, as apposed to treating them as two individual entities. Reasoning about such a concept is spatio-temporal reasoning. A spatio-temporal object is an object with a spatial extent and whose properties change over time. Change can take many forms. One such form is motion, which is the result of continuous change of location over time. Another form is the change of shape imposed on an object via continuous transformations, merging or splitting. Also, as the object shrinks or grows, the resulting change is that of size. Since the definition of spatio-temporal change implies change of one or more aspects of space over time, a spatio-temporal theory that needs to model change of a certain aspect of space must be based upon a spatial theory that captures that aspect of space for the objects at hand. For example, a spatio-temporal theory that models change of the shape of an object over time must be built upon a spatial theory capable of describing the shape of the object. Also, as a spatio-temporal object undergoes change, the spatial relations between it and its surroundings may change. Therefore, a spatio-temporal theory must not only be able to model the relations among its objects, but also the relations among the relations. This means being able to describe how one spatial relation among two or more objects is gradually transformed into another spatial relation among the same objects when change occurs [59]. To represent spatio-temporal notions in a qualitative manner and formulate ways to reason about them is qualitative spatio-temporal representation and reasoning.

2.2

Importance of Qualitative Representation and Reasoning

Quantitative means of representing and reasoning about concepts such as space and time exist and have been found useful in many types of applications [103, 104, 137]. University of Windsor, 2004

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The aim is not to demonstrate that quantitative methods are not adequate is not the case. Rather, we are trying to illustrate the existence of applications as well as situations where qualitative approaches would be highly preferred. In applications that require modelling dynamic environments, describing the com-mon-sense relations among the objects in the world eases the representation and reasoning process a great deal. For example, having to specify the exact coordinates of a chair and that of a desk implies changing the coordinates of the objects every time one of them is moved, resulting in a lot of overhead in terms of calculations if we are only interested in the position of the chair relative to the desk. However, if we describe the relation between the two objects qualitatively by saying the chair is behind the desk, then as long as the chair is indeed behind the desk and the qualitative relation between the two objects holds, the statement remains true regardless of any change in the coordinates of the chair or the desk. Uncertainty is another issue that speaks to the importance of qualitative Representation and reasoning. For instance, it is not always possible to obtain the exact time of occurrence or exact location of an event, but that does not prevent answering questions regarding them. As an example, one may know that snow started falling before it became dark but is not aware of the exact instant at which snow started falling or the instance at which it became dark. So the aim is to capture the intuitive relations between events or time instances and not their exact time of occurrence. Also, we may not know the exact coordinates of southern Ontario on the map but we are perfectly capable of formulating useful relations that enable us to deduce information such as: Southern Ontario is to the east of British Columbia and is to the west of Quebec.

2.3 2.3.1

Applications of Qualitative Representations Temporal Applications

Planning

A plan is a finite sequence of actions that transforms an initial state to a state that satisfies a given goal [9]. The involvement of the aspect of time in planning is obvious as the actions force the objects affected by the plan to change from one state to another. A qualitative representation of a plan is composed by capturing the


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temporal relations governing the actions. Natural Language Processing

A sentence is usually composed of nouns, verbs and other constructs, where verbs usually stand for actions that manipulate and change the nouns of the phrase. Understanding how objects referenced by nouns are transformed from one state to another by the actions induced by verbs calls for a temporal reasoner, because change and time are interrelated. Also, words such as “before, after” need to be semantically represented in order for their effect on the sentence to be understood. For example, ”John entered before Mary did” requires a temporal representation [68]. This is not a trivial problem as the following example illustrates. “Mr. Jones was alive after Dr. Smith operated on him”. Does it follow that “Dr. Smith operated on Mr. Jones before Mr. Jones was alive?” [124] . This example illustrates a semantical problem in the use of temporal relations such as ’before’ and ’after’ as the meaning of these relations in this example does not permit ’before’ to be the inverse of ’after’. Hence, an adequate temporal representation that is accountable for such pitfalls must be present. Medical Applications

The order and timing of symptoms is important in diagnosis, hence we need to capture temporal relations of temporal events, intervals and points. For this we need a qualitative reasoner. Other

Temporal Reasoning is vital to many other types of applications as time must be represented in any application that needs to model change. This includes temporal databases [7], time seriation in archeology [82], knowledge-base systems [25], finances and genetics [18] to list a few.


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2.3.2


Spatial Applications

Geographic Information Systems (GIS)

The first generations of GIS systems were not very user-friendly. Queries submitted by the users involved a lot of formulations and numerical details that took a long time for an average person to become acquainted with. They are usually constructed by applying what is known as an analysis function [110], which is a highly-complex formulation. The hope for the next generations is for the users to submit perfectly qualitative queries such as: Output all the countries adjacent to Thailand. Such queries are intuitive and encourage what is now known as Na¨ıve Geography, “a field of study concerned with formal models of the common-sense geographic world” [53]. In order to perform qualitative queries, the geographical model must be represented by a qualitative model of space expressive enough to handle such queries. QSR is expected to construct the infrastructure for the next generation of GIS systems and is currently active as many GIS user-friendly interfaces are being developed such as the one found in [110], a query-by-sketch [49] and GeoVIBE [27]. Robotics

Navigating robots may be faced with many circumstances that introduce flaws to their navigation process. As a robot traverses the real world, uncertainty, unexpected conditions or even calculation errors may lead to blunders in the decision making process. Mechanical errors may occur leading to a slight imprecision which can propagate as motion is taking place. In turn, this may cause drastic changes in the behavior of a robot. Also, imperfection may be present in the world model the robot has constructed for itself. In this case, the quality of the decisions a moving robot makes deteriorates due to the errors present in its own map [54]. A qualitative model of space can solve most of the problems that a numerical model may introduce. Having a qualitative map instead of a numerical one forces the robot to know only the spatial relations among the objects in the map and not their exact positions. As a result, the numerical details that may cause errors are not included in the reasoning process. Also, since only the qualitative relations between the spatial objects are present, mechanical errors would not have a great impact on movement [86].


