Bennett, B

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This is a repository copy of Spatial Reasoning.White Rose Research Online URL for this paper:http://eprints.whiterose.ac.uk/8509/Article:Bennett, B (2008) Spatial Reasoning. Encyclopedia of Geographic Information Science. pp. 426-432. eprints@whiterose.ac.ukhttps://eprints.whiterose.ac.uk/Reuse See Attached Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing eprints@whiterose.ac.uk including the URL of the record and the reason for the withdrawal request. Spatial ReasoningBrandon BennettDecember 10, 20061 The Field of Spatial ReasoningThe field of spatial reasoning is concerned with representations and inferencemechanisms that enable one to draw conclusions from spatial information.The the study of principles of spatial reasoning dates back to ancient times.It is known that the Egyptians used a variety of systematic procedures for spatialinference, especially with regard to measuring and demarcating areas of land.Geometry (literally meaning ‘Earth measurement’) was also a central topic ofancient Greek mathematics. This investigation culminated in the publication(around 300 B.C.) of Euclid’s Elements, an axiomatic system that provides amore or less comprehensive set of rules for reasoning about geometrical figuresas described by points and lines and the relationships between them.A major shift in the analysis and representation of spatial information was in-stigated by Descartes (1596–1650), who showed how point locations in space canbe represented by means of numerical coordinates. This idea can be generalisedto specify complex figures in terms of sets of numerical values and equations.The coordinate-based approach provides an extremely powerful mathematicaltool for representing and manipulating spatial information, and enables certainuseful kinds of spatial reasoning to be carried out by algebraic numerical meth-ods. Consequently, the spatial representations used in modern scientific modelsand computational information systems (such as GIS) are predominantly Carte-sian in nature.While the Cartesian analysis of space is certainly both profoundly illuminat-ing and of immense practical value, it does have limitations, and is ill-suited todescribing many intuitively natural forms of spatial inference. The main limi-tation stems from the fact that concise and mathematically simple descriptionsin terms of coordinates can only be given for specific instances of spatial fig-ures and configurations, whose geometry is fully determined; whereas, naturalreasoning about space is often couched in terms of general qualitative proper-ties and relationships of spatial objects. For instance, we may know that aspatial region is convex or that one spatial region is part of another, withoutknowing the particular geometry of the regions involved. Qualitative reasoningis required whenever, spatial information is partial or is presented in terms ofabstract, high-level concepts.1Recently, arising out from the Knowledge Representation strand of ArtificialIntelligence (AI) and also from the need for more flexible interaction with GIS,much research has been directed towards the study of reasoning with qualita-tive spatial information. The field of Qualitative Spatial Reasoning is now anestablished branch of AI research and elements of this work are beginning to beincorporated into GIS.2 Aspects of Spatial InformationThe realm of spatial information encompasses a wide variety of different entities,properties and relationships. Because reasoning with all these aspects togetheris extremely complex, representations designed for computing spatial inferencestypically handle only sub-domains of spatial information, comprising specifictypes of entity and a limited range of related concepts. This section summarisesthe most significant ways in which the domain of spatial information can bedivided into more restricted sub-domains.2.1 Basic Entities: Points, Lines and RegionsEuclidean geometry takes points and lines as the basic entities and Cartesiangeometry is based on points. More complex entities are constructed as con-figurations or sets of these elements. By contrast, many of the more recentlydeveloped qualitative spatial calculi deal directly with extended regions (either2 or 3 dimensional) and often do not explicitly refer to points at all. This can beadvantageous for implementing procedures for reasoning about regions, since itabstracts away from their complex structure. Some representations handle enti-ties of mixed dimensionality but this makes computation of inferences extremelycomplex.2.2 MereologyThe domain of mereology concerns the fundamental relation of parthood whichcan hold among extended entities and spatial regions. Parthood can be usedto define other relations such as overlap and disjointness: two regions overlapif they share a common part, otherwise they are disjoint. The notion of amereological sum can also defined: the sum of a set of regions is that (unique)region such that all of its parts overlap some region in the original set.2.3 TopologyTopology is concerned with those properties of a spatial configuration thatremain constant if the configuration is continuously deformed (i.e. stretched,squashed or bent). In mathematics, topology is often studied using very expres-sive formalisms (such as set-theory) which can refer to both points and regions.However, in order to achieve effective computation, researchers in spatial reason-ing have mostly concentrated on (binary) topological relations between regions.2The well-known Region Connection Calculus is a theory expressed in classicalfirst-order logic (see below), in which spatial properties and relations are definedin terms of the primitive relation of connection. For instance one can specifythat: region x is part of region y if and only if every region z that is connectedto x is also connected to y. (Consequently, the domain of topology is generallyconsidered to include mereology as a sub-domain, although some analyses treatthese as separate aspects, which combine to form a mereo-topology.)Figure 1 illustrates a set of eight basic topological relations definable fromconnection. Here, standard symbolic names are used for the relations holdingbetween regions a and b: disconnected (DC), externally connected (EC), tangen-tial proper part (TPP), tangential proper part inverse (TPPi), partially overlap-ping (PO), equal (EQ), non-tangential proper part (NTPP) and non-tangentialproper part inverse (NTPPi). This set, known as RCC-8, is particularly sig-nificant, since any two regions must be related by exactly one of these eightrelations.Figure 1: the RCC-8 topological relations2.4 Betweenness, Convexity and Affine GeometryAffine geometry concerns those properties that are definable from the relation ofbetweenness. This includes the notion of convexity, since a convex region is onesuch that any point that lies between two points of the region is also includedin the region.Betweenness is a very useful concept for describing or querying geographicinformation. For instance, one might be interested in settlements lying betweensome range of mountains and the sea, or the land between two rivers. Theconcept of convexity is also very expressive. The notion of a convex-hull of aregion r can be defined as the smallest convex region that contains r. Thisnotion can be very useful in describing forms and relationships of geographicfeatures.3For example, in Figure 2 we see an island i and three smaller islands a, b andc. Of these only a is convex. The convex-hull of i consists of i itself together withthe dark regions surrounding i (which correspond to bays around the island).We see that a is completely within the convex-hull of i, c is completely outsideit and b is partly in and partly out.Figure 2: example of a convex-hull2.5 Direction and OrientationA class of spatial relationships that is particularly significant for GIS applica-tions is those relating to compass direction. The standard nomenclature fordescribing directions (N, S, W, E, NW, NE etc.) may be used to represent thedirection of one point location relative to another; but, since the points maynot be aligned exactly along a designated direction, this is normally adapted sothat directions are associated with segments, rather than lines, as illustrated inFigure 3.Figure 3: cardinal directions associated with segmentsThis scheme must also be modified if it is to be applied to extended regions,which do not lie on a unique line. To handle this case, it has been proposedthat directions should be defined in terms of a grid, as illustrated in Fig 4. Herethe white region lies to the west of the dark reference region, which defines thegrid; and the mid-tone region overlaps each of the N, NE, E and SE sectors ofthe grid.Relative locations may also be described in terms of orientation. Here weconsider the position of one object in relation to another, from the perspectiveof a particular observation point. Fig 5 illustrates how a given viewpoint (the4Figure 4: cardinal direction grid determined by a reference regionblack disc) and reference landmark (the black triangle) determine a grid, bymeans of which we can characterise the position of another object in terms ofwhether it is to the left or right of the landmark and whether it is nearer orfurther away. Here, the white disc lies on the “right perpendicular” (rp) relativeto the viewpoint and reference point; and the cross lies in the left centre (lc)region.Figure 5: orentation relative to a landmark2.6 Metric GeometryTo fully describe spatial structure one needs to utilise some notion of the dis-tance. In fact, (as demonstrated by Tarski’s formulation of the theory of El-ementary Geometry) the relation of equidistance (holding when the distancebetween points a and b is the same as that between points c and d) is suf-ficient to completely define the