The Challenge of Qualitative Spatial Reasoning

A. G. CC)HN

School of Computer Studies, Uru.erslty of Leeds, Leeds LS2 9JT

(age@ scs.leeds.ac.uk )

The principal goal of qualitative reason-ing (QR,) is to represent not only oureveryda<y commonsense knowledge aboutthe physical world, but also the underly-ing abstractions used by engineers andscientists when they create quantitativemodels. Endowed with such knowledgeand appropriate reasoning methods, acomputer could make predictions and di-agnoses and explain the behavior ofphysical systems in a qualitative man-ner, even when a precise quantitativedescription is unavailable or computa-tionally intractable. The key to a qualita-tive representation is not simply that itis symbolic and utilizes discrete quantityspaces, lbut that the distinctions made inthese discretizations are relevant to thebehavior being modeled.

QR has now become a mature subfieldof AI, as evidenced by its tenth annualinternational workshop, several books

(e.g., Weld and DeKleer [1990]) and awealth of conference and journal publica-tions. Although the field has broadenedto become more than just qualitativephysics (as it was first known), the bulkof the work has dealt with reasoningabout scalar quantities, whether they de-note the level of a liquid in a tank, theoperatirlg region of a transistor, or theamount of unemployment in a model ofan economy. Space, which is multidimen-sional and not adequately represented bysingle scalar quantities, has only re-cently become a significant research areawith the field of QR and, more generally,in the knowledge representation commu-nity. In part this may be due to the

Poverty Conjecture promulgated by For-bus et al. [Weld and DeKleer, 1990]:

“there is no purely qualitative, general-purpose kinematics.” Of course, qualita-tive spatial reasoning (QSR) is more thanjust kinematics, but it is instructive torecall their third (and strongest) argw-ment for the conjecture—”No total order:

Quantity spaces don’t work in more thanone dimension, leaving little hope forconcluding much about combining weakinformation about spatial properties.”They correctly identify transitivity of val-

ues as a key feature of a qualitative

quantity space but doubt that this can beexploited much in higher dimensions andconclude: “We suspect the space of repre-

sentations in higher dimensions is sparse;that for spatial reasoning almost nothingweaker than numbers will do.”

Happily, over the last few yearsan increasing amount of research hastended to refute or at least weaken thisconjecture. There is a surprisingly richdiversity of qualitative spatial represen-tations, and these can exploit transitiv-ity, as demonstrated by the relatively

sparse transitivity tables (cf. the wellknown table for Allen’s interval temporallogic [Weld and DelKleer 1990]) whichhave been built for these representations

(actually “composition tables” is a bettername for these structures). Below, Ibriefly survey the current state of the artin QSR (see Herrkindez [1994] for a par-tial survey and bibliography) and at-tempt to indicate some directions forfuture research in this area.

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ACM Computmg Surveys, Vol 27, No. 3, September 1995

324 ~ A. G. Cohn

Although broadly agreeing with

Davis [ 1990] in his chapter on spatialreasoning that there are relatively fewfundamental ontological problems in rep-resenting space, I would take issue withhis claim that there are no axiomatic orrepresentational problems “in the senseof new concepts that need a formal defi-nition or axiomatic systems that need tobe formulated or evaluated.” Although itis true that mathematics provides us withboth topology and geometry, the formeris often too abstract to be useful as abasis for commonsense and qualitativereasoning about space, whilst the lattergoes too far in the other direction. Whatis needed, not only in AI (e. g., to providea semantics for spatial expressions innatural language [Aurnague and Vieu1993]) but also in geographical informa-tion systems [Clementini et al. 1994] (e.g.,to provide a basis for natural query lan-guages), are representations that allowextended spatial entities and the rela-tions between them to be described at anappropriate level of abstraction; further-more, these representations should havegood computational properties.

What, then, are the key areas andchallenges of qualitative spatial reason-ing research? Although it is superficiallyattractive to attempt to model space byconsidering each dimension separately,this is easily seen to be inadequate: forexample, two individuals may overlapwhen projected to both the .-c and y axesindividually, when in fact they do notoverlap at all. Following earlier work inthe philosophical logic literature datingback to Whitehead, Tarski, and others,much recent work has concentrated onspatial calculi for extended regions of

space (rather than points or other lower-dimensional entities)—the work con-ducted at Leeds on the Region Connec-tion Calculus (RCC) [Cohn et al. 1995]has been of this kind. It turns out to bepossible to define a rich spatial calculus(with hundreds of distinctions possible)with very few primitives. For example,RCC uses just two: a predicate to testwhen two regions are connected and afunction to denote the convex hull of a

region. The first primitive alone allowsmany topological spatial relations to bedefined, and the second allows a muchbetter handle on the thorny problem ofdescribing shape qualitatively—a prob-lem that Davis highlights and that is atthe heart of the problems in qualitativekinematics mentioned earlier. The ad-vantages of having few primitives are notonly of elegance and theoretical simplic-ity, but also of interfacing to a perceptualcomponent.

