Fusing Procedural and Declarative Planning Goals for Nondeterministic Domains

Transcription

1 Proceedings of the wenty-hird AAAI Conference on Artificial Intelligence (2008) Fusing Procedural and Declarative Planning Goals for Nondeterministic Domains Dzmitry Shaarau FBK-IRS, rento, Italy Marco Pistore FBK-IRS, rento, Italy Paolo raverso FBK-IRS, rento, Italy Abstract While in most lanning aroaches goals and lans are different objects, it is often useful to secify goals that combine declarative conditions with rocedural lans. In this aer, we roose a novel language for exressing temorally exted goals for lanning in nondeterministic domains. hekeyfeatureofthislanguageisthatitallows for an arbitrary combination of declarative goals exressed in temoral logic and rocedural goals exressed as lan fragments. We rovide a formal definition of the language and its semantics, and we roose an aroach to lanning with this language in nondeterministic domains. We imlement the lanning framework and erform a set of exerimental evaluations that show the otentialities of our aroach. Introduction In most lanning aroaches, goals and lans are different objects: goals are declarative requirements on what has to be achieved, and lans are rocedural secification on how to achieve goals. his is the case of classical lanning, where goalsareconditionsonstatestobereachedandlanssecifysequencesofactions. hisisalsothecaseofmoreexressive and comlex settings, such as lanning with temorally exted goals in nondeterministic domains, where goals are, e.g., formulas in a temoral logic, and lans are, e.g., olicies or conditional and iterative combinations of actions, see, e.g. (Kabanza, Barbeau, and St-Denis 1997; Pistore and raverso 2001; Dal Lago, Pistore, and raverso 2002; Kabanza and hiébaux 2005). However, it is often imortant to have the ossibility to combine declarative goals and rocedural lans, and this is esecially useful for lanning in nondeterministic domains for temorally exted goals. Indeed, in nondeterministic domains, it is often useful to secify artial lans, i.e. lansofactionstobeexecutedonlyforasubsetoftheossible outcomes of the lan execution, while we might need to secify declarative goals to be achieved when uncovered states are reached. For instance, we can directly secify nominal lans that are interleaved with declarative conditionstobesatisfiedincaseofure. Viceversa,wecan interleave a declarative goal secification with rocedures Coyright c 2008, Association for the Advancement of Artificial Intelligence( All rights reserved. to be executed as excetion handling routines that recover from dangerous ures. Inthecaseoftemorallyextedgoals,artsofthegoals can be better secified directly as rocedures to be executed than as temoral formulas to be satisfied. For instance, considerthegoalforarobotthathastovisiteriodicallysome locationsinabuilding.hisgoalcanbeeasilysecifiedasa rocedural iteration interleaved with a declarative secificationofthestatestobereached,ratherthanasatemoralformula with nested maintenance and reachability conditions. In this aer, we roose a novel language for exressing temorally exted goals for lanning in nondeterministicdomains. hekeyfeatureofthislanguageisthatit allows for an arbitrary combination of declarative goals exressed in temoral logic and rocedural goals exressed as lan fragments. It combines declarative temoral goals of the EAGLE language(dal Lago, Pistore, and raverso 2002) with rocedural secifications such as conditional and iterative lans, olicies and ure recovery control constructs. Werovideaformaldefinitionofthelanguageanditssemantics. We also roose an aroach for lanning with this language in nondeterministic domains. he idea is to construct control automata that reresent roerly the interleaving of rocedural and declarative goals. We can then use such automatatocontrolthesearchforalaninasimilarway as in(dal Lago, Pistore, and raverso 2002). We imlement the roosed aroach and erform a reliminary set of exerimental evaluations with some examles taken from the robot navigation domain. We evaluate the erformances w.r.t. the dimension of the domain. he exerimental evaluation shows the otentialities of our aroach: a rather simle and natural combination of rocedural and declarative goalsallowsustoscaleuoforderofmagnitudesw.r.t.fully declarative goals. heaerisstructuredasfollows. Wefirstgiveashort background on lanning in nondeterministic domains. We then define the goal language and give its semantics. Next weshowhowwecanconstructcontrolautomatathatcan guidethesearchforalan.wefinallyreorttheresultsof our exerimental evaluation and discuss a comarison with related work. 983

