(Some) Default and Non-Monotonic Aspects of Qualitative Spatial ReasoningA Preliminary Report

Mehul Bhatt

SFB/TR 8 Spatial CognitionUniversitat Bremen, Germany

bhatt@informatik.uni-bremen.de

Abstract

We present some scenarios where default and/or non-monotonic reasoning patterns are either necessary or usefulfor the modelling of dynamic spatial environments. The iden-tified instances bear a direct relationship to the fundamen-tal epistemological issues relevant to the frame and ramifi-cation problems; these are cases where a typical use of non-monotonicity is necessary at a meta-theoretical or domain-independent level. Furthermore, non-monotonic reasoningis also necessary whilst modelling the appearance and dis-appearance of spatial objects – these phenomena, consid-ered inherent to dynamic spatial systems, essentially involvethe representation of an incompletely known domain of dis-course. The final case, from the viewpoint of this paper,involves the use of non-monotonic reasoning for modellingcausal explanation tasks in an abductive manner. Indeed,the non-monotonic patterns we illustrate are only indicativeof the ones encountered and accounted for in the context ofour key task of developing a situation calculus based domain-independent qualitative spatial theory that is usable in diversedynamic spatial domains. The identification of other similarpatterns and the general utility of non-monotonic reasoningfrom a specific spatial reasoning viewpoint is an important re-search agenda, and this paper calls for a further investigationof the same within the mainstream qualitative spatial reason-ing domain.

IntroductionThe integration of specialised spatial representation andreasoning techniques within general common-sense and/orlogic-based reasoning frameworks is an important next-stepfor their applicability in realistic domains. This integrationis non-trivial and requires unification along ontological, rep-resentational and computational fronts, i.e., a paradigm suchas ‘Reasoning about Space, Actions and Change’ needs tobe pursued and promoted. Indeed, this is also closely re-lated to the general problem pertaining to the sub-division ofendeavours in AI and the development of a unifying seman-tics for logic-based common-sense reasoning and other spe-cialised reasoning domains such as qualitative spatial rea-soning. The point is adequately summed up by Shanahan[1995]:

Copyright c© 2008, Association for the Advancement of ArtificialIntelligence (www.aaai.org). All rights reserved.

‘If we are to develop a formal theory of commonsense,we need a precisely defined language for talking aboutshape, spatial location and change. The theory willinclude axioms, expressed in that language, that cap-ture domain-independent truths about shape, locationand change, and will also incorporate a formal accountof any non-deductive forms of commonsense inferencethat arise in reasoning about the spatial properties ofobjects and how they vary over time’

Past work has pursued the broad agenda of integrating thespecialisation of ‘qualitative spatial reasoning’1 within gen-eral logic-based frameworks in AI. Specifically, as a meansto narrow down this problem of integration, the emphasishas been on operationalising a ‘dynamic spatial systems’ ap-proach for the modelling of changing spatial environments[Bhatt and Loke, 2008, Bhatt, 2008]. Here, a dynamic spa-tial system is regarded as a specialisation of the dynamicsystems [Sandewall, 1994] concept for the case where spa-tial configurations undergo change as a result of interac-tion within the environment, i.e., as a result of explicitlyidentified events and actions within the system being mod-elled. In this context, we have investigated the construc-tion of a ‘domain-independent qualitative spatial theory’,which may be utilised in diverse application scenarios thatinvolve the modelling of dynamically varying spatial infor-mation. Grounded in this previous work, we identify and il-lustrate some instances where default and/or non-monotonicreasoning patterns are either necessary or useful for repre-senting and reasoning about dynamic spatial environments.The identified instances are a direct product of our attemptto operationalise the aforementioned notion of a qualitativespatial theory within the framework of the situation calcu-lus, which is a general formalism for modelling dynamicdomains [McCarthy and Hayes, 1969]. Specifically, and inso far as the scope of this paper is concerned, the identi-fied requirements for non-monotonic reasoning fall withinthe following conceptual categories:

• Epistemological: These instances bear a direct relation-ship to the fundamental epistemological issues relevant tothe frame and ramification problems, i.e., these are cases