Apart from topology and shape descrip-tion, other aspects of qualitative spatialrepresentation that have been investi-gated include distance, orientation, size,granularity, and uncertainty (both of po-sition and extent). Progress has also beenmade in qualitative kinematics (e.g., al-lowing computer-aided mechanism de-sign [Faltings and Sun 1993]). Althoughsome of these are relatively straightfor-ward, there are still many interestingissues to be addressed, including inte-grating these in a single system in anefficient, understandable, and completemanner.

However, it is of equal importance,given that many applications are notconcerned with a purely static view ofthe world, to build theories of how spa-tial descriptions may change over time,given certain underlying or domain-specific constraints. The most obvioussuch constraint is continuity (already ex-ploited in traditional QR): one can builda conceptual neighborhood graph of spa-tial relations specifying which transi-tions are possible, assuming continuoustranslation and deformation of regions.Using this information and a compositiontable to check for consistency, one canbuild a simple qualitative spatial simu-lator. A desirable extension, by analogywith earlier QR work, would be to incor-porate a proper theory of spatial pro-cesses couched in a language of QSR.

Another problem is that the conceptualneighborhood and composition table areusually built manually for each new cal-culus—an arduous and error-prone oper-ation; techniques to derive either or bothof these automatically are being investi-

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The Challenge of Qualitative Spatial Reasoning “ 325

gated but still represent a considerablechallenge in general.

Just as one normally builds an eventcalculus on top of a pure temporal the-ory, so a theory of physicall objects shouldbe integrated with a pure spatial repre-sentation: properties of physical objects,such as rigidity, will affect the spatialinferences possible. Some research hasbeen dcne here but much more remains.

Although much work in QSR has con-centrated on representational aspects,various computational paradigms are be-ing investigated, including constraint-based reasoning and the interesting ideaof transforming first-order theories to

(decidable) zero-order ones by interpret-ing “propositional” letters as regions: thelogical symbols now become spatial oper-ators and one has a true spatial logicrather than a logical theory of space

[Cohn et al., 19951. Of course, decidablesystems are still not necessarily efficient,so fUrther research is needed on the com-putational aspects of QSR.

As in so many other fields of knowl-edge representation, it is unlikely that asingle universal spatial representationlanguage will emerge—rather, the bestwe can hope for is that the field willdevelop a library of representational andreasoni mg devices and some criteria fortheir most successful application. Fi-nally, QSR must not eschew quantitativeknowledge and reasoning when available

1 I mean physical objects in the broadest possiblesense here: e.g., not only bicycles but also geograph-ical entit] es such as countries and habitats.

and appropriate; for example, interfacingsymbolic QSR to the techniques beingdeveloped by the diagrammatic reason-ing community is another interesting andimportant challenge.

ACKNOWLEDGMENTS

The support of the SERC under grants GR/G36852,

GR/H 78955 is gratefully acknowledged. This work

has also been partially supported by a CEC ES-

PRIT basic research action, MEDLAR II, 6471. I

would like to acknowledge all the useful discussions

I have had with the rest of the qualitative spatial

reasoning group at Leeds and the Spacenet commu-

nity.

REFERENCES

AURNAGUE, M. AND VIEU, L. 1993. A three-level

approach to the semantics of space. In TheSemantics of Prepos~t~ons—From Mental Pro-

eessmg to Natural Language Processing, C.Zelinsky-Wibbelt, Ed., Mouton de Gruyter,

Berlin.

CLEMENTINI, E., SFIARNIA, J., AND EGENHOFER, M. J.1994. Modeling topological spatial relations:

strategies for query processing. Cornput. Graph.18, 6, 815-822,

COHN, A. G., GOODAY, J. M., BENNETT, B., AND GOTTS,N. M. 1995. A logical approach to represent-ing and reasoning about space. Al Retxew (toappear).

DAVIS, E. 1990. Representations of CommonsenseKnowledge. Morgan-Kaufmann, San Mateo, CA.

FALTINGS, B. AND SUN, K. 1993. Computer-aided

creative mechanism design. In Proceedings IJ-CAZ93. Morgan-Kaufmann, San Mateo, CA.,1451-1457.

HERN.LNDEZ, D. 1994. Qualitative Representationof Spatial Knowledge. Lecture Notes in Artifi-

cial Intelligence, Vol. 804, Springer-Verlag.

WELD, D. A. AND DE KLEER, J., Ed. 1990. Read-ings in Quahtative Reasoning About PhysicalSystems. Morgan-Kaufmann, San Mateo, CA.

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