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It is important to note that the robot would eventually need numerical information to make some of its decisions. This information however, is not needed until late stages of the movement plan and may not be required throughout many of the movement stages. For this purpose, the concept of a hierarchial structure of movement can be implemented where a qualitative model is used until a stage is reached where numerical details are in fact needed [127]. Natural Language Understading

Many applications in NLU make use of QSR. For example, in narrative story-telling, the reader must not only visualize the spatial objects in the story but also the spatial relationships among these objects and how they change as the theme of the story progresses [93, 75]. Moreover, Natural Language Processing is full of ambiguities that can only be resolved with a representation that makes an account for them. This especially applies to spatial prepositions. For example, the preposition in can have many meanings depending on its location in the sentence and intended use [77]. Therefore, a spatial representation that takes into account all the different meanings of the prepositions and proper ways to distinguish them and reason about them is required. Also, since sentences are interpreted by the brain in a qualitative fashion, qualitative spatial representations are preferred. Cognitive Vision

In reference to scenes containing spatial objects, the process of reasoning about the objects in a scene with the aim of interpreting it involves complex image segmentation operations to capture the relative location of objects [40]. The use of a qualitative representation of visual scenes helps simplify such tasks by adding meaning to a scene, which helps identifying scenes that are semantically related to facilitate the interpretation task. Other

QSR is being used in many fields [11]. For example, design tasks for engineering disciplines make use of qualitative spatial representations in its initial stages [57] and only adapts numerical representations in its final stages to ease the design process and to capture the relations among the objects in the design more uniformly. University of Windsor, 2004

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The intuitive, high-level nature of QSR has made it the prime choice for modelling AI games by constructing the worlds of the games using qualitative techniques [58]. 2.3.3

Spatio-temporal Applications

Location-aware Assistants

Ubiquitous or pervasive computing [136] is a paradigm which aims to deploy computing devices in areas of our daily lives and make them capable of discovering other ubiquitous devices to provide a network of information providing various services. A special type of ubiquitous devices is location-aware assistants [123] whose task is to facilitate coordinating the time of the holders by connecting to each other. They schedule appointments and meetings without the need for intervention from the holders themselves. The tasks the assistant performs involve a large amount of constraints and dynamic spatio-data such as the current schedule of the holder and its conflicts with the schedules of the other holders. The tasks can also involve deciding the location of the meeting based on criteria that the holders may define dynamically or by letting the assistant choose the most adequate place based on spatial, temporal and changing (spatio-temporal) constraints. The large amount of information and constraints calls for a qualitative reasoner to deploy such applications. Robots that are location-aware can also serve in coordinating tasks for suppliers of goods. For instance, a truck supplying product x may be informed by its built-in robot which has connected to a store the truck has passed that the store’s stock of the product it is supplying is running out. Hence, the truck driver may stop and supply the store before the store makes an order. Project Oxygen by MIT1 is an effort to make this type of applications a living reality. The project is comprehensive to all applications with a context-aware nature. Natural Language Understanding

It is important to have a formalization of the objects in a sentence and means to reason about them as motion verbs manipulate the position of the objects and as a result, modify their location relative to other objects [121]. The set of verbs that 1 http://oxygen.lcs.mit.edu/


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describe motion, leave, reach, cross, hit, internal and external are made formal in [98]. This formalization is based on a spatio-temporal theory that aims at providing a general qualitative platform to represent moving objects described in [97] and will be described later on in the chapter. Intelligent Highway Systems

The main mechanisms involved here are those of spatio-temporal tracking and prediction such as attempting to find vehicle x at time t1 when it was at region y at time t0 . Such reasoners are useful in tasks such as monitoring traffic flow and taking preventive measures against accidents and traffic jam. They also serve in areas of security and criminal investigations. Victims of kidnapping may be tracked based on signals sent with the description of the kidnapping car and possible locations being sent to police stations. Qualitative approaches are useful in this domain because the highway can be split into regions and relations based on topology can be used to perform the tasks required. Storm Tracking

Given the spatio-temporal nature of storms and hurricanes, scientists are always tracking their movement attempting to predict their future behavior in order to take preventive measures. However, data about natural phenomena in general are dataintensive, making the task difficult. Because of this property, qualitative approaches to storm tracking systems are being found quite useful [101]. Fully-autonomous Supply Chains

A fully-automated supply chain is one that possesses all the information relevant to its function without the need for external interference. The chain connects the suppliers of all items necessary to make a product and places orders on behalf of the manufacturer to the suppliers when an order for a shipment of the product is requested from an external consumer. Since the chain is automated, when shipment s is requested with due date t1 , the supply chain not only automatically places the order to the suppliers for the parts needed, but also checks the feasibility of having the orders in time to manufacture the product and ship it to the customer. This is done by connecting to the other suppliers and performing complex reasoning involving rapidly-changing spatio-temporal data. University of Windsor, 2004

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Hence, the supply chain is context-aware and makes judgement based on its own prediction of what may be the possible choice [89].