1A comprehensive review of qualitative spatial reasoning canbe found in [Cohn and Hazarika, 2001]

where appeal to non-monotonicity is necessary at a meta-theoretical level

• Phenomenal: Non-monotonicity is also necessary whilstmodelling phenomena involving appearance and disap-pearance of spatial objects – this essentially involves rep-resenting an incompletely known domain of discourse.We hypothesize that modelling of other phenomenal as-pects that are considered intrinsic to dynamic spatialsystems will lead to the identification of other scenar-ios whereas non-monotonic reasoning is necessary and/oruseful

• Reasoning tasks: The final case, given the scope of thispaper, involves the use of non-monotonic reasoning formodelling causal explanation tasks in an abductive man-ner. Here, the application of non-monotonic reasoningis intrinsic to an abductive formalisation of explanation[Shanahan, 1993, 1997]

Indeed, the non-monotonic patterns we identify and il-lustrate are only indicative of the ones encountered andaccounted for in the context of a limited task involvingthe development of the situation calculus based domain-independent qualitative spatial theory [Bhatt and Loke,2008, Bhatt, 2008]. The identification of other patterns andthe general utility of non-monotonic reasoning from a spe-cific spatial reasoning viewpoint is an open and importantresearch agenda, and this paper calls for a further investi-gation of the same within the mainstream qualitative spatialreasoning domain.

Dynamic Spatial Environments and Need forNon-Monotonicity

Application domains such as intelligent systems, spatial in-formation systems, temporal and event-based GIS and cog-nitive robotics involve the modelling of dynamically vary-ing spatial information in one way or another. These appli-cation domains require modes of reasoning that are richerthan those provided by conventional constraint-based rea-soning techniques that are prevalent in the qualitative spatialreasoning domain [Renz and Nebel, 2007]. Of course, theutility of the conventional apparatus is not disputed, rather,the point here is that a level-of-abstraction higher than thatprovided by such techniques is essential. For instance, atlower-levels, constraint-based reasoning techniques remainapplicable, but at higher (application) levels, other forms ofreasoning, e.g., in the form of spatial planning and explana-tion, in the context of general logic-based reasoning frame-works are both necessary as well as useful for applicabilityin realistic domains such as the ones aforementioned.

Consider, for instance, a logic-driven intelligent systemthat, amongst other things, has a spatial component in-volving representing and reasoning with qualitative spatialknowledge. One application requirement here is to modelreasoning tasks that involve non-monotonic knowledge up-dates in the form of spatial information assimilation (e.g.,by way of explanation) and other forms of useful reason-ing tasks involving spatial property projection and spatialplanning. Indeed, if the spatial component is to be based

on existing (qualitative) theories of space, it is essential thatsuch theories (precisely, qualitative spatial calculi pertain-ing to differing aspects of space such as topology [Randellet al., 1992], orientation [Moratz et al., 2000]) be embed-ded with the general logic-based framework under consid-eration. Most importantly, it is essential that the followinghigh-level axiomatic semantics of the calculi being embed-ded be preserved:

1. jointly exhaustive and pair-wise disjoint property of therelations

2. other basic properties of the relations including symmetry,asymmetry and transitivity

3. compositional inference and consistency maintenance4. a primitive notion of change based on the principle of con-

ceptual neighbourhoods [Freksa, 1991]5. relative entailments between calculi when more than one

spatial calculus is being modelled in a non-integratedmanner (i.e., there exist separate composition tables)Additionally, it is also desired that phenomenal aspects

(e.g., appearance of new objects) considered inherent in typ-ical dynamic spatial systems be accounted for within thespatial theory that is required to be embedded within thegeneral logic-based reasoning framework. The main objec-tive of such an embedding is that all modes of reasoning thatare supported by the containing logical framework, whichin our case is the situation calculus, can then be directly ex-ploited for modelling useful reasoning tasks in the contextof the spatial component. However, the endeavour is fraughtwith several complexities – if a true integration of the spe-cialisation of qualitative spatial reasoning within a generallogic-based reasoning framework is to be achieved, key re-quirements from a specific ‘dynamic spatial systems’ view-point need to be accounted for:

1. Qualitative physics: seamless integration of a domain-independent ‘qualitative physics’ that is based on existingqualitative theories of space

2. Causal and teleological aspects: support for modellingand reasoning with dynamic teleological and causal ac-counts of a system or process in addition to the repre-sentation of the underlying (qualitative) physics (i.e., anintegration of the ’how’ and the ’why’ aspects of spatialchange)

3. Epistemological issues: investigation of the implicationsof some of the fundamental epistemological problems(frame, ramification and qualification), which have other-wise assumed a primary significance within the symbolicartificial intelligence domain, for the special case of dy-namic spatial systems

4. Non-monotonic reasoning: incorporation of non-monotonic or default forms of inference, which isnecessitated by the requirement to model human-likecommon-sense reasoning patterns and the need to accountfor the fundamental epistemological issues in (3), and

5. Concurrency in the spatial domain: an account of concur-rency and continuity for the specialised spatial domain inthe context of existing temporal reasoning approaches

Figure 1: Spatial Property Persistence

Indeed, there exist no clear benchmarks for the fulfilmentof each of the above-stated requirements. In so far as thescope of this paper is concerned, we focus on the aspectsthat are related to requirements (1), (3) and (4) from above.

Default and Non-monotonic Aspects of SpatialReasoning

Several forms of non-monotonic inference are useful, and insome cases even necessary, when reasoning about changingspatial relationships within a dynamic spatial environment.The objective in this paper is to intuitively illustrate the casesusing examples; axiomatisation and treatment using the sit-uation calculus formalism have been covered in detail else-where [Bhatt and Loke, 2008, Bhatt, 2008]. Although notstrictly necessary, basic familiarity with the representationalaspects of situation calculus will certainly aid the discussionto follow:

Spatial Property Persistence (‘the frame problem’)Spatial property persistence, i.e., the intuition that the spa-tial relationship between two objects typically remains thesame, is one default reasoning pattern rooted in the frameproblem that is identifiable within the spatial context. For in-stance, assuming that dynamic topological and orientationalinformation constitutes the state descriptions correspondingto the unique ‘situations’2, the problem is that of formal-ising the intuition that the topological relationship betweentwo objects or the orientation of an object relative to another‘typically’ remains the same, unless if there is ‘cause’, what-ever be the nature of such cause, to believe to the contrary.Consider Fig. 1, which qualitative depicts the relationship ofan agent, modelled as a directed line-segment (‘b’) to a con-taining object (‘a’) that is interpreted as a room. Given thatthe spatial relationship of the agent with that of the room isthat of containment, i.e., inside(b, a), the problem of spatialproperty persistence is that of formalising the intuition thatthis containment relationship persists in the situation result-ing from the occurrence of an action such as turn around.At least one other instance, addressing this line of investiga-tion, can be found in the work of Shanahan [1995]. Withina real-valued co-ordinate system, Shanahan investigates the

2A situation can be interpreted as a unique node within abranching-tree structured situation space. Corresponding to eachsuch node or ‘situation’ is a state description denoting which dy-namic properties hold in that situation.

Figure 2: Compositional Constraints and Ramifications

default reasoning pattern required to model a different kindof property persistence, also connected to the frame prob-lem, required to model the common-sense law that ‘spaceis typically empty’.3 For instance, an agent would needto make such a default assumption before moving itself ormoving other objects to a certain region of space or whenother domain specific occurrences have happened.