metrical structure of space. Reasoning aboutmetric information in a general way is extremely complex and inferences cannotbe computed effectively. However, representation of specific configurations interms of Cartesian coordinates does enable efficient numerical computation ofmetrical properties.53 Spatial Inference MechanismsOne of the main obstacles to the employment of automated spatial reasoning inpractical applications is the complexity of computing deductions. Theoreticalresults show that even when dealing with information expressed using seeminglyvery restricted vocabularies of spatial concepts, the number of computationalsteps required to determine whether a given consequence follows is typicallyan exponential function of the amount of information involved. Thus inferencecomputations performed on large amounts of data are likely to require hugeamounts of CPU time and memory. To mitigate this problem, a number ofspecial purpose procedures have been developed that are effective for computingcertain kinds of spatial inference. But there is always a trade-off between thegenerality and computational tractability of any reasoning algorithm.The rest of this section summarises some of the most widely used reasoningmechanisms that have been applied to spatial information.3.1 Reasoning with Cartesian Coordinate InformationCartesian coordinate representations enable qualitative relationships to be com-puted by numerical calculation. For instance one may compute whether a pointlies within a polygon or circle, or whether two polygons touch or overlap. Suchfunctions are widely implemented in spatial information systems. However, thisis only a limited form of reasoning, in that the representation does not allowpartial or general information to be expressed and does not support extendedchains of inference.3.2 Reasoning with Classical Formal LogicThe the notation of formal logic was developed by mathematicians during theend of the 19th and the first half of the 20th century (in particular by Got-tlob Frege (1848–1925 and Bertrand Russell (1872–1970). It provides generalpurpose mathematical tool for representing and reasoning with information for-mulated in terms of properties and relationships holding among any domainof entities. This representation can be used to formulate axioms constrainingthe meanings of the conceptual terms and the structure of the domain, as wellas for specifying particular facts. Moreover, it provides a general mechanismfor determining valid inferences that follow from any set of theoretical axiomsand/or specific facts.A number of theories of spatial entities and properties have been formu-lated using classical formal logic. For instance: Tarski’s Elementary Geometryis a fully formal specification of the Euclidean geometry of points; the RegionConnection Calculus is a theory of topological (and in its extended form alsoaffine) properties of spatial regions; and Region-Based Geometry is a highly ex-pressive theory of spatial regions. Typically such theories employ the restricted(but still very expressive) sub-language of full classical logic known as first-order6logic. However, not all spatial properties can be adequately expressed in purelyfirst-order logic (e.g. the continuity axiom of Euclidean geometry).An advantage of using classical logic is that it has a well-defined semantics(formulated in terms of set-theory), and for first-order formulae a complete in-ference procedure can be specified. A major disadvantage is that the complexityof computing inferences it typically extremely high. Hence, unless the formalismis severely restricted, it is not usually possible to implement practical reasoningalgorithms.3.3 Relational Constraints and Compositional ReasoningOne simplification of the representational framework that has proved very ef-fective in making spatial reasoning problems more effective is to limit the rangeof spatial facts under consideration to those expressed using a fixed vocabularyof spatial relations among entities. These are typically binary relations such as“a is connected to b” or “r is to the north of s”. These can be seen as forming anetwork of constraints between entities. The reasoning problem is then confinedto determining whether such a network is consistent or deriving new relationsentailed by the given constraints.Compositional reasoning is a general mode of deduction applicable to rela-tional information. Most often it is employed with binary relations, in whichcase deductions take the following form: from two relations R1 holding betweenx and y and R2 holding between y and z, we infer a relation R3 between x andz. Such reasoning has been found to be very useful when dealing with spatial in-formation. For instance, in the realm of mereological and topological relations,one can make compositional inferences such as:• If x is part of y and y is part of z, then x is part of z.• If x overlaps y and y is part of z, then x overlaps z.