2.4

Representing Qualitative Spatial & Temporal Knowledge

In order to be able to reason about spatio-temporal notions, they must be made formal. However, one cannot rush to construct a formal representation without paying attention to the two components of space-time, namely space and time. A lot of efforts have been invested to represent time (e.g. [1, 96]) in the past two decades. Similarly some qualitative representations of space ( e.g. [29, 38]) have been proposed. To represent a concept is to identify its components and connect them in a logical fashion to facilitate reasoning about them. In other words, to represent a concept is to explore its nature and make it explicit, which is done by identifying two aspects that define the concept. The first is the ontological commitment made, which identifies the building blocks that make up the concept. The second are the set of relations that connect the ontological primitives we are committed to. These two aspects are discussed next. 2.4.1

Ontology

Ontology is the study of things, their nature and means to categorize them by identifying their building blocks and searching for properties that distinguish one object from another [120]. There are several ontological questions that we are interested in. For example, with respect to time, studying its ontology means trying to identify the basic primitives that make up what we know as time. The same applies for space as the answers to these questions shape how our theory perceives the nature of spatial objects. We are also trying to answer questions that outline properties that may differ according to the view point of the theory and distinguish one school of thought from another. Temporal Ontology

Two views on the building blocks of time exist. The first assumes points to be the primitive making up time [96] and argues that points are the only way to model the continuity of time by defining continuous time as a “dense set of partially ordered and non-circular time points” [63]. In the light of this claim, many temporal University of Windsor, 2004

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representations choose points as their ontological primitives [6, 135]. The second view assumes that time is made up of intervals [1]. This view refutes representing time as points and claims that introducing instances can result in many semantical problems as nothing happens in an instant [4] and that points are somehow artificial [5] and do not form an intuitive way to represent time. Moreover, this view claims that an interval is the most suitable way of representing both states and events [1]. Theories supporting such a view usually regard instances as very short temporal intervals. A third view objects to the previous two by claiming that the concept of time is too complex to be completely represented by one type of ontological primitive and aims at using a hybrid of the two views. The hybrid approaches have been motivated either by the need to fix the semantical problems of points presented in [4] or by the deficiencies of interval-based theories in not being suitable for representing continuous events . The Instant-Period theory (IP) [133] aims at fixing the first problem by placing instances at the beginning and the end of an interval, while [63, 5] use points to connect intervals to fix the continuity problem of interval-based logic. Furthermore, whether time is linear or circular, finite or infinite, open or closed, continuous or discrete, relative (before, concurrent, after) or absolute (past, present, future) decide the structure of time. The topology of the temporal relations is either linear (total topology) or branching (partial topology). Spatial Ontology

There are also two schools of thought regarding the buliding blocks of space. The first views space as being made up of points and consequently defines several other constructs such as regions and lines as being made up of a set of points [47, 51]. This assumption is used to formulate the relations which are defined by examining different modes of intersection between point-sets [48]. Point-based geometry supporters argue their position by examining the success of coordinate geometry and its ability to construct regions out of points. Also, they claim that there is no need to have regions as ontological primitives because at a certain minimum level, region-based geometry is as complex as the point-based ones [108]. Region-based geometry advocates that points are not perceivable and hence cannot form a base for any qualitative theory of space [125]. Instead, it assumes University of Windsor, 2004

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that extended regions are the building blocks to what we know as space. This view forms what is known as mereology, or calculus of Individuals [29, 30]. It is favored in qualitative spatial reasoning due to its expressive power and intuitive nature and has been adopted as the base for many spatial theories [38, 8]. Apart from the nature of the primitive entities making up space, the continuity of space is another aspects which defines the ontology adopted. Discrete space is usually assumed in quantitative applications with large amounts of numerical details. This is due to the fact that numbers corresponding to a spatial representation eventually map to pixels on raster devices, which are discrete and hence modelling them can be performed better in a discrete environment [126]. Qualitative approaches assume continuous space. 2.4.2

Formulating the Relations

After committing to a set of ontological primitives, the theory must decide the relations that may hold between these primitives. For example, if an interval-based theory of time is assumed then how are the intervals related? What guides the construction of the relations among the intervals? Similar questions can be formulated for the spatial domain. For the concept of time, whether it is dense or discrete, bounded or unbounded, linear [1], branching [15] or circular [78] are all issues to be considered to fully understand its nature and then formulate relations among the ontological primitives we are committed to. Also, whether or not the length of the durations of the events are important [137, 109]. The answers to these questions add more concreteness to the concept of time and makes the ontological commitment more well-defined. As for spatial objects, the relations among them are decided by addressing one or more aspects of space and using them to come up with ways to link spatial objects. These aspects of space include topology, shape, orientation, direction and distance, and are all discussed next. Topology

Topology defines spatial relations among objects that do not change under transformations such as translation, rotation or scaling of the objects as a whole. Therefore, topological relations are not affected by the shape, size or exact location of the ob-


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jects. This is why it is good to define the position of a spatial object relative to its surroundings [97], without being concerned by its coordinate location or size. The work in topology and its incorporation into qualitative spatial models has its roots in Clarke’s calculus of individuals [29, 30]. Apart from being the first to consider regions as primitives of space (as opposed to point-based approaches), Clarke’s work also formulates a set of completely topological relations and axiomatize them. The resulting set is the base for many of the calculi to be discussed in this chapter including [8, 111]. It is important to note that although the pioneer work associates topology with region/pointless geometry, these two aspects of space are independent concepts. It is possible to define topological relations connecting points [46, 47]. It has been argued that topology by itself is not sufficient for a complete qualitative representation of space as it does not add sufficient constraints to the description of the object. There are applications such as robot navigation [137] and shape recognition [42] that require information that topology cannot provide because it is too abstract and some details are needed in order to solve real-world problems. For this reason, other aspects of space, such as the following, can be incorporated into the theory. Size