In the context of the situation calculus formalism, ageneric frame assumption incorporating the principal of in-ertia whilst deriving the successor state axioms [Reiter,1991] is sufficient to handle this and other similar scenar-ios that involve spatial property persistence.4

Global Consistency of Spatial Information (‘theramification problem’)Spatial situation descriptions denoting configurations of do-main objects, i.e., by way qualitative spatial relationshipsrelevant to one or more spatial dimensions that hold betweenthe objects of the domain, must be globally consistent in ad-herence to the compositional constraints of the underlyingqualitative space, i.e., all composition theorems as appli-cable for a particular spatial domain being modelled (e.g.,topology, orientation) should be satisfiable for the spatialsituation descriptions. The notion of compositional consis-tency also includes those scenarios when more than one as-pect of space is being modelled in a non-integrated way, i.e.,relative dependencies between mutually dependent spatialdimensions that are modelled explicitly too should be satis-fiable. Ensuring these two aspects of global consistency ofspatial information is non-trivial because the compositionalconstraints contain indirect effects in them thereby necessi-

3Depending on the purpose for which such a default assumptionis applied, one could argue that this also has connections to thequalification problem. But this is a different issue altogether sincethe common-sense law remains the same.

4Successor state axioms constitute the causal laws of the do-main – these determine what changes as a result of an occurrencewithin the (spatial) system being modelled [Reiter, 1991].

tating a solution to the ramification problem [Finger, 1987],i.e., the problem of indirect effects. In the context of thesituation calculus, Lin [1995] illustrates the need to distin-guish ordinary state constraints from indirect effect yieldingones, the latter being also referred to as ramification con-straints. This is because when ramification constraints arepresent, it is possible to infer new effect axioms (or simplyeffects) from explicitly formulated (direct) effect axioms to-gether with the ramification constraints. Simply speaking,ramification constraints lead to what can be referred to as’unexplained changes’, which is clearly undesirable withina theory of change. In the spatial representation task, i.e.,the embedding of qualitative spatial calculi within the spa-tial theory [Bhatt, 2008], indirect effect yielding constraintsare a recurring problem – as mentioned already, modellingcomposition theorems and axioms of interaction (using or-dinary state constraints) leads to unexplained changes sincethe resulting constraints contain indirect effects in them. Forinstance, this is evident whilst performing compositional in-ference with the (three) mutual spatial relationships involv-ing the trivial case of three objects o1, o2 and o3 – wheno1 and o2 undergo a transition to a different qualitative state,this also has an indirect effect on the relationship between o1

and o3 since the relationship between the latter two is con-strained by the compositional constraints of the relationalspace. As an example, consider the illustration in Fig. 2 –the scenario depicted herein consists of the topological rela-tionships between three objects ‘a’, ‘b’ and ‘c’. In the initialsituation ‘S0’, the spatial extension of ‘a’ is a non-tangentialpart of that of ‘b’. Further, assume that there is a change inthe relationship between ‘a’ and ‘b’, as depicted in Fig. 2,as a result of a direct effect of an event such as growth or anaction involving the motion of ‘a’. Indeed, as is clear fromFig. 2, for the spatial situation description in the resultingsituation (either ‘S1’ or ‘S2’), the compositional dependen-cies between ‘a’, ‘b’ and ‘c’ must be adhered to, i.e., thechange of relationship between ‘a’ and ‘c’ must be derivableas an indirect effect. In a trivial scenario, such as the presentone, consisting of few objects, it could be correctly arguedthat the indirect effects can be completely formulated as di-rect effects. However, for a more involved scene descriptionn objects and complete n-clique descriptions consisting ofn(n − 1)/2 spatial relationships for every spatial domain(e.g., topology, orientation, size) being modelled is impracti-cal and error prone. The situation is only complicated giventhat fact that some of the spatial domains being modelledcould be inter-dependent.

Ramification, Causality and Non-Monotonic ReasoningWhilst the details are not relevant here, it suffices to pointout that a solution to the problem of ramifications for thisparticular case (of ensuring global compositional consis-tency of spatial scene descriptions) is obtainable from thegeneral works of Lin and Reiter [1994], Lin [1995]. Thesolution basically involves appeal to causality (i.e., mod-elling all ramification yielding constraints in the form ofcausal rules) and applying non-monotonic reasoning (us-ing circumscription) to minimise the effects of occurrenceswhilst deriving the successor state axioms or the causal laws

Figure 3: Appearance Events - Delivery Example

of the domain. Note that this manner of deriving the succes-sor state axioms is an extension to the original approach pro-posed by Reiter [1991], where only a solution to the frameproblem is included under a general ‘completeness assump-tion’ stipulating that there are no indirect effects within thedomain theory. The application of this approach to ensur-ing global consistency of spatial scene descriptions has beenillustrated in detail in [Bhatt and Loke, 2008].

Phenomenal Aspect - Appearance andDisappearance of ObjectsAppearance of new objects and disappearance of existingones, either abruptly or explicitly formulated in the domaintheory, is characteristic of non-trivial dynamic spatial sys-tems. In robotic applications, it is necessary to introducenew objects into the model, since it is unlikely that a com-plete description of the robot’s environment is either speci-fiable or even available. Similarly, it is also typical for amobile robot operating in a dynamic environment, with lim-ited perceptual or sensory capability, to lose track of certainobjects because of issues such as noisy sensors or a lim-ited field-of-vision. As an example, consider a ‘delivery sce-nario’ in which a vehicle/robot is assigned the task of deliv-ering ‘object(s)’ from one ‘way-station’ to another (see Fig.3). In the initial situation description, the domain consists ofa finite number of ‘way-stations’ and deliverable ‘objects’.However, the scheduling of new objects for delivery in fu-ture situations will involve introducing new ‘objects’ intothe domain theory. For example, an external event5 suchas ‘schedule delivery(new load, loc1, loc3)’ introducesa new object, namely ‘new load’, into the domain.

Appearance and disappearance events involving the modi-fication of the domain of discourse are not unique to applica-tions in robotics. Even within the projected next-generationof event-based and temporal geographic information sys-tems, appearance and disappearance events are regarded tobe an important typological element for the modelling of dy-

5External events are those occurrences that do not have an as-sociated occurrence criteria and may therefore occur abruptly.

Figure 4: Appearance Events - Propagating ExistentialFacts

namic geospatial processes [Claramunt and Theriault, 1995,Worboys, 2005]. For instance, Claramunt and Theriault[1995] identify the basic processes used to define a setof low-order spatio-temporal events which, among otherthings, include appearance and disappearance events as fun-damental. Similarly, toward event-based models of dynamicgeographic phenomena, Worboys [2005] suggests the use ofappearance and disappearance events at least in so far as sin-gle object behaviours are concerned. We regard that suchphenomena, being intrinsic to a typical dynamic spatial sys-tem, merit systematic treatment.

Maintaining and Propagating Existential Facts From arepresentational viewpoint, introducing new objects in thedomain poses a problem since there is no general way todeal with an incompletely known domain of discourse. Forinstance, let < s0, s1, s2, . . . , sn > denote a situation-based linear history or one branch within the branching-treestructure of the overall situation space (see Fig. 4). From adynamic spatial system perspective, each state correspond-ing to every situation with this history is primarily a set de-noting the spatial configuration of objects in that situation.Further assume that an object ‘b’, that is unknown or nota part of the dynamic ‘spatial configuration set’ in the ini-tial situation ‘s0’, comes into existence (by an appearanceevent) in a later situation, say ‘s2’. At this point, it is nec-essary to incorporate the non-existence of ‘b’ in the situa-tions preceding ‘s2’ by (non-monotonically) propagating itsnon-existence backwards into the situation-based history. Infact, appearance of previously unknown objects is the onlyreason ‘existential facts’ about objects need to be includedas propositional fluents / dynamic properties at a domain-independent level. The case of disappearing objects is trivialand simply involves negating and object’s existential statusupon the occurrence of disappearance events. Indeed, anobject that is known but has disappeared may not participatein spatial relationships with other objects, until such a timewhen it reappears.6

Reasoning Requirement – Explanation asAbductionExplanation, in general, is regarded as a converse operationto temporal projection essentially involving reasoning from

6Presently, upon reappearance, it is presumed that an object’sidentity is maintained.

