Open World Planning as SCSPAlexander Nareyek

GMD FIRSTGerman National Research Center for Information Technology

Research Institute for Computer Architecture and Software TechnologyKekul~strafle 7

D - 12489 Berlin, Germany

alex%ai-center.com

Abstract

Planning for the real world requires the ability to senseand reason about an arbitrary number of entities andrelations that are not known in advance. However,most satisfaction-based planning systems only reasonabout a fixed number of given unique fiuents. Thisarticle presents the planning model of the constraint-based EXCALIBUR agent’s planning system. The modelis based on structural constraint satisfaction, whichmakes it possible to use arbitrary plan structures andto handle open world planning.

IntroductionConventional planning approaches use highly specificrepresentations and algorithms, e.g., STRIPS (Fikes& Nilsson 1971), UCPOP (Penberthy & Weld 1992)and PRODIGY (Veloso et al. 1995). Newer approachesare often based on more general search frameworks likepropositional satisfiability (SAT), operations research(OR) and constraint programming (CP). Examples clude SATPLAN (Kautz & Selman 1996), ILP-PLAN(Kautz & Walser 1999) and CPlan (Van Beek & Chen1999). The advantage of using a general framework in-stead of specific approaches is the availability of ready-to-use off-the-shelf methods and the more general ap-plicability of the new methods developed for the spe-cific domain. In addition, future changes in the prob-lem specification must be reflected only at the modelinglevel and not in the underlying search algorithms.

However, the general search frameworks require agiven structure of variables and constraints and search"only" for consistent values for the variables. For thetask of planning, though, there are lots of possible struc-tures. Thus, a kind of maximal structure is used inwhich unnecessary parts can be deactivated by certainvalue assignments. If a solution cannot be found, thestructure is further expanded to allow for longer plansand the search .starts again. But the (exponential)structure expansion is only feasible for small problems.For an open planning world, where it is not clear in ad-vance how many and which types of fluents/resourcesare involved1, the use of these maximal structures is apointless approach.

1Incomplete knowledge can be related to various other

The approach of structural constraint satisfaction(Nareyek 1999a; Nareyek 1999b) combines conventionalconstraint satisfaction with structural requirements andenables us to formulate and solve combinatorial searchproblems without explicitly giving the solution’s struc-ture. The approach is used in this paper to tackle plan-ning problems without a closed-world assumption.

The planning problem is specified as a so-called struc-tural constraint saris]action problem (SCSP). From theSCSP, productions can be automatically derived thatcan be used to create/modify the structure of con-ventional constraint satisfaction problems (CSPs). the search for constraint satisfaction, local search tech-niques are applied. The constraints are enhanced bydomain knowledge and can vary their variables~ valuesand apply productions to change the CSP’s structure soas to improve their current satisfaction. Specific struc-tural constraints must be maintained during this pro-cess.

(Nareyek 2000) presents an SCSP formulation forplanning that is restricted to finite planning worlds.This article presents an extended version capable ofhandling open planning worlds.

The Model’s Basics

This section introduces the model’s basic concepts. Aformal specification will be given in a later section.

The model focuses on resources: A resource (alsocalled state variable or fluent)2 is a temporal projectionof a specific property’s state, which may be subject toconstraints such as preconditions and changes. Numer-ical as well as symbolic properties are uniformly treatedas resources. For example, a battery’s POWER and thestate of a DOOR are resources:

features, like an uncertain outcome of actions. Here, how-ever, we consider only issues related to the existence of enti-ties. Other aspects of incomplete knowledge are handled bycertain value assignments within the model (see (Nareyek1998))

2The term resource is a common term in the CP/ORcommunity and is used here because of the planningsystem’s close connection with applications for resourceallocation/optimization.

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From: AAAI Technical Report WS-00-02. Compilation copyright © 2000, AAAI (www.aaai.org). All rights reserved.

POWER is [ 0: t E [0..5], 10-- 0.75 x t: t E [6..13],0: t e ],

Dooa is [ OPEN: t E [0_45] , CLOSED: ~ e [46..60],UNKNOWN: ~ e [61..c~[ ].