Size is very popular in representing qualitative information about spatial objects as it is easily represented. A good example is the idea of the predicate canConnect(x,y,z) [43] that holds true if region z can connect regions x and y. Such predicates can be used to compare the size of two objects. To illustrate, take any two objects z1 and z2 and another two objects x and y. It may be possible to make claims regarding the relative size of z1 and z2 by seeing if one of them can connect x and y while the second cannot. Several calculi have been proposed in this regard, one of which is the delta calculus [141]. The idea is based upon a triadic relation, x(>,d)y that holds if region x is larger than region y by amount d. However, size by itself is not very useful to fully describe the relations between spatial objects. Hence, it has been mainly used in combination with other aspects of space, mainly topology [66], where metric size information is combined with a topological spatial theory (such as [8, 29, 112]) to add additional constraints that make the representation more concrete. University of Windsor, 2004

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Orientation

Orientation aims at relating two objects by finding their positions with respect to a fixed frame of reference and deducing the resulting relations. The choice of a frame of reference enables formulating deictic relations ( x to the left of y), intrinsic relations (x to the front of y) and absolute ones (x to the north of y) [28]. Hence, we deal with triadic relations , which makes it possible to speak about things such as ’clockwise’ and ’anticlockwise’ with respect to the frame of reference object. This adds more detail and rigidness to the qualitative representation. By taking this idea further several calculi were developed. For example, three points can be mapped to one of three values, -,0 or + to denote clockwise, collinear or anti-clockwise orientation [122]. Another calculi formulates 24 relations that describe the orientation of two line segments in a plane [80]. Shape

One can use topology to introduce simple shape descriptions of the objects such as whether the object has holes or not, or whether it is one piece or not. In general, shape is the hardest aspect of space to represent and usually metric information is needed when it is important to represent shape precisely. Combining more than One Aspect of Space

Usually, the tasks required for real applications are too complex to consider only one aspect of space [81]. Topology, orientation and cardinal directions have been combined in [81]; [76] uses orientation and position, [142] uses orientation and distance, [54] uses orientation, distance and cardinal directions, and finally [66] incorporates size into topological relations.

2.5

Qualitative Calculi for Space and Time

Based on the building blocks we have discussed earlier, several qualitative calculi have been formulated in the spatial and temporal domain. Here, we discuss these calculi stating the ontological commitment made by each and the bases for formulating their relations. We also discuss the reasoning aspects of each theory, how it is done and impact. University of Windsor, 2004

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2.5.1


Qualitative Temporal Calculi

Allen’s Interval Algebra

This theory [1] represents time as being made of intervals. It assumes that nothing happens in an instance and a totally-ordered, non-circular and linear set of intervals makes up what we know as time. The theory uses the notion of two intervals meeting each other to formulate the relations that may hold between the intervals. Allen defines meets as the relation that holds between two intervals if one precedes the other but there is no time between them [3]. This notion is used as the base for 13 relations that represent the only form of relations between any two intervals. These relations are given in figure 2.1. Mainly, they are six relations, their inverses as well as the equality relation. The properties of

Figure 2.1: Allen’s Relations , pp. 835 of [1]

the intervals have been axiomatized to enable reasoning. The axioms governing the behavior of intervals are: 1. Given an interval, there exist another interval related to it by one of the 13 relations 2. The relations are mutually exclusive 3. The relations can be used to perform composition on pairs of intervals to obtain new relations [2]. The idea behind Allen’s reasoning procedure is that since an event takes place in an interval, it is possible to deduce the relations between two events from the relations between their intervals. For example, if event e 1 takes place during interval i 1 , event e 2 takes place during interval i 2 , and event e 3 takes places during intervals i 3 and we University of Windsor, 2004

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also know that i 1 has relation R 1 with interval i 2 and that i 2 has relation R 2 with interval i 3 , then we can deduce the temporal relations between interval i 1 and i 3 and as a result know the temporal relations between events e 1 and e 3 . The reasoning method explained above has been made formal by Allen and placed in what is known as the composition table given in Figure 2.2. The composition of any two relations r 1 and r 2 can be obtained from the intersection of their entries in the composition table. The results are fed into the table using an algorithm that computes the composition of a set of relations. The algorithm that finds the solution to all of Allen’s relations is NP-complete [135]. However, subsets for which the algorithm perform in sub-polynomial time, i.e. tractable subsets of the algebra have been found. In total 18 tractable subsets of Allen’s algebra exist [44, 45, 90, 102] and these are the only forms of tractability in Allen’s algebra [84], i.e. they are ’maximal’ tractable subsets and other subsets of the algebra not belonging to any of those should have at least an exponential performance in the worst case. Being a milestone theory of time is not the only importance of Allen’s theory. It is considered the start of a new age for qualitative temporal reasoning as it is the first to propose an explicit representation of time and not having it implicitly embedded in logic [134]. Composition tables have become a common reasoning technique for the qualitative community and are used in many theories of time, space and space-time [38, 97, 70, 142, 138]. Allen’s relations have also been used as a base for formulating calculi to represent and reason about space [70].