Figure 5: GIS Domain - Deforestation

effects to causes, i.e., reasoning about the past [Shanahan,1989]. Precisely, given a set of time-stamped observationsor snap-shots (e.g., observation of a mobile-robot or time-stamped GIS data), the objective is to explain which eventsand/or actions may have caused the resulting state-of-affairs.In the context of the situation calculus formalism, Shana-han [1993, 1997] proposes a non-monotonic approach thatutilises circumscription as a basis of minimisation (of ef-fects) and explanation derivation (in terms of potential oc-currences). We have specialised this approach toward theformulation of an abductive occurrence-driven causal expla-nation task, where a set of time-ordered observations (e.g.,pertaining to spatial configurations) may be explained interms of the spatial actions and events that may have causedthe observed state-of-affairs [Bhatt, 2008].

The non-monotonicity required in modelling explanationtasks is characteristic to modelling explanation problems ab-ductively in general, rather than being peculiar to spatial rea-soning tasks. However, one aspect of this non-monotonicityis characteristic to a spatial reasoning task – in derivingminimal models or explanations of observations consistingof changing spatial configurations, it is possible that thederived explanations may be inadequate, i.e., may not in-clude domain-specific occurrences that have caused the ob-served changes. For instance, consider a geographic infor-mation system domain / scenario as depicted in Fig. 5. Ata domain-independent level (i.e., at the level of a generalspatial theory), the scene may be described using topolog-ical and qualitative size relationships. Consequently, theonly changes that are identifiable at the level of the spa-tial theory are shrinkage and eventual disappearance – thisis because a domain-independent spatial theory may onlyinclude a generic typology of spatial change at the most.However, at a domain-specific level, these changes couldcharacterise a specific event (or process) such as, for in-stance, deforestation. The hypotheses or explanations thatare generated during a explanation process should necessar-ily consist of the domain-level occurrences in addition to theunderlying (associated) spatial changes (as per the generictypology) that are identifiable. That is to say, that the de-rived explanations be ‘adequate’ and more or less take aform such as: ‘Between time-points ti and ti, the process ofdeforestation is abducible as one potential hypothesis’. Toachieve this adequacy, a model-filtration heuristic that dis-regards those models (i.e., explanations) that do not include

any domain-specific (spatial) occurrences (actions or events)leads to explanations that are adequate, if such explanationexists per se – this is because minimal models that only con-sist of a domain-independent explanation (e.g., in the formof shrinkage, disappearance and a temporal-order betweenthese two) would be excluded by such a filtration heuristic.

Other potential solution to achieve adequacy is to in-clude high-level or domain-specific predicates that relate thedomain-independent occurrences (as per the typology) to ar-bitrary high-level processes that have a domain-dependentinterpretation. Notwithstanding the fact that we regard bothpotential solutions to the problem of achieving adequacyto be rather rudimentary or ad-hoc solutions, it must bepointed out that the model-filtration approach is more gen-eral and does not presuppose any information of the domain-independent typology on the part of a domain modeller.

OutlookWe have identified and illustrated some instances wheredefault and/or non-monotonic reasoning patterns are eithernecessary or useful for the modelling of dynamic spatial en-vironments. Although the details have not been includedhere, each of the identified instances has been grounded us-ing the situation calculus formalism, using a broader con-text of modelling dynamic spatial systems [Bhatt and Loke,2008]. Property persistence, i.e., the spatial relationship be-tween two objects ‘typically’ remains the same, is one de-fault reasoning pattern connected to the frame problem thatwas encountered here. Similarly, the ramification problemarises whilst modelling constraints containing indirect ef-fects (e.g., compositional constraints, axioms of interactionbetween inter-dependent spatial domains). It is also evi-dent that several forms of non-monotonic inference are use-ful (e.g., in deriving the successor state axioms and also inexplanation tasks), and in some cases even necessary (e.g.,compositional constraints), when reasoning about changingspatial relationships between objects. Non-monotonic rea-soning is also required to model explanation tasks in an ab-ductive manner. Clearly, in addition to aspects already ac-counted for, a closer look at application-level use-cases thatmay benefit from default or non-monotonic reasoning pat-terns is an important next step and deserves utmost prior-ity. The identification of ‘common-sense’ laws or reasoningpatterns encountered in qualitative reasoning about space –patterns that are applicable in general, or for specific spatialreasoning tasks, is one of the most interesting and importantresearch agendas, at least from the viewpoint of the futuredirection of this work.

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