Actions, Action Tasks and ActionResources

The execution of an action (like EAT PEANUT) in-cludes action task subcomponents. These action tasksrepresent different operations that are necessary tocarry out the actions. Each of the action tasks utilizesan action resource for its execution. For instance, theaction EAT PEANUT requires action tasks on a MOUTHand a LEFT HAND or RIGHT HAND action resource.Figure 1 visualizes the assignment of action tasks toaction resources.

Feet

Mouth

Left Hand

Right Hand

Time

Action Resource

Current Time Action Task

@@

28 sea

I I30 sea 35 sea

Figure 1: The Assignment of Action Tasks tb ActionResources

It is forbidden for action tasks on the same actionresource to overlap, as simultaneous executions of taskswould interfere with each other. For example, the agentis not allowed to talk and eat with his MOUTH at thesame time.

The tasks of an action are subject to action-specificconditions. For example, the action tasks of the actionEAT PEANUT must begin and end at the same time, andthe begin and end values must be four seconds apart.

State Resources, State Tasks andPrecondition Tasks

A state resource is similar to an action resource. Itdoes not manage actively planned actions, but ratherthe development of a specific property of the environ-ment or the agent itself. For example, an OWN PEANUTstate resource with a Boolean assignment for any onetime can provide information about the possession of apeanut (see Figure 2).

The status of the state resources can restrict theapplication of actions. To execute the action EAT

Precondltlon Task

Own Peanut C~n

Mouth ~~

~" , ...........i,,

Time I I

- ~ 30 sec. 35 sec.28 sea

State Resource

Figure 2: A State Resource

PEANUT, it is first necessary to have a peanut. Theserelations are checked by precondition tasks of ac-tions. A precondition task includes a state value (orvalue ranges) that must correspond with the state of related state resource at a specific time.

The effects of actions are more complicated to real-ize, as multiple actions and events may have synergeticeffects. For example, a state resource HUNGER withassignments of natural numbers can be influenced by abeneficial action EAT PEANUT and a detrimental WALKat the same time.

It is the job of state tasks to describe an action’seffects. For instance, a state task of the action EATPEANUT is responsible for a decreasing contributionof-3 to the state resource HUNGER during the action’sexecution (see Figure 3). Each state resource has a spe-cific state mapping that maps the contributions of thestate tasks to values of the state resource’s domain. Inthe case of the HUNGER resource, the synergetic effectis a simple addition of the state tasks’ single gradients.

Feet

Mouth

Hunger

Time

Current Time_ ._]=,,.

~ BBI~~’~ State Task

I

-- ~ 30 sec. 35 eec.28 sec.

Figure 3: The Mapping Mechanism of State Resources

There can be further effects, which may be caused bysynergetic effects within a state resource. Adding waterto a bathtub may result in its overflowing and wettingthe bathroom. The actions cannot provide state tasksto realize these further effects because an action hasonly the limited view of its state task contributions.Thus, dependency effects of specific state resourcestates must be expressed in addition. The dependen-

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cies are special actions that are beyond the agent’s con-trol. Expected external events can also be integratedby these dependencies.

Objects and References

In a finite and known world, there is a fixed set of re-sources to be considered. Static relations to specificstate resources can be used to realize the effects ofactions, e.g., that the state tasks of the action EATPEANUT affect the state resources PEANUT LOCATIONand PEANUT NUTRITIVE VALUE. But such static rela-tions are no longer possible in an open world, where itis unclear which and how many resources exist 3. Thus,as a consequence of the open world assumption, an ac-tion’s tasks must be specified with variable referencesfor the state resources involved (e.g., EAT X instead ofEAT PEANUT).

The next problem that arises by dropping the closed-world assumption is that the relations between the stateresources themselves are no longer fixed. For example,there could be two peanuts, a big and a small one, andthus multiple state resources PEANUT LOCATION andPEANUT NUTRITIVE VALUE. The resources of the sametype are indistinguishable, and it is not clear which twobelong to a specific peanut. If the EAT PEANUT actionwere applied, it is not clear which resources would beaffected, and the big peanut might vanish, while thesmall one would be used to decrease hunger. Thus, inaddition to an action’s tasks, the state resources’ statesmay involve references, too.