McDermott’s Theory

This theory regards time as a discrete set of time points whose elements form possible histories each corresponding to a different way through which the universe may evolve. The time points are connected via a hierarchial structure called the chronicle tree [96] that represents the evolution of time from past to future. The higher nodes represent the past and the lower nodes represent the future. The branches capture the different possible evolutions from the past (higher nodes) to the future (lower nodes). A fact is defined by the first order predicate that describes it and two chronical tree nodes that represent the end points of the time during which it holds, e.g.


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B r2 C A r1 B “before”
oi mi df oi mi df

d

o oi d di =

d

O di fi

Oi di si

O di fi

Di fi o

Di

mi

oi

> oi mi di si >

ods

mi

mi

d

oi d f

d

ods

oi di si

m

Si oi di

o

> oi mi di

f

> oi mi di si

od s o

> oi mi

oi mi di si di

“overlaps” o

“overlapped by” oi “meets” m

“met-by” mi “starts” s

“started by” si “finishes” f “finished by” fi

◦ q)

(2.4)

Modal temporal logic has been found useful in many applications that require temporal reasoning [64, 91, 56], not only for its desirable representational properties but also for its computational effectiveness [62]. 2.5.3

Qualitative Spatial Calculi

Extending Allen’s Intervals to Space

After the success of Allen’s intervals, efforts were directed towards coming up with relations similar to those of Allen’s for the spacial dimension. One such effort is Guesgen’s work in [70]. The theory uses points as primitives of space, defines regions out of this primitive and depends on relative orientation and position to define 8 relations between any two spatial entities a and b. These relations are a is left of b (a is right of b), a is attached to b (b is attached to a), a is overlapping b (a is overlapped by b) and a is inside of b (a contains b) [70]. These relations mimic the ones Allen defines for temporal intervals. Reasoning is performed using a composition table with inconsistencies removed using constraint satisfaction techniques. The technique followed by this work has serious limitations. Because only relative orientation and position are used, there is no absolute reference system and hence it is not possible to restrict it to one uniform coordinate system. It is therefore not possible to compare the distance between two regions qualitatively because finding the distance between any two spatial entities requires them to be within the same coordinate system. The Region Connection Calculus (RCC):

The RCC calculus is one of the first to recognize the notion of Clarke’s Calculus of Individuals [29, 30] by refuting the existence of points and admitting the primitive of an extended region of space [112].


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The relations among the spatial entities are built from the base topological notion of ’connectedness’, where two regions x and y are connected C(x,y) if the distance between them is 0 [112]. This purely topological representation is taken further and used to define a set of relations that may hold between two regions of space. There are two versions of the RCC relations, a boundary-sensitive version, which takes into account the topological connection between the two regions and a boundary-insensitive version. The RCC-8 boundary-sensitive relations are eight jointly-exhaustive pairwisedisjoint relations. They are Disconnected (DC), Externally Connected (EC), Partially Overlap (PO), Tangent Proper Part (TPP) and its inverse (TPPI), non-tangent Proper Part (NTTP) and its inverse (NTPPI) and the equality relation (EQ) [112] shown in the upper part of figure 2.5. DC

EC

PO

TPP

NTPP

TPPI

NTPPI

EQ

RCC-8 A

B

A

A

B

DR

B

A

B

A

B

PO

A

B

A

A

B

B

PP

B

A

B

B

PPI

A

A

A,B

A,B

RCC-5

EQ

Figure 2.5: RCC-5 & RCC-8 Relations pp. 2 of [41], and pp. 9 of [35]

A boundary-insensitive variation of the RCC set is the RCC-5 [36, 67], obtained by collapsing the RCC-8 relations to obtain 5 jointly exhaustive pairwise disjoint relations instead of 8. The process of collapsing the former is performed by merging the relations that differ because of the ’connectedness’ of the boundaries of the regions they represent and is done as follows: The notion of parthood in RCC-5 is represented by the relation PP which is a generalization of Tangent Proper Part (TPP) and nontangent Proper Part (NTPP). The generalization is also performed for the inverses of these two relations, TPPI and NTPPI to obtain PPI (Proper Part Inverse). The EC and DC relations are generalized into one relation, DR (distinct regions). Figure 2.5 below shows the RCC-8 relations and the corresponding RCC-5. Reasoning is performed via a composition table similar to that of Allen. Figure 2.6 shows the composition table for the RCC-8 set. The RCC theory has gone through many stages of maturation in a series of papers [112, 34, 17, 16] and has become a widely-accepted theory in the QSR community. Both RCC-8 and RCC-5 are closed under composition, negation and disjuncUniversity of Windsor, 2004

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R2(b,c) R1(a,b) DC


EC

PO

No info

DR, PO, PP

DR, PO, PP

DR, PO, PP

DR, PO, TPP, TPPi

DR, PO, PP

PO

DR, PO, TPPi, NTPPi DR,PO,PPi

DR,PO,NTTP i, TPPi

TPP

DC

DR

NTPP

DC

DC

TPPi

DR,PO, TPPi, NTPPi DR,PO, TPPi, NTPPi DC

EC

NTPPI

EQ

DC

TPP

NTPP

TPPi

NTPPi

EQ

DC

DC

DC

EC, PO, PP

DR, PO, PP PO,PP

DR

DC

EC

No info

PO,PP

PO,PP

DR,PO, TPPi, NTPPi

PO

TPP,NTPP

NTPP

DR,PO,TPPI

TPP

NTPP

NTPP

No info

NTPP

EC,PO, TPPi, NTPPi

DR,PO,TPP, NTPP DR,PO,TPP, NTPP PO, TPPi, NTPPi

PO,TPP,TPPI

PO,TPP, NTPP

DR,PO,T PPi, NTPPi DR,PO,T PP,TPPI DR,PO,N TPP,TPP TPPi, NTPPi

NTPPI

TPPI

PO, TPPi, NTPPi

PO, TPPi, NTPPi

PO, TPPi, NTPPi

O

NTPPI

NTPPI

NTPPi

EC

PO

TPP

NTPP

TPPi

NTPPi

EQ

Figure 2.6: Composition Table for RCC-8, pp. 20 of [35]

tion [83]. The RCC-8 theory has been extended to apply the 8 relations not only to single spatial relations, but also to boolean combinations of the relations. This work forms what is known as the BRCC-8 [138]. The efficiency of RCC-8 has been investigated [114, 69, 118, 115] and reasoning about the entire set has been found to be NP-Hard [117]. However, maximal tractable subsets have been identified [117, 114]. The N-Intersection Model

The N-Intersection model [47, 46] takes points as its ontological primitives and defines further constructs, such as regions in R2 or lines in R1 as a point set. The first proposed model, the 4-intersection model establishes the relations between two spatial constructs (regions or lines) by distinguishing two types of points in the point set, which are the set of boundary points and the set of interior points. For a spatial construct A, the boundary of A is denoted by σA and its interior by A◦ . The relations that may hold between two spatial constructs A and B are identified by seeing which of the sets (σA∩σB , A◦ ∩B◦ ,σA ∩ B◦ ,A◦ ∩ σB ) are empty and which are non-empty. Therefore, the relations are purely topological in nature [47, 52]. Combining the four possibilities results 24 = 16 relations. However, only 8 of them are perceivable. They are disjoint, contains, inside, equal, meet, cover, coveredBy and overlap [47] and form what is known as the 4-intersection set. Figure 2.7 taken from [51] shows the 8 relations for two regions in R2 . The same results are University of Windsor, 2004

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obtained for two lines in R1 or a region in R2 and a line in R1 [51]. There is a direct correspondence between this set and the RCC-8 set.

Figure 2.7: The 8 Relations of 4-Intersection Model, pp. 3 of [51]

The 4-intersection model has been later found to be too coarse to accommodate smooth transitions of the relations between regions. This problem has been tackled by introducing the 9-intersection model [48, 46, 50], which defines three instead of two sets to make a region. They are the border, the interior and the complement (exterior) of the region. The result is a 3 × 3 matrix with 29 = 512 combinations possible. For two regions in R2 , the resulting set of relations is equal to that of the 4-Intersection [47]. The result also applies to two lines in R1 . However, for a region and a line in R2 , 19 relations instead of 8 are found [92]. In either model, composition can be used to reason about point sets using Allen’s composition table [1]. The 9-Intersection model is problematic when observed closely because the complement of an entity is infinite [55], which prevents the model from being usable. This is because finding the intersection between any set and the complement of an entity is difficult if not impossible. A modification of the 9-intersection model aims at eliminating this problem by replacing the complements of spatial entities with their Voronoi regions [55]. The 9-Intersection model has also been extended to include regions with holes [50]. LOS-14 and OCC

These theories are motivated by the need for qualitative representations for computer vision as existing calculi are not expressive enough for this field [83]. The first is the University of Windsor, 2004

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line of site (LOS-14) calculus [65]. It is concerned with objects in the visual field of a viewer and defines their location relative to other objects in the same visual field. The resulting set is composed of 14 topological relations. These relations are C (A is clear of B ), JC(A is just clear of B ), PH(A partially hides B ), PHI(A is partially hidden by B ),JH(A just hides B ), JHI(A is just hidden by B ), H (A hides B ), HI (A is hidden by B ), EH(A exactly hides B ), EHI(A is exactly hidden by B ), F(A is in front of B ), FI(A has B in front of it), JF(A is just in front of B ) and JFI(A has B just in front of it). Twelve of the fourteen relations are given in figure 2.8 taken from [65]. The relations not shown are EH and EHI because in these two case, A and B are identical and the two cases are drawn the same.

Figure 2.8: Lines of Site (LOS-14), pp. 2 of [65]

The LOS-14 relations are easily composed using the composition table of Allen [1] and can be reasoned about accordingly. The LOS-14 has been extended in [113] in an attempt to incorporate occlusion in the relations of LOS-14. This work is known as the ROC-20 and it only works for non-convex objects. Later on, the occlusion calculus (OCC) has been proposed to overcome the inability of the ROC-20 calculus to reason about convex objects. It incorporates the concept of visual occlusion into the RCC-5 calculi to obtain a new set of 8 relations that may hold between two convex objects in the 3-D visual field of an observer. The topology of occluded regions results in the relations given in Figure 2.9, taken from [83].


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O

I

D

F

O^T

B

F^T

B^T

Figure 2.9: Occlusion Topological Relations, pp. 1 of [83]

2.6

Qualitative Spatio-temporal Representation & Reasoning

As the previous sections have shown, qualitative theories of space and time are quite mature. Formalisms as well as applications have been developed and have proven useful in many fields of Artificial Intelligence. However, there exist applications that cannot make use of separate representations of space and time as they require a combined spatio-temporal representation. These applications usually require modelling spatial objects that change over time [140], such as applications of motion [97] or deformation [106]. This requirement, along with the maturity of theories of space and time have encouraged combining space and time towards a general qualitative theory of spatiotemporal objects, or a general theory of change. The general form of spatio-temporal theories has been composed of a qualitative spatial and a temporal theory, which are combined in a logical manner to form a model used to represent spatio-temporal change that can be reasoned about using some well-known qualitative algorithms. We start by explaining a concept important to continuity in the spatio-temporal domain. 2.6.1

The Conceptual Neighborhood

“Two relations between a pair of events/intervals are conceptual neighbors if they can be directly transformed into one another by continuous deformation (i.e. shortening or lengthening) of the events/intervals ”[60] without the need to go through another relation. For example, Allen’s relations meets and overlaps are conceptual neighbors; if interval A meets interval B, then by increasing the length of interval A we may get the overlaps relation as shown in figure 2.10. The relations before and equals are not conceptual neighbors as two intervals must go through the relations meets, overlaps and during in order to go from one to University of Windsor, 2004

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another as shown in figure 2.11. A set of relations form a conceptual neighborhood if one can visually arrange the elements (relations) in the set such that each two adjacent relations are conceptual neighbors [60, 59].

A

B

A

B

Figure 2.10: Meets and Overlaps are Conceptual Neighbors

A

B

B A

Figure 2.11: Before and Equals are Not Conceptual Neighbors

Conceptual Neighborhoods in Qualitative Calculi

It has been found that the conceptual neighborhood applies not only to Allen’s intervals, but to many calculi in the spatial domain [61]. Figure 2.12 shows the neighborhood structure of the RCC-5 relations. It is obvious from the figure that the relations PP and PPi are direct neighbors of EQ, so is PO. PO in turn is a direct neighbor of DR. Figure 2.13 shows the neighborhood structure for the RCC-8 relation and also makes the fact that the RCC-8 set is an expansion of the RCC-5 set more visible. Figure 2.14 is another pictorial representation showing the conceptual neighborhood of the OCC relations and finally, figure 2.15 displays the structure for the LOS-14 relations, all discussed in section 2.5.3. The Importance of Conceptual Neighborhoods for Motion

The idea of the conceptual neighborhood embodies the continuity of transforming one relation to another, which is important in modelling motion which is known as a form of continuous change. With regards to motion, having such neighborhood structure makes predicting the spatial relations among the moving objects at a later time possible. Therefore, the spatial theory chosen to be part of the spatio-temporal University of Windsor, 2004

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PP a b a a b

a

b EQ

b

DR

a

PO

b PPi Figure 2.12: The RCC-5 Conceptual Neighborhood, pp. 275 of [88]

TPP a

NTPP

b

b a

b

a b

a a b

a b EQ

b a

DC

EC

PO

a NTPPi b TPPi

Figure 2.13: The RCC-8 Conceptual Neighborhood, pp. 6 of [38], and pp. 23 or [35]

platform must obey Freska’s neighborhood structure. 2.6.2

Formulating a Spatio-Temporal Theory

As with space and time, a spatio-temporal theory is formulated by defining its ontological primitives and the relations that exist among them. Moreover, a spatio-temporal representation must capture the nature of change itself and classify it formally. The Nature of Change

There are two types of spatio-temporal change, discrete and continuous [106, 39]. Continuous change is defined by separate states that cannot be easily distinguished, as University of Windsor, 2004

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F

O B^T I B

D

F^T O^T Figure 2.14: The OCC Conceptual Neighborhood, pp. 6 of [83]

is the case with motion. For this type of change, continuity is enforced by constructing strict models for it where any abnormal transitions [39] are not allowed. In discrete change, one can usually identify the stages through which change occurs and is usually associated with changes in shape, which is common with geographical objects [106] such as rivers and mountains. Ontology

The first choice to make when it comes to qualitative spatio-temporal ontology is whether represented objects are continuants or occurants [8, 97]. Traditionally, all world objects have been viewed as continuants, i.e. eternal, as opposed to events which are considered to be occurants, i.e. having a temporary existence. The first to object to this view was Patrick Hayes [74] who addressed the problem of modelling change for an object(continuant) which contains temporary(occurrents) parts. The ontology of spatio-temporal objects also depends upon the choice of spatial and temporal theories that the spatio-temporal theory is based on. Point and regionbased theories have been used in spatio-temporal theories. However, region-based calculi were the favorite choice for spatio-temporal formalisms [37, 99, 97, 98, 139, 20]. The same applies for the temporal compartment as there exists a wide variety of algebras and temporal logics to choose from. To give examples, [97, 98, 99] uses the RCC-8 [112] extended regions of space and Allen’s intervals [1] for time. The RCC-8 [112] is used as the topological base and propositional temporal logic is used as the University of Windsor, 2004

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C

JC

JH

PH

JF

JHI

PHI

JFI

H

EH

F

HI

EHI

FI

Figure 2.15: The LOS-14 Conceptual Neighborhood, pp. 5 of [65]

temporal base in [139] . The choice of the spatial and temporal components dictates whether motion is absolute or relative and whether it is discrete or dense. 2.6.3

Theories of Space-Time

In the light of the above, spatio-temporal theories have been developed and formulated in the past few years. Some of the important contributions in this area are: Wolter and Zakharyaschev’s Spatio-temporal Theory

The spatial theory chosen is the RCC-8 theory (section 2.5.3) due to its expressive power and success in modelling space for various applications such as GIS [139] along with the fact that it is decidable [13]. The temporal component used is prepositional temporal logic (PTL) with modal operators (section 2.5.2), which takes as primitive points of time that form a set of partially ordered elements {N, ◦(DC(x,y),(EC(x,y))) [139], the statement is read as, it will always hold that if two regions x and y are disconnected then in the next moment


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they will either stay disconnected or become externally connected. The same way the statement: ¤+(EC(x,y)-> ◦(DC(x,y),(EC(x,y),( PO(x,y))) [139] states that it will always hold that if two regions x and y are externally connected, then in the next moment of time, they will either be disconnected, stay externally connected or partially overlap each other. In ST0 , the modal operators are being applied to the RCC-8 set, i.e. what we are capturing is the temporal aspect of the relations among the regions and how these relations change over time. This is not very useful if the dynamic nature of the region itself needs to be captured and not only that of the relations it has with other regions. In other words, we need a way of modelling the evolution of a region by capturing its states at two time points. For example, if we would like to say that the universe is always expanding, then a formula such as ¤+P(Universe, ◦Universe) which states that the universe at moment n is always a part of the universe at the next moment (n+1), where P represents parthood [29]. The ◦ operator is applied to the region to represent its state at the following time point, which is exactly what is done in ST1. ST1: Extends ST0 “by allowing applications of the next-time operator ◦ not only to formulas but also to region variables” [139]. Hence, it is now possible to describe an object that will never change via the following sentence: ¤+EQ(x,◦x) [139], which is interpreted as x being an object that will always be equal to itself at any two consecutive moments of time. ST1 allows modelling two consecutive states of a region. However, modelling two arbitrary states that may not be consecutive is a different matter and requires the ability to apply the future operators (¤+ and ♦+) to the regions, which is what is done in ST2. ST2: The terms used as regions and supplied to the RCC-8 relations are expanded by applying all possible modal operators to them. In other words, if d is a term then so is ¤+d,♦+d and ◦+d. These three levels of formalisms present a well-formed spatio-temporal theory that can be expanded in several directions and may be used to represent and reason about dynamic objects of all kinds.


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Muller’s Spatiotemporal Theory

The primitive entities in Muller’s theory are spatiotemporal regions. The relations among these regions can be spatiotemporal as well as temporal. A spatiotemporal relationship describes the topological relations among the regions over time, while the temporal relationship between two regions describes how the two spatiotemporal evolutions are related in time [97, 98, 99]. The theory follows a general trend in spatiotemporal representation by combining a spatial theory as a topological base, along with a temporal theory and accordingly formulates the spatiotemporal interactions between the two. The RCC-8 set [112] is chosen as the topological base due to its wide acceptance and inherent conceptual neighborhood structure. The temporal relations are those of Allen. The resulting theory is mereotopological and is based on assumptions compatible with RCC-8. The theory does not perceive points as primitives and relies on intervals/regions in representing spatial/temporal quantities. The choice of Allen’s interval logic and RCC-8 produces a coherent spatiotemporal theory from this perspective. As a side note, since the theory has the RCC-8 set as its spatial base, and the RCC-8 is a boundary-sensitive set of relations, then like all spatio-temporal theories, Muller’s theory assumes that the boundaries of the regions under study are well-defined. The spatial extents of two regions x and y are connected (C x y) if the distance between them is zero. The same is also true for the temporal connection between two intervals a and b denoted by (a ∂ b). We use ∂ instead of the original symbol ♦ to avoid confusion because ♦ has been used in this thesis to denote the modal operator “eventually”2 . Binding the spatio-temporal objects is performed by formulating spatial and temporal relations among them and using the two types of relations to come up with a new set of spatio-temporal relations. The spatio-temporal relations are essential because the collaboration between the spatial and temporal domain must be represented In this theory, this collaboration takes the form of axioms that add structure to the theory, some of which are given below. Lemma 2.1. C x y → x ∂ y . Lemma 2.1, taken from [97], captures the relationship between spatiotemporal connectedness and temporal connectedness. Note that for notational clarity the following convention is adapted: 2 See

sections 2.5.2 and 2.6.3).


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1. Spatiotemporal operators precede the regions they operate upon (prefix notation) 2. Temporal operators come between the regions they operate upon (infix notation) Lemma 2.1 is the base for all other types of interaction between the topological (spatial) and temporal aspects of the theory. Another important concept in Muller’s theory is the notion of a ’temporal slice’ denoted by (x |y), which refers to the subinterval of x corresponding to the “lifetime” of y when y ⊆t x, as shown in Definition 2.1 below taken from [97]. Definition 2.1. x | y is a continuous section of the trajectory of region x such that y ⊆t x . TEMP IN xy describes the notion of temporary parthood. The following defines the meaning of region x being a part of region y temporarily, also taken from [97]. Definition 2.2. TEMP IN xy if ∃ z,u: ( z|x ∧ Pzy) ∧ (u|x ∧ ( u ⊆t y) ∧ ¬ Cuy). The ultimate aim of the theory is to define patterns that govern the behavior of what is known as motion. Accordingly, the theory identifies and formally defines 6 patterns of motion according to the possible interactions between spatiotemporal objects [97]. For example, region x is said to have reached region y during an interval z if x finishes y and x is temporarily in y. The patterns, as well as notions necessary for their definition, are given in Definitions (2.3-2.12) below taken from [97]. Definition 2.3. REACH zxy if TEMP IN x|z y|z ∧ (x|z . y|z FINISHES z ). In (2.3), FINISHES is based on Allen’s interval logic and is defined as: Definition 2.4. x FINISHES y if x ⊆t y ∧ ∀ z(x