2 Background heaimofthissectionistoreviewthebasicdefinitions of lanning in nondeterministic domains which we use in therestoftheaer. Allofthemaretaken,withminor modifications, from(dal Lago, Pistore, and raverso 2002; Cimatti et al. 2003). We model a(nondeterministic) lanning domain in terms of roositions, which characterize system states, of actions, and of a transition relation describing system evolution from one state to ossible many different states. Definition 1(Planning Domain) A lanning domain D is a4-tule P, S, A, R where Pisthefinitesetofbasicroositions, S 2 P isthesetofstates, Aisthefinitesetofactions, R S A Sisthetransitionrelation. Intuitively, temorally exted goals are the goals that exress the conditions on the whole execution of lans, not just on the final states. Alternatively, they can be considered as goals that dynamically change during execution. raditionally, such evolving goals are modeled through different states of the lan executor, where each state corresonds to a secific current goal. In the following, these execution states are called contexts, and lans are reresented by two relations, one which defines the action to be executed deing on the current domain state and execution context, and one which defines the evolution of the execution context deing on the outcome of the action execution. Definition2(Plan)Alan πforalanningdomain Disa tule C,c 0,act,ctxt where Cisasetofcontexts, c 0 istheinitialcontext, act : S C Aistheactionfunction, ctxt : S C S Cisthecontextfunction. Ifweareinastate sandinanexecutioncontext c,then act(s,c)returnstheactiontobeexecutedbythelan,while ctxt(s,c,s )associatestoeachreachedstate s anewexecutioncontext. hetule s,c S Cdefinesthelan execution state. Weuse σtodenoteafinitelanexecutionath,whichis a asequenceofstate-actions s 1,c i 1 a... n sn,c k. We denote with f irst(σ) and last(σ) the first and the last lanstatesintheexecutionaths.wewrite s σifstate s aearsintheath σ.weuseanotation σ = σ 1 ;σ 2 ifthe ath σisobtainedbytheconcatenationof σ 1 and σ 2. We write σ 1 σ 2 if σ 1 isarefixfor σ 2,i.e σ : σ 2 = σ 1 ;σ. odefinewhenagoalissatisfiedweassumethatthe semantics ofthegoallanguagedefineswhenagoal g is satisfiedinalanexecutionstate s,c,andwerequirethat thegoalissatisfiedintheinitiallanexecutionstate s 0,c 0 hedefinitionof s,c = gdesonthesecificgoal language. For instance, in the case of CL goals(pistore and raverso 2001), the standard semantics can be used(emerson 1990), while ad-hoc semantics have been defined for goal languages secifically designed for lanning, e.g., EA- GLE goals(dal Lago, Pistore, and raverso 2002). he Goal Language We roose an extension of the existing aroaches for exted goal language which combines the benefits of temoral logic formulas with a rocedural goal definition aroach. Let Bbeasetofbasicroositionsand bea roositionalformulaover B. Let Abeasetofactions, and Gbeasetoftemorallyextedgoals.heexted goaltasks tover Aand Garedefinedasfollows. goal g doaction a t 1 ;t 2 if do t 1 else t 2 while do t 1 check try t 0 catch 1 do t 1 olicy fdo t 1 temorally exted goal rimitive action call sequences conditionals cycles check-oint urerecovery searchcontrololicy We now rovide the intuition and motivation for this language. First, the language defines the oerator goal g, where g is a temorally exted goal. Currently, we suort CL (Pistore and raverso 2001) and EAGLE(Dal Lago, Pistore, and raverso 2002) goals, but our aroach can be generalized to suort any kind of temorally exted goals. In order to make this aer self-contained we describe the managementoftwobasictyesofgoalsexressedineagle language: DoReach and ryreach. DoReach reresentsareachabilitygoalwhichhastobesatisfiedinsiteof nondeterminism. ryreach is a weak version of DoReach. Itrequiresthatatleastoneexecutionathhastouin aessfulstate,butitalsorequiresthatthelanhastodo everything ossible to satisfy the goal. he other constructs of the language ext temorally exted goals with rocedural declarations. he rimitive action call oerator doaction is a basic oerator in the lan definition, therefore it is critical to suort such construction in the goal language. he wise usage of rimitive actions can eliminate critical branching oints in the lan search and effectively imrove the lanning time. hekeyideaforthegoalotimizationistoeliminatelanningifitisnotneeded. Ifwerequireorknowinadvance that some intermediate sub-goal Put down the box has to besatisfiedbyonlyonerimitiveactionthenwecanutit in the goal definition: doaction ut down box. hesequenceofgoaltasks t 1 ;t 2 isusedtosuortlan fragments in the goal. he simlest examle is a sequence of rimitive action calls. But the sequence oerator is a owerful attern to manage the lanning rocess and to increase the resulting lan quality. One of the most intuitive and effective ideas for lan search imrovements is to slit a comlexgoalintoasequenceofsimlergoals.forexamle,a goal DelivertheboxfromtheroomAtotheroomB can beredefinedassequence ReachtheroomA;Findthebox; akethebox;reachtheroomb;putdownthebox.goals ReachtheroomA and Findthebox canberesolved indeently, as they require the different search strategies (athfinding and scanning), therefore the right goal slit in a sequences can significantly reduce lanning search sace. he conditional goal task if-else rovides the ossibility to switch between different goals based on the reached do- 984

3 mainstate.moreover,weoftenneedtodefineacyclicgoal task while which has to be erformed until some condition istrue.hisisthecaseforinstanceforthegoal Whilethere existsaboxintheroomaickuanyboxanddeliveritto theroomb.insuchcasewecanaddsomesearchknowledgeinformationaboutthefactthatboxeshavetobedeliveredonebyone.hence,thegoalcanbedefinedas while ( theroomaisnotemty )dodoreach therobotloaded aboxina ;DoReach therobotisinb ;doaction unload the robot. he goal task check verifies whether the current domain state is allowed according to the check-oint condition. For examle,ifwehaveataskgoal t 1 ;check() itmeans thatwerequirealanwhichsatisfies t 1 insuchwaythat guarantiesthatfinallythedomainhastobeinthestate. hemostintuitivereasontousecheck-ointistocheckthe correctness of the lan fragment in the goal definition. Construction try-catch is used to model recovery from ure due to domain nondeterminism. his oerator requires a lan that does everything ossible to achieve the goaltask t 1 andonlyifitbecomestrulyunreachablethen satisfies t 2.Moreover,foronetrywecandefinemanycatch blocks to secify different recovery goal tasks for different ure conditions. he last construction of the roosed language is olicy whichcanbedefinedforanygoaltask t. heolicyisa formula on the current states, actions, and next states(oerator next). Intuitively, olicy is an additional restriction to the domain transition function, which defines the relation between the current state and the next state of basic roositions.hereforeweallowtheusageofactionsintheolicy definitions to make them simler and readable. For examle,foragoaltask ReachroomA wecandefineaolicy (next(osition) = osition) action=look-around. It meansthatwerestrictthelannertoconsideronlytheactions which change the robot osition or the articular action named look-around, and therefore reduce the search sace. Effectiveness of the olicies increases if we associate them to simle sub-goals of comlex goals. he reason is that simler goals can be restricted by stronger olicies. Wenowshowkeybenefitsoftheroosedgoallanguage by the examle of a simle robot navigation domain. Examle 1(Exerimental domain) Consider the domain reresentedinfigure1. ItconsistsofasequenceofN roomsconnectedbydoors. Eachroomcancontainabox, whichcanbeabombornot. Arobotmaymovebetween adjacent rooms, scan the room to check whether there is aboxornot,investigatetheboxtocheckwhetheritisa bombornot,disarmabomb,anddestroythebox. Astate ofthedomainisdefinedintermsoffluentositionthat rangesfrom1tonanddescribestherobotosition,offluentroom[i]thatcanbeinoneoffivestates { unknown, emty, withbox, withbomb, or safe }, of fluent room contentthatcanbeinoneoftwostates { clean, damaged }. he actions are goright, goleft, scan, investigate, disarm, destroy. Actions goright, goleft deterministically change the robot osition. Action scan nondeterministically changes the room state from unknown to emty or withbox. scan investigate destroy disarm Room 1 Room 2 Room N-1 Room N goleft Robot goright Box Figure 1: A robot navigation domain Action investigate nondeterministically changes the room state from withbox to safe or withbomb. Action disarm deterministically changes the room state from withbomb to safe andcanbealiedonly2times.action destroy deterministically changes the room state to safe and sets the room content to damaged. Intuitively, disarm and destroy actions can be used to removethethreatoftheboxexlosion,butdestroyhasto bealiedonlyif disarmisnotallowed.basedonthese actions we will see how the lanner can cature intentions of the lanning goal and generate the best ossible lan, i.e. toreventthecontentofroomtobedamaged,thelanner hastoroosetoerform destroyonlyif disarmcan not be alied. helanninggoalistoreachastatewhereallroomsare emty or safe and the content of some(very imortant) roomsisnotdamaged.wecandefinesuchgoalinournew goal language as follows(as examle we consider a domain with3rooms): Examle 2(Procedural goal task) while (room[osition]=unknown ) do olicy (next(osition)=osition) do try goal ryreach (room[osition]=emty) catch (room[osition]=withbomb) do goal DoReach (room[osition]=safe) doaction goright check(content[3]=clean) Inthisgoaltherobotmovementsbetweentheroomsare hardcoded, but the robot activities in the room require a combination of reachability goals. Moreover, we refer to reach a state room[osition]=emty, and only if it is not ossible we require to achieve room[osition]=safe. In order to reduce the search sace for lanning robot activities in the room we define a olicy next(osition)=osition that does not allow for considering robot movement actions. We note that the roosed rocedural goal language does notforcetheusertohardcodethecomletesolutioninthe goal definition. It rovides a toolkit for embedding domain secific knowledge in the goal definition and to guide the lan search rocess. In the examle above we hardcoded robot movements because it is quite intuitive and easy. Consider another goal examle where, for instance, we need to removethethreatoftheboxexlosionintheroom3only: Box 985

4 olicy (next(osition)!=osition) do goal DoReach osition=3 doaction investigate if (room[3]==box) do doaction destroy In this examle, the rocedure of the robot activities in the roomisfixedbuttherobotmovementtotheroomisdefined asareachabilitygoal.inthisexamleweencodethestrategythatiftherobotdetectsaboxithastobedestroyed. In comarison with CL goals, the roosed goal language is more exressive because it rovides such constructionsastry-catchorwhilethatcannotbeexressedincl. Moreover, the oerator goal rovides a native suort for the goals exressed in CL. In comarison with the EaGLe goals, the roosed goal language can be considered as a wide extension that contains a lot of new rocedural constructs such as doaction, if-else, while, olicy, and goal. Moreover, the syntax of our language allows the usage of lan fragments in the definition of lanning goals, i.e. reviously generated or written manually lans can be used as arts of new goals. We now rovide a formal semantics for the roosed language. In order to formalize whether the lan π satisfies thegoaltask t,weassigntoeachlanexecutionstate s,c twosets: S π t ( s,c )and F π t ( s,c ). hesesetscontainfinite lan execution aths which satisfy or the goal task tfromthelanstate s,c followingthelan π.hence,the lansatisfiesthegoaltask tifthereisnoexecutionath which leads the domain to the ure state. Definition3Alan πsatisfiesagoaltask tinastate s,c ifandonlyif F π t ( s,c ) =. We now consider all kinds of goal tasks and define St π ( s,c )and Ft π ( s,c )byinductiononthestructureof the goal task. t =goal gissatisfiedby πif πsatisfiesthegoal gwritten either in CL(Pistore and raverso 2001) or EaGLe(Dal Lago, Pistore, and raverso 2002) goal languages. t =doaction aissatisfiedby πinastate s,c ifthere existsanaction afrom s,c andtherearenoothersactions, otherwiseits. Formally, St π ( s,c ) = {σ : s,c a.σ = s,c s,c }. Ft π ( s,c ) = { s,c }if s, c is a terminal state of the execution structure, otherwise Ft π ( s,c ) = {σ : b a, s,c b.σ = s,c s,c }. t =check requiresthatformula holdsinthecurrentstate ofthelanexecution s,c. Formally, St π ( s,c ) = {σ : σ = s,c s = }. Ft π ( s,c ) = {σ : σ = s,c s = }. t = t 1 ;t 2 requirestosatisfyfirstsub-goaltask t 1 and,once t 1 eeds, tosatisfynextsub-goaltask t 2. Formally, St π ( s,c ) = {σ = σ 1 ;σ 2 : σ 1 St π 1 ( s,c ) σ 2 St π 2 (last(σ 1 ))}. Ft π ( s,c ) = {σ 1 : σ 1 Ft π 1 ( s,c )} {σ = σ 1 ;σ 2 : σ 1 St π 1 ( s,c ) σ 2 Ft π 2 (last(σ 1 ))} t =if do t 1 else t 2 requirestosatisfythesub-goal task t 1 ifformula holdsinthecurrentstate,orthealternativesub-goaltask t 2 if doesnotholdinthecurrent state. Formally,if s = then St π ( s,c ) = St π 1 ( s,c ) and F π t ( s,c ) = F π t 1 ( s,c ). Otherwise, S π t ( s,c ) = S π t 2 ( s,c )and F π t ( s,c ) = F π t 2 ( s,c ). t =while do t 1 requirescyclicsatisfiabilityofthe sub-goaltask t 1 whilecondition holds. Moreover,itrequiresthattheloohastobefinite. Formally,if s = then S π t ( s,c ) = { s,c }and F π t ( s,c ) =. Otherwise, S π t ( s,c ) = {σ : last(σ) = σ 1 ;...;σ n = σ. i = 1..n : first(σ i ) = σ i S π t 1 (first(σ i ))}and F π t (s) = {σ : σ 1 ;...;σ n = σ. i = 1..n 1 : first(σ i ) = σ i S π t 1 (first(σ i )) σ n F π t 1 (last(σ n 1 ))}. t =try t 0 catch 1 do t 1...catch n do t n requiresto satisfythe main sub-goaltask t 0,butincaseif t 0 s, it requires to satisfy one of the recovery sub-goal tasks t 1,...,t n accordingtothedomainstateobtainedafter t 0 s. Formally, S π t ( s,c ) = {σ : σ S π t 0 ( s,c )} {σ = σ 0 ;σ : σ 0 F π t 0 ( s,c ) i.last(σ 0 ) = i σ S π t i (last(σ 0 ))}and F π t ( s,c ) = {σ : σ F π t 0 ( s,c ) i = 1..n : last(σ) = i } {σ = σ 0 ;σ : σ 0 F π t 0 ( s,c ) i.last(σ 0 ) = i σ F π t i (last(σ 0 ))}. t =olicy fdo t 1 requirestosatisfythesub-goaltask t 1 followingthetransitionrule f, thatdoesnotchange a set of ure execution aths, but narrows the set of essful execution aths in order to reduce the search sace. Formally, S π t ( s,c ) = {σ : σ S π t 1 ( s,c ) s i,a,s i+1.f si,a,s i+1 = }and F π t ( s,c ) = F π t 1 ( s,c ), where s i,s i+1 σarestatesconnectedbytheaction a. Planning Aroach Inthisworkwere-usethelanningalgorithmthatwasroosed in(dal Lago, Pistore, and raverso 2002) with minor imrovements. It is based on the control automata and erformed in three stes: Initially, we transform the goal task into a control automaton that encodes the goal requirements. Each control state describes some intermediate sub-goal that the lan ints to achieve. he transitions between control states define constraints on the domain states to satisfy the source subgoal and to start satisfiability of the target sub-goal. Once the control automaton has been built we associate to eachcontrolstateasetofdomainstatesforwhichalan exists with resect to the sub-goal encoded in this control state. he lan search rocess starts from assumtion that all domain states are considered comatible with all the control states, and this initial assignment is then iteratively refined by discarding those domain states that are recognized incomatible with a given control state until a fix-oint is reached. Finally, we consider each control state as a lan execution contextandextracttheresultinglanbasedonthesetof domain states associated to the control states. We re-use this aroach because it is very scalable to suort newgoallanguages.infact,lasttwostesofthelanningalgorithm are indeent from the goal language. herefore, we only need to define the control automaton construction rocess for the roosed goal language. 986

5 Definition4(ControlAutomaton)LetFbeaolicyformula defined over domain states S, next states next(s), anddomainactions A. Acontrolautomatonisatule C,c 0,,RB,where Cisasetofcontrolstates; c 0 istheinitialcontrolstate; (c) = t 1,t 2,...,t n istheorderedlistoftransitions from control state c. Each transition can be either normal, i.e. t i B F (C {, }),orimmediate,i.e. t i B (C {,}; RB= rb 1,...,rb m isthelistofsetsofcontrolstates marked as red blocks. he order of the transition list reresents the reference among these transitions. he normal transitions corresond totheactionexecutioninthelan.domainstate ssatisfies the normal transition guarded by roosition formula if it satisfies andthereisanaction afrom ssuchthatallossible action outcomes are comatible with some of the target control states. Moreover, a normal transition is additionally guarded by the olicy formula, which restricts the set of actions that can be erformed according to this transition. he immediate transitions describe the internal changes in the lanexecutionandcauseonlythechangeofthelanexecution context, therefore they cannot be guarded by the olicy formulas. he red block states indicate that any lan execution ath should eventually leave these control states, i.e. the lanexecutioncannotstayforeverintheredblockstate. In the following, we use the grahical notations of(pistore, Bettin, and raverso 2001) for control automata definitions. Westartfromthegoaloerator. Aswementioned above, for simlicity we consider only ryreach and DoReach temorally exted goals: DoReach c0 c1 ryreach c0 c1 Bothautomataconsistoftwostates: c 0 (initialcontrolstate) and c 1 (reachabilityredblockstate). herearetwotransitionsoutgoingfromtheredblockstate c 1 indoreachautomaton. hefirstoneisguardedbythecondition. It is a ess transition that corresonds to the domain states where holds. his transition is immediate because our goal task immediately becomes satisfied whenever the domain aears in the state where holds. he second transition isguardedbycondition.itreresentsthecasewhere doesnotholdinthecurrentstate,andtherefore,inorderto achievegoaldoreach,wehavetoassurethatthegoalcan beachievedinallthenextstates.histransitionisanormal transition since it requires the execution of an action in the lan. ryreach differs from DoReach only in definition of thetransitionfrom c 1 guardedbycondition.inthiscase wedonotrequirethatthegoalryreach holdsforallthe nextstates,butonlyforsomeofthem.herefore,thetransitionhastwoossibletargets,namelythecontrolstate c 1 (corresonding to the next states were we exect to achieve ryreach)and c 0 (fortheothernextstates).hesemantics of the goal ryreach requires that there should always beatleastonenextstatethatsatisfiesryreach ;thatis, target c 1 ofthetransitionismarkedby inthecontrolautomaton. his non-emtiness requirement is reresented in thediagramwiththe onthearrowleadingbackto c 1.he referredtransitionfromcontrolstate c 0 istheonethatleads to c 1.hisensuresthatthealgorithmwilltrytoachievegoal ryreach whenever ossible. he control automaton for doaction a and check are the following: doaction a c0 c1 action =a check c0 We notice that, in doaction a, the transition from the control state c 1 guaranteesthatadomainstateisaccetableonly ifthenextstateisachievedbytheexecutionoftheaction a. In check all transitions are immediate, because action erforming is not required. he automaton only checks that the current domain state satisfies formula. he control automata for comound goal tasks are modeled as a comosition of automata designed for sub-tasks. he conditional goal task if-else is modeled as follows: if then c 0 t1elset2 A t1 A t2 hecontext c 0 immediatelymovesthelanexecutiontothe initialcontextofoneofthecontrolautomataforthegoal tasks t 1 or t 2 accordingtothecurrentdomainstate,i.e. whethertheroerty holdsinthecurrentdomainstateor not.wenoticethatifelseartoftheif-elseconstructionis absent then the transition leads directly to ess. hecontrolautomatonforthecyclicgoaltaskwhileisthe following: while do c 0 t 1 A t1 hecontext c 0 hastwoimmediatetransitionsguardedbythe conditions and.heformerleadstotheinitialcontext oftheautomatonfor t 1,i.e.thebodyofthecycle,andthe later leads to the ess of the comound automaton. he essfultransitionsoftheautomatonfor t 1 returnback tocontext c 0,buttheuretransitionfor t 1 falsifiesthe comoundautomaton.wenoticethat c 0 ismarkedasared block. It guaranties that the loo is finite. he control automaton for the try-catch task with two catch oerators is the following: 987

6 try t catch 0 1 do t catch 1 2 do t2 A t0 1 A t1 ( c ) Planning time Reachability goal ask with olicies ask without olicies A t2 0,1 0,01 Number of rooms Allessfultransitionsof t 0 leadtotheessofthe comoundautomaton,butalluretransitionsof t 0 lead tothecontext c 0 whichisresonsibleformanagementof the recovery goals. c 0 hasanimmediatetransitiontoeach t i automatonthatisguardedbytheroerty i.ifthereexistsadomainstate swhichsatisfiesmorethanonecatch roerty i thentherecoverygoalisselectednondeterministically.helasttransitionfrom c 0 toureisguardedby theroerty ( 1... n ).Itmanagesthedomainstates that are not catched by the recovery goals. he oerator olicy causes the refinement of the control automatonofthegoaltasktowhichtheolicyisalied.in ordertobuildtheautomatonforagoaltaskolicy fdo t 1 wedothefollowing: constructtheautomatonforthegoaltask t 1 consider all normal transitions of the control automaton builtfor t 1 andconjunctivelyadd ftotheirguardingolicy formulas. Exerimental Evaluation We imlemented the roosed lanning aroach on to of thembplanner(bertolietal. 2001). Inordertotestthe scalability of the roosed technique, we conducted a set of tests in some exerimental domains. All exeriments have been executed on a 1.6GHz Intel Centrino machine with 512MB memory and running a Linux oerating system. We consider the robot navigation domain defined in Examle 1 and evaluate the lanning time for different kind of lanning goals, i.e. sequential, conditional, cyclic, recovering and reachablity goals. We tested the erformance of the lanning algorithm for goaltaskssimilartotheoneonexamle2withresectto thenumberofroomsinthedomainandcomaredwithlanningforasimlereachabilitygoalthatcanbedefinedas Reachthestatewhereallroomsinthedomainaresafeor emty. We encode the reachability goal using the EaGLe oerator DoReach, therefore the lanner generates the lan inbothcasesusingthesamelanningaroachbasedonthe controlautomata.itallowsustoclaimthatouraroachis usefulduetotheroceduralgoallanguagebutnotduetothe esecial lan search rocedure. he average exerimental resultsareshownonfigure2. he exeriments showed that for simle domains(less than 6 rooms) lanning for a reachability goal is faster, because the memory management and the oerations on the control automaton take more time than lanning for reach- Figure 2: Evaluation results for the robot domain ability goal. But in more comlex domains our lanning technique solves the roblem much faster, esecially with the systematic usage of olicies. Wealsotestedouraroachwiththegoalthatdoesnot contains the last oerator check(content[3]=clean). In this casetherobotactivitiesintheroomarenotlinkedwiththe robotosition.itmeansthatwecanresolvethegoal try goal ryreach (room[osition]=emty) catch (room[osition]=withbomb) do goal DoReach (room[osition]=safe) ; inthedomainwithoneroom,i.e. inaverysmalldomain. Such lanning takes about 0.01 second. After that we can re-usegeneratedlaninthereviousgoalandobtainagoal task which does not contain goal oerators. Hence, all normal transitions corresond only to doaction oerators. In fact,wetransformthelansearchroblemtoalanverification roblem which is incomarably easier. For instance, lanningforsuchkindofgoalsinthedomainwith50rooms takes less than 1 second. his exeriment shows that the rocedural decomosition of the lanning goal and re-usage of the reviously generated lans can significantly simlify the lan generation rocess. he second exerimental domain is insired by a real alication, namely automatic web service comosition(pistore, raverso, and Bertoli 2005; Pistore et al. 2005). Examle 3 For testing reasons we model a simlified version of the bookstore Web Service alication. In articular, we consider only 2 Web Services: BookSearch and Book- Cart which are shown on Figure 3(a,b). BookSearch describesaworkflowtofindabookinthebookstore. Book- Cartdescribesaworkflowtoaddtheobtainedbooktothe usercartand,finally,tocheckoutit. AllossibleUseractivities for communications with the bookstore are shown on Figure3(c).Weuseafluent message toemulatethemessage exchange between Web Services and User. We write? beforetheactionnameifthisactionrequires(asarecondition) secific message to be sent, and! if this action initiates(as a ostcondition) the message sing. For instance, the User action? getbookid can be erformed onlyifcurrentmessageis bookid thatcanbesetbythe 988

7 initial Initial! login! logout! search? login?additem? checkout?clearcart loginfail loginack add checkout cleared!loginfail!loginsucc!itemadded checkout waitmsg? search? logout Error Ok logout!checkoutfail!checkoutsucc search searchresult tosucc Fail notfound result!notfound!bookid?getbookid User!addItem! checkout!clearcart a) BookSearch Web Service b) BookCart Web Service c)possible User Actions Figure 3: Exerimental domain for WS comosition action!bookid ofbooksearch.duetolackofsacewe do not rovide a formal definition for the lanning domain and the lanning roblem. Intuitively, each actor in the Web Service comosition is reresented by a state-transition system(ss)andthelanningdomainisobtainedasaroduct of these SSs. N airs of BookSeach and BookCart services emulatenbookstoresandourgoalistolanuseractivities tofindandbuyatleastonebookinatleastonebookstore. We tested the erformance of the lanning algorithm with resecttothenumberofbookstoresinthedomainandcomared with lanning for a simle reachability goal that can bedefinedas Reachthestatewhereuserfoundandbought a book in exactly one bookstore. Using the roosed goal language we can encode some search knowledge in the goal definition.forinstance,tosatisfysuchkindofgoalwecan consider bookstores one by one iteratively until the book is bought. Ineachbookstorewecantrytofindthebookand buyit.ifitisimossibleduetodomainnondeterminismwe need to erform some recovering actions to leave the bookstore in a consistent state. herefore, the lanning goal can be written as follows: while ( book is not bought && available bookstore ) do doaction choose available bookstore try goal ryreach achieve search results if ( book is found ) do goal ryreach cart is checked out else goal DoReach user is logged out catch ( login is ed ) { goal DoReach BookSerch is in && BookCart is in initial catch ( cart checkout is ed ) { goal DoReach user is logged out && BookCart is in heaverageresultsareshownonfigure4.asintherobot Planning time 1000,00 100,00 10,00 1,00 0,10 0,01 Plan-as-goal Reachability goal Number of WS Figure 4: Evaluation results for the WS comosition domain navigation domain we can see that our aroach is effective in large domains. Wenotethatallsimlereachabilitygoalsusedintheexeriments can not be significantly imroved using EaGLe goal language. It confirms the fact that the exressiveness and erformance of the roosed rocedural goal language isbetterincomarisonwitheagle. Itisverydifficultto comare our aroach fairly with CL goals because CL can not exress try-catch goals that are almost obligatory for the real-life roblems in nondeterministic domains. All exeriments show that fusing rocedural and declarative aroaches for the goal definition allows the user to define a lanning goal more clearly and detailed. Moreover, suchdetailedgoalcanbesolvedbythelannerinaveryefficientwayincomarisonwiththemoregeneralgoalsuchas a simle reachability goal. It is esecially effective in large domains. Conclusions and Related Work Inthisaer,wehaveresentedanovelaroachtolanning in nondeterministic domains, where temorally exted goals can be exressed as arbitrary combinations of 989

8 declarative secifications and of rocedural lan fragments. he advantage of the aroach is twofold. First, the exressiveness of the language allows for easier secifications of comlex goals. Second, rather simle and natural combinations of rocedural and declarative goals oen u the ossibility to imrove erformances significantly, as shown by our reliminary exerimental evaluation. Several aroaches have been roosed to deal with the roblem of lanning in nondeterministic domains, where actions are modeled with different ossible outcomes, see, e.g., (Rintanen 1999; Cimatti et al. 2003; Hoffmann and Brafman 2005; Kabanza and hiébaux 2005; Kuter et al. 2005; KuterandNau2004). Someofthemdealalsowiththe roblem of lanning for temorally exted goals(bacchus and Kabanza 2000; Pistore and raverso 2001; Dal Lago, Pistore, and raverso 2002; Kabanza and hiébaux 2005). However, none of these works can combine rocedural and declarative goals like the lanning language that is roosed inthisaer.hisisthecasealsooftherecentworkthatexts HN lanning to nