On the Instantiation of ADL Operators Involving Arbitrary First-Order Formulas

OntheInstantiationofADLOp eratorsInvolvingArbitraryFirst-OrderFormulasJanaKo ehlerSchindlerLiftsLtd.CH-6031EbikonSwitzerlandjanako ehler@ch.schindler. comJorgHo mannInstituteforComputerScienceAlb ertLudwigsUniversity79110Freiburg,Germanyho mann@informatik.uni-freiburg.deAbstractThegenerationofthesetallgroundactionsforagivensetofADLop erators,whichareallowedtohaveconditionale ectsandpreconditionsthatcanb erepre-sentedusingarbitrary rst-orderformulasisacomplexpro cesswhichheavilyinuencesthep erformanceofanyplannerorpre-planninganalysismetho d.Thepap erdescrib esasophisticatedinstantiationpro-cedurethatdeterminesso-calledinertiainagivenproblemrepresentationandusesthemtop erformsim-pli cationsofformulasduringtheinstantiationpro-cess.Asaresult,manyinapplicableactionsarede-tectedandruledoutfromthedomainrepresentationyieldingamuchsmallersearcspacefortheplanner.Intro ductionAplanningsystemthathandlesamoreexpressivelan-guagethanSTRIPSrequiressophisticatedalgorithmicsolutionstoquiteanumb erofproblems,whichhavenothingtodowiththeactualsearchpro cessforaplan.Oneoftheseproblemsconcernsthecomputationsetofactionsasallgroundinstancesagivenop erators.Theaimoftheinstantiationpro cessistogenerateallthosegroundinstancesoftheplanningop eratorsthatareapplicableinsomelegalworldstate.Thismeans,thatthepreconditionofop eratorshouldb esatis ableanditse ectsshouldconsistent.Ononehand,anaiveinstantiationpro cedurethatsimplyexpandslogicalquanti ersandenumeratesallp ossibleinstantiationsofop eratorparameterswillquicklyren-derevensimpleplanningproblemsunsolvable.Ontheotherhand,arathersophisticatedinstantiationpro ce-durecanruleoutmanyactions,whichwillneverb eapplicableinanyreachableworldstateorthatould|ifapplied|yieldaninconsistentstate.Itshouldalsoreturnthemostsimplesyntacticrepresentationofpre-conditionsande ects.Manyplanningsystemsdogeneratethecompletesetofactionsb eforeplanningactuallystarts.Theyusethisseteitherfortheenco dingofdomaininotherCopyrightc2000,AmericanAsso ciationforArti cialIn-telligence(www.aaai.org). Allrightsreserved.representationformalismssuchasSAT(Kautz&Sel-man1996)orforthederivationofusefulinformationthatcanhelpduringplanning,e.g.,distanceheuristics(Ge ner1999),symmetries(Fox&Longrelevantactions(Neb el,Dimop oulos,&Ko ehler1997),andgoalorderings(Ko ehler1998). WhenusingthePDDLlanguage(McDermott&oth-ers1998)torepresentADLop erators(Pednault1989),quitecomplexdescriptionsofpreconditionsande ectsarep ossible:arbitraryfunction-symb olfree rst-orderlogicformu-lasrepresentpreconditions,conditionale ectshaveform(whenantecedentconsequent)wheretheantecedentcanb earbitrarypreconditionandtheconsequentisaconjunctionofliterals,i.e.,anatomeithero ccursp ositivelyorneg-ativelyinit.conditionale ectcanalsob euniver-sallyquanti ed.Givensuchanop erator,theinstantiationhastore-placeallo ccurringvariables,whichareeitherquanti edoro ccurasparametersoftheop erator,bythosetyp e-consistentconstants,whichhaveb eendeclaredintheplanningproblem.Inordertoreplaceallvariablesbyconstants,theinstantiationpro cesspro ceedsinthreephases:1.Theexpansionofuniversalandexistentialquanti erso ccurringinthe rst-orderformulasrepresentingpre-conditionsorantecedentsofconditionale ectselim-inatesmostofthequanti edvariables,2.Theexpansionuniversallyquanti edconditionale ectseliminatesremainingquanti edvariables,3.theinstantiationofop eratorparameterseliminatesthevariableparameters.1Ineachphase,thefollowingatomicinstantiationtasko ccurs:Givenavariable? x,constantandanatomicformulap,determineresultinginstantiationp[? x=c]. 1InIPPtheassumptionismadethatdi erentop eratorparametersareinstantiatedwithdi erenconstants,i.e.,theplannernevergeneratesactionslikemove(a,a).