• If x is part of y and y is disconnected from z, then x is disconnectedfrom z.3.4 Reasoning about TransitionsAs well as providing a static description of the world, spatial relations mayalso be used to describe changes in the state of the world. Such change can berepresented simply by a series of snapshots describing successive spatial config-urations. For example, at one time a desert region may be disconnected froma certain province, but later it might externally connect and then overlap thatprovince.For many real world situations, the regions involved cannot change arbitrar-ily but can only deform continuously over time. Moreover, there are strongconstraints on the transitions in spatial relations that can occur during contin-uous processes. Thus, the analysis of possible transitions can provide a usefulmechanism for spatial reasoning. Given a set of spatial relations, the possible7continuous transitions among them can be represented as a graph, which is oftencalled a transition network.3.5 Reasoning with Ontologies of Spatial ConceptsAn ontology represents the meanings of a conceptual vocabulary by expressinglogical relationships between concepts, usually in some kind of formal language.Such formalisms will typically support inference by means of computational al-gorithm. A number of standardised formalisms have been devised for expressingontologies. In particular, OWL (the Web Ontology Language) is an XML com-pliant representation based on Description Logic, that is widely used to encodeontologies for use via the World Wide Web.One limitation of standard ontology languages, such as OWL, is that theydo not in themselves support distinctively spatial aspects of reasoning. Logicalconnections among spatial properties and relations can only be captured in sofar as they can be encoded in terms of the somewhat restricted representationprovided and this may not always be possible.4 Spatial Reasoning in GISThe essentially spatial nature of geographic information means that there isconsiderable potential for the application of spatial reasoning within geographicinformation systems.Traditionally GIS applications have represented spatial information primar-ily in a Cartesian form (i.e. as coordinatised points, lines and polygons). Thus,they do not support representation of partial or general information and cannotcarry out extended chains of inference. However, it is increasingly apparentthat more sophisticated GIS applications require more powerful spatial reason-ing capabilities. Representation of partial information is particularly needed forplanning and hypothesis testing, where spatial constraints typically occur in aqualitative form (“the toxic waste dump may not be adjacent to a residentialarea”).Spatial reasoning is also required for flexible query interpretation. Whereas atraditional GIS requires queries to be framed in a limited number of ways thatare closely aligned to the way that data is actually stored, spatial reasoningcan enable interpretation of queries involving complex combinations of a largevocabulary of spatial terminology.Another important use of spatial reasoning in GIS is for data consistencyand integrity checking. A good example of the kind of spatial consistency con-straint that could be handled is: “a building may not overlap a road.” Clearlysuch conditions could be hand coded into a GIS, but spatial reasoning allowsthem to be formulated by users in a general spatial constraint language, andenables automatic inference to detect constraint violations. A spatial constraintlanguage can be made more powerful by combining it with an ontological theoryof types of geographic object. This would include a specification of sub-type8relationships such as “a house is a type of building,” which would mean thathouse object would satisfy any constraints applicable to buildings.The use of spatial reasoning in GIS is still (in 2006) at an early stage ofdevelopment. But its potential is widely recognised and increasingly sophisti-cated reasoning components are already appearing in commercial GIS systems.There are still many challenges to meet in implementing flexible and effectivereasoning algorithms. However, it seems certain that the use of automated spa-tial reasoning will transform the way that we use computers to interact withgeographic information.Suggested Further ReadingCohn, A. G. and Hazarika, S. M.: 2001, Qualitative spatial representation andreasoning: An overview, Fundamenta Informaticae 46(1-2), 1–29.Egenhofer, M. J. and Golledge, R. G. (eds): 1998, Spatial and Temporal Rea-soning in Geographic Information Systems, Oxford University Press, NewYork.Pratt-Hartmann, I., Aiello, M. and van Benthem, J. F. A. K. (eds): 2007, TheLogic of Space, Kluwer.Stock, O. (ed.): 1997, Temporal and Spatial Reasoning, Kluwer.9

Spatial Reasoning

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Spatial Reasoning

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