The most common relation between state resources isan aggregation -- they form objects. As this is a veryimportant relation, it is not handled by references, butexplicitly represented in the model. For example, thestate resources PEANUT LOCATION and PEANUT NU-TRITIVE VALUE form an object PEANUT. Figure 4 illus-trates the application of the EAT action to a PEANUTobject.

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

Sensors and Existence

The agent must be capable of acquiring new informa-tion about the environment, which must somehow beintegrated into the planning process. The real-worlddata is collected by so-called sensors. The sensors re-port actual data, like the current level of hunger or theproperties of a sighted peanut. We assume high-levelsensoring that provides ready-structured objects. Sen-sors are related to the virtual objects of the plan. Figure5 shows an example of a plan to put one block on topof another where only one of the blocks has already besensed.

Having introduced the concept of sensors, we arefaced with the question of whether a planning objectwill ever be connected to a sensor, i.e., if a counterpartin the real world actually exists or will exist. For

3We assume that all possible types of resources areknown, but not the number of instances.

Mouth

Reference

Hunger

Time

l

W~

----]=,’- 30sea28sac.

m__w

I35 sac.

Figure 4: References and Objects

Current Time

Hand(.~

Time I30 sea 35 sac.

28 sea

Figure 5: Sensors

Sensor

’i

example, it is pointless optimistically creating/revisinga plan so that a matching KEY is always found next to alocked DooR. However, the existence of objects is not ayes/no matter. It is temporally dependent (a KEY maybecome available after a LOCK i a installed) and a proba-bilistic matter (the KEY may become available). Thus,we need a temporally projected probabilistic measurefor every object, which expresses the confidence thatthis object really exists -- an existence projection.

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Structural Constraint SatisfactionThis section describes the concept of structural con-straint satisfaction. For a more detailed presentation,please refer to (Nareyek 1999a) and (Nareyek 1999b).The concept is based on algebraic graph grammars(Ehrig, Pfender gc Schneider 1973; Rozenberg 1997).

A structural constraint satisfaction problem can beused to overcome the deficiencies of conventional CSPswith respect to structural variety. In an SCSP, the con-straint graph is not explicitly given. Only the types ofconstraints (together with their possible connections)and structural constraints that restrict certain graphconstellations are specified.

A CSP can be represented by a graph, in which thevariables are circular nodes and the constraints rectan-gular nodes, an edge between a variable and a constraintindicating the variable’s involvement in the constraint.Since a constraint may involve multiple variables, theremust be a way to indicate a variable’s role in the con-straint. The direction -- and a possible label -- of anedge can be used for this purpose.

For structural constraint satisfaction, a distinction ismade between extensible constraints and nonextensibleconstraints, as there may be constraint types that allowa variable number of elements to be included. Further-more, an SCSP allows the existence of so-called objectconstraints. These do not restrict the variables’ values,but merely provide structural context information. Ob-ject constraints are represented by a rectangular vertexwith a dashed outline (see Figure 6).

Extenstble ExtensibleObject Constraint Conventional Constraint

(variable number of (variable number ofnon-overlap constraints) task constraints)

Start Duration

ExtensibleConventional Constraint

(variable number ofincoming variables)

Start Duration

Variable

NonextensibleConventional Constraint

Figure 6: An Example Graph

Structural constraints allow us to formulate restric-tions on admissible constraint graphs, e.g., that thesame person is not allowed to drive a car and read apaper at the same time (see Figure 7). A structural con-straint consists of a docking part and a set of testing-part alternatives. If the docking part of a structural

constraint matches the constraint graph at some point,an alternative of the testing part must match, too. Oth-erwise, the graph is structurally inconsistent. An iden-tity of objects in the docking part and the testing partis marked by identifiers like : 1.

Testing Part. Alternative I

Docking Part

Testing Part. Alternative 2

Figure 7: An Example of a Structural Constraint

Structural constraints may involve application condi-tions. A negative application condition (NAC) specifiesa structure that is not allowed to match the graph. AnNAC is indicated by a convex dark area[e.g., see Figure11). For multiple application conditions, the conjunc-tion of the conditions must hold.

A structural constraint satis-faction problem SCSP = (CD, S) consists of a tupleof sets of constraint descriptions CD = (Cn,Ce, On, Oe)and a set of structural constraints S. The constraintdescriptions of gn and On are pairs (c, pba,e) with anonextensible conventional (or object) constraint c andits embedding graph Pbaae. The constraint descriptionsof ge and Oe are 4-tuples (C, Pbase, E, pma=) with an ex-tensible conventional (o~ object) constraint c, its mini-mal embedding graph Phase, a set of extension graphs Eand the constraint’s maximal embedding graph P,na=.

An embedding graph shows the constraint with allits directly connected neighbor vertices. If an extensi-ble constraint has no maximal embedding, Pma= is theempty graph. An extension graph shows the constraintconnected to the vertices that can be added in one step.

From an SCSP formulation, graph productions andfurther structural constraints to prevent redundanciescan be automatically derived (Nareyek 1999a; Nareyek1999b). The productions generate the structural searchspace (see Figure 8). However, certain parts of thesearch space are inconsistent because of the structuralconstraints.

The search mechanism used for the EXCALIBUR.agent’s planning system is based on local search. Con-ventional constraints have constraint-specific cost/goalfunctions which express the constraints’ satisfac-tion/optimization. In addition, a constraint has in-

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Start

Pa dl

Pal ~,] Pdl

Pal

Pa3 Pa3

qpOperation ExecuUo~State Begin End ResourceType Rellou.ceType

Figure 8: An Example of a Structural Search Space

ternal heuristics to improve its current state. Theseheuristics can perform knowledge-based changes in thevariables’ values and can also use productions for mak-ing structural changes in the graph. The constraint thatis to improve the current state is selected by a so-calledglobal search control.

In this way, the plan is improved step by step towarda consistent and optimal plan. Because of the applica-tion domain, computer games, the available reasoningtime is very limited. Thus, the structural constraintsare not checked during search, but only heuristics thatdo not threaten any structural constraints by structuralchanges may be used. This property of the heuristicsmust be ensured a priori.

The Planning Model as SCSPThis section defines the planning model in terms ofstructural constraint satisfaction. The representationof the planning model as an SCSP allows us to applythe techniques of (Nareyek 1999a; Nareyek 1999b) generate the structural search space. Figure 9 givesan overview of possible relations between the planningSCSP’s elements.

The Current TimeThe very first thing we need is a variable for the currenttime because the constraints’ heuristics and cost/goalfunctions use this as input. For example, actions thatare still to be executed should not be placed in thepast. The variable is marked by a CURl:tENT TIMEobject constraint (see Figure 10). As there can onlybe one current time, we need a structural constraint toprevent there being multiple variables representing thecurrent time (see Figure 11).

ActionsAn action consists of a set of different preconditions,operations and resulting state changes. These elementsare represented by tasks, i.e., there are PRECONDITION

Figure 9: Possible Type Relations

P~ CurrentTIme =

Figure 10: The Extensible Object Constraint CURRENTTIME

~S CurrentTIme~

Figure 11: The Structural Constraint CURRENT TIME

39

TASKS for precondition tests, ACTION TASKS for oper-ations and STATE TASKS for state changes. All tasksare represented by object constraints and must be con-nected to a TASK CONSTRAINT.

A TASK CONSTRAINT enforces a certain task config-uration (i.e., it specifies which tasks are to be connectedto the TASK CONSTRAINT and what kind of restrictionsapply to the tasks’ variables) for a specific action, in-cluding the temporal order of the tasks. The specificaction to be expressed is determined by the value ofa connected ActionType variable (see Figure 12). Thecost function of the’TASK CONSTRAINT describes thedistance from the current task configuration to the con-figuration that is demanded by the ActionType variable.In addition, the Begin of nonexecuted ACTION TASKSbefore the CURRENT TIME is penalized.

Pb~¢ TaskConstralnt

Figure 12: The Extensible Conventional TASK CON-STRAINT

An action’s task may not also be part of another ac-tion. This is ensured by the structural constraint inFigure 13 (an ellipse with a dotted outline matches anynode).

f~% CurrentTime ~

~~ S TaskConstralnt

Figure 13: The Structural TASK CONSTRAINT

Operations

An ACTION TASK specifies a concrete operation thatmust be executed within an action (see Figure 14).

An ACTION TASK’s operation uses a specific actionresource. For an ACTION TASK’s duration, other tasksare required to leave enough of the action resource’scapacity to carry out the task’s operation. An ACTIONRESOURCE CONSTRAINT (ARC) internally projects action resource’s capacity and reflects an overload ofthe resource by its cost function4 (see Figure 15).

4 Multiple incorporation of the same ACTION TASK is pre-vented by structural constraints that prohibit redundancies.

=

Figure 14: The Extensible Object Constraint ACTIONTASK

P b=o Action ReeouroeConstralnt =

P exteadon ActlonResoumeConstra[nt =

Figure 15: The Extensible Conventional ACTION RE-SOURCE CONSTRAINT

Each ACTION TASK must be linked to a specific ARCof the required ResourceType (see Figure 16). The AC-TION TASK that is to be executed at the CURRENTTIME on an action resource is determined by the ARCby demanding a special value for the task’s Execution-State variable.

~ SAot~nTuk

Figure 16: The Structural Constraint ACTION TASK

In addition, all ARCs must have a different Resource-Type. To realize this, we can use an ALL DIFFERENTconstraint (see Figure 17). This constraint ensures thatall connected variables have different values. An ARCmust have its ResourceType variable connected to anALL DIFFERENT constraint (see Figure 18). The re-quirement that there be no other ALL DIFFERENT con-straint for different ARCs is formulated later in thefollowing sections.

The structural constraints also ensure that an AC-TION TASK cannot be connected to two ARCs at thesame time.

States.

Besides ACTION TASKS, an action consists of PRECON-DITION TASKS (see Figure 19) and STATE TASKS (seeFigure 20). The reason why STATE TASKS do not have

These structural constraints are automatically deduced fromthe SCSP formulation (see (Nareyek 1999a)).

4O

P base AIIDlfferent

Pcxtension AIIDlfferent

Figure 17: The Extensible Conventional ConstraintALL DIFFERENT

SActionResourceConstralnt

Figure 18: The StructUral ACTION RESOURCE CON-STRAINT

to be linked to a TASK CONSTRAINT will become clearlater.

Figure 19: The Extensible Object Constraint PRECON-DITION TASK

Unlike an action resource’s structures, which are onlyinternally represented in an ARC, a state resource’sstructures are explicitly stored in the model. Thisis because other constraints must access the state in-formation as well. Thus, PRECONDITION TASKS andSTATE TASKS are linked to a STATE RESOURCE objectconstraint which specifies the property that is to betested/changed. A STATE RESOURCE object constraintrelates PRECONDITION TASKs and STATE TASKs to aResourceType variable, a STATE PROJECTION and aCURRENT STATE (see Figure 21).

All PRECONDITION TASKS and STATE TASKS arerequired to be connected to exactly one STATE RE-SOURCE (see Figures 22 and 23).

The CURRENT STATE references a variable (see Fig-ure 24) that contains the STATE RESOURCE’S state atthe CURRENT TIME. The STATE PROJECTION refer-ences a variable (see Figure 25) that stores the temporalprojection of the resource’s state for the whole timeline.

P ba~ 8(steTask =

Pextenzion StateTask =

Figure 20:TASK

The Extensible Object Constraint STATE

P hue 8tateReaoume =

StateReso,Jrce=P~xt~ndonl

Poxtonsion2 StateResoume "

Figure 21: The Extensible Object Constraint STATERESOURCE

~ ~ S PrecondilionTMk

Figure 22: The Structural Constraint PRECONDITIONTASK

~ S StateTask

Figure 23: The Structural Constraint STATE TASK

Pba~e CurrentState =

Figure 24: The Extensible Object Constraint CURRENTSTATE

Pb~ StateProJecUon =

Figure 25: The Extensible Object Constraint STATEPROJECTION

41

Both, the CURRENT STATE and the STATE PROJEC-TION, must be connected to exactly one STATE RE-SOURCE (see Figures 26 and 27).

Figure 26: The Structural Constraint CURRENT STATE

S StaloProJ~tlon

Figure 27: The Structural Constraint STATE PROJEC-TION

A STATE RESOURCE CONSTRAINT (SAC) is linked the STATE RESOURCE to ensure a correct STATE PRO-JECTION (see Figure 28). The SRC uses the STATERESOURCE’S CURRENT STATE and the Contributionsof the STATE RESOURCE’s STATE TASKs to project theproperty’s state development according to the STATETASKS’ TemporalReferences on a timeline, which isstored in the STATE PROJECTION’S variable. The con-straint’s costs are computed according to satisfaction ofthe assigned precondition tests.

Pbase St ate ResourceConetralnt=

P extension : ~ ’ ’’ StateResourceCon.traint ~ StateR.ourceConstralnt ~ StateTask__~

Figure 28: The Extensible Conventional STATE P~E-SOURCE CONSTRAINT

The structural constraint of Figure 29 ensures thateach STATE RESOURCE is connected to exactly one OB-JECT and that an SRC is connected. Although it is notnecessary that all ResourceType variables of the STATE

RESOURCES have different values, this facilitates search-ing, e.g., for all NUTRITIVE VALUE STATE RESOURCESof the current plan. For this purpose, all Resource Typevariables must be connected to an ALL DIFFERENT con-straint.

The SRC is also responsible for maintaining the de-pendency effects of the resource’s state development.To accomplish this, STATE TASKS can be connected toan SRC. Thus, a STATE TASK must either be connectedto exactly one STATE RESOURCE CONSTRAINT or oneTASK CONSTRAINT. This is ensured by the structuralconstraint of Figure 30 (together with that of Figure

S 8tate~emuroe

Figure 29: The Structural Constraint STATE RE-SOURCE

13). A configuration of an SRC’s STATE TASKs thatdoes not represent the correct dependency effects hasan impact on the value of the SRC’s cost function.

~ S DependenoyEffeot

Figure 30: The Structural Constraint DEPENDENCYEFFECT ̄

ObjectsThe OBJECT aggregation is shown in Figure 31. Therole of the ExistenceProjection variable and the EXZS-TENCE CONSTRAINT are explained in a later section.

Pb.o Object "

P©xtonsionl Object =

P extension2 Object =

Figure 31: The Extensible Object Constraint OBJECT

The ObjectType variable specifies the type of the OB-JECT -- for example, a DOOR. The ObjectType variableis not directly included in the Pbaae graph, as this allows

42

the search (using the production generation of (Nareyek1999a; Nareyek 1999b)) to exchange an OBJECT’s Ob-jectType for a similar one without deleting the wholeOBJECT, e.g., if the search decides to consider a PICK-LOCK instead of a KEY. However, this necessitates thestructural constraint of Figure 32, which ensures thatan OBJECT has exactly one ObjectType variable. In ad-dition -- as with the SRC -- all ObjectType variablesare connected to an ALL DIFFERENT constraint.

~S Object

t--

Figure 32: The Structural Constraint OBJECT

Instead of a general unique identifier, the Resource-Type of a STATE RESOURCE has the task of specifyingthe STATE RESOURCE’S role within the OBJECT, e.g.,a DOOR’s LOCATION, LOCK or COLOR. To preventambiguities, all Resource Type variables of an OBJECT’sSTATE RESOURCE must be different (see Figure 32).

This means, for an ALL DIFFERENT constraint, thatthe constraint must either be connected to an ARC’s orSTATE RESOURCE’S ResourceType variable or an OB-JECT’S ObjectType variable, and that there must be noother ALL DIFFERENT constraint for the same kind oftype variables (see Figure 33).

Any STATE RESOURCE must be linked to an OBJECT(see Figure 29). Agent-internal STATE RESOURCEs,e.g., the agent’s HUNGER, can be linked to a uniqueEGO OBJECT.

A TASK CONSTRAINT (or SRC in the case of de-pendency effects) must also ensure that the connec-tion of ResourceTypes and Object Types with the TASKCONSTRAINT’S (SRC’s) STATE TASKS and PRECONDI-TION TASKs is correct with respect to the TASK CON-STRAINT’S AetionType ( Resource Type of the SRC’sSTATE RESOURCE). For example, an action EAT should link its STATE TASK with a Vanish contributionto the same OBJECT that its PRECONDITION TASK withthe location test is linked to.

References

All tasks can use OBJECT REFERENCES to OBJECTs,e.g., a general STATE TASK that causes one OBJECTX to be on top of another OBJECT V can make use ofOBJECT REFERENCES to assign its variables X and Y.Likewise, the CURRENT STATE and the STATE PRO-JECTION can have OBJECT REFERENCES (see Figure34). An OBJECT REFERENCE can also be linked toanother OBJECT REFERENCE to realize a list. For ex-ample, the STATE TASK of the previous example would

S; AIIDlfferent

Figure 33: The Structural Constraint ALL DIFFERENT

have to use a list (instead of a set) with two OBJECTREFERENCEs to distinguish the two OBJECTS (whichOBJECT has to be placed onto which). The TASK CON-STRAINT (or SRC in the case of dependency effects)must ensure that the reference structures are valid.

P bzse ObJeot Referent== ~

Pexten=lonl ObJ{tctReforence =

P oxtenslon20bjecJtReferenae ~

P extension30bJectReference =~

P ~xl*~n~ion40bleciRaferln¢e=

P ~xlenuion~ ObJQctReferQnco= ’~

P ext--=ion60bjec~tRoferenGe=

P 6xt~n=io.70bJectReferen¢a=

Figure 34: The Extensible Object Constraint OBJECTREFERENCE

43

All OBJECT REFERENCES are required to be con-nected to exactly one OBJECT that is to be referenced(see Figure 35).

Sob|eotReference

Figure 35: The Structural Constraint OBJECT REFER-ENCE

SensorsThe sensors provide data that is structured accordingto the planning model’s OBJECTS. The linking of asensor with an appropriate OBJECT is done by a SEN-SOR CONSTRAINT that links a SensorID variable, whichspecifies the real-world’s sensor, with an OBJECT (seeFigure 36). The SENSOR CONSTRAINT ensures that theOBJECT’s ObjectType variable corresponds to the sen-sor data and that the OBJECT has all necessary STATERESOURCEs. The productions for an addition/deletionof a SENSOR CONSTRAINT are not allowed to be ap-plied by any improvement heuristic. If the SENSORCONSTRAINT is satisfied, the sensor data can providevalues for the CurrentState variables of the connectedOBJECTS’ SRCs.

~ba~o 8ensofC:orls|~Jnt =

Pextcnsion SensorConstralnt=

P max Senso;’Constralnt=

Figure 36: The Extensible Conventional SENSOR CON-STRAINT

Again -- as with an OBJECT’S Objec$Type -- a SEN-SOR CONSTRAINT’S OBJECT is not directly included inthe Pbase graph, as this allows the search to exchange aSENSOR CONSTRAINT’8 OBJECT for a similar one with-out deleting the whole SENSOR CONSTRAINT, e.g., ifthe search decides that a sensed PEANUT is a new oneinstead of the PEANUTthat has been eaten. However,this necessitates the structural constraint of Figure 37,which ensures that a sensor is linked to an OBJECT.

Existence ProjectionsThe confidence projection that an OBJECT really ex-ists is realized by using an ExistenceProjection variable,which is similar to the temporal StateProjection vari-ables of SRCs, taking values between 0 and 1 at eachtime point (see Figure 31).

~ S SensolCon~mln!

Figure 37: The Structural SENSOR CONSTRAINT

The ExisteneeProjection is not only dependent onOBJECT-local features but on the whole world state.For example, it is improbable that there is a DOOR if theagent’s LOCATION is PARK. Given these global condi-tions, an EXISTENCE CONSTRAINT must be linked to allOBJECTS to be capable of accessing all relevant STATERESOURCEs. To keep the linking costs reasonably low,only one EXISTENCE CONSTRAINT is responsible formaintaining all OBJECTs’ ExistenceProjections.

Pba.~ ExistenoeConeValnt m

Figure 38: The Extensible Conventional EXISTENCECONSTRAINT

S ExlstenceC~atmlnt ~~

Figure 39: The Structural EXISTENCE CONSTRAINT

The EXISTENCE CONSTRAINT must also ensure cor-rect OBJECT configurations. For example, an OBJECTwith a DOOR ObjectType and a NUTRITIVE VALUESTATE RESOURCE gets a very poor ExistenceProjec-tion. However, OBJECTS do not have to be specifiedcompletely. For example, the DOOR’s COLOR STATERESOURCE may not be important for a plan to open it.

The ExistenceProjection of an OBJECT has a~l impacton the satisfaction of TASK CONSTRAINTS that havetasl~s connected to the OBJECT.

Problem FormulationThe SCSP of the previous sections specified a gen-eral planning problem. For a specific problem, theconstraints must be able to handle the specific do-main values and must have appropriate cost/goal func-tions and improvement heuristics, e.g., the TASK CON-STRAINT must know about the permitted action con-figurations and the EXISTENCE CONSTRAINT must beable to project the existence confidence for the domain’sOBJECTs.

A specific planning problem can include satisfactiongoals (e.g., the door is to be open at time point 2507)and optimization goals (e.g., the maximal hunger levelover time is to be as low as possible). Satisfaction goals

44

can be represented by TASK CONSTRAINTs that con-sist only of PRECONDITION TASKS, and optimizationgoals can be realized by initialization of the resourceconstraints’ goal functions5.

The satisfaction goals and optimization goals mustbe embedded in the search process. Thus, we can de-fine additional constraints that restrict variables to aspecific value (a constant) and require that these con-straints exist in a solution where they, for example,restrict the TASK CONSTRAINTS’ ActionType variablesand STATE RESOURCES’ TlesourceType variables (e.g.,see Figure 40).

P ba,~e Equals:door_open_goaJ

P b~e Equals:hunger_resource

=

=

~ SGoals

Figure 40: Specification of Satisfaction Goals

Conclusion

This paper presented the planning model for the EX-CALIBUR agents. It combines temporal planning withresource reasoning and makes it possible to situate theagents in an environment where they have only a re-stricted view of the world state.

The planning model is based on the SCSP ap-proach and features a domain-independent representa-tion. In contrast, constraint-based planning systemslike parcPLAN (E1-Kholy & Richards 1996) and CPlan(Van Beek & Chen 1999), besides their inability search in a way that is not focused on the criterion ofplan length, do not have a model for general planning,but rely instead on domain-dependent encodings.

The past decade has seen the development of afew planning systems capable of reasoning about in-complete world states, e.g., xII (Golden, Etzioni

~Of course, the whole expressiveness of the SCSP ap-proach can be used to formulate much more complicatedgoals, but for most cases, a representation using specificTASK CONSTRAINTs / resources’ goal functions is adequateto model the goals.

Weld 1994) and PSIPLAN (Babaian & Schmolze 1999).Satisfaction-based planners in this domain are veryrare. The lack of satisfaction-based approaches toopen world planning can be explained by the mas-sive explosion of the search space under consideration.Satisfaction-based planners usually work on maximalstructures, which become unmanageable for more com-plex problems. The presented planning model avoidsthe use of maximal structures by including the searchfor the structure as part of the satisfaction process.

The planning model borrows from typical constraint-based applications for resource allocation/optimization.The power of global constraints for constraint-specificrepresentation and reasoning mechanisms for specificresource types was recognized here very early on and ledto significant speed-ups in the solution process. Generalframeworks for planning and scheduling like Muscet-tola’s HSTS (Muscettola 1994) lack such specializedrepresentation and reasoning capabilities.

More information on the underlying EXCALIBURproject is available at:

http://www, ai-center, com/proj ects/excalibur/

AcknowledgmentsThe work reported here is supported by the German Re-search Foundation (DFG), NICOSIO, Conitec Daten-systeme GmbH and Cross Platform Research Germany(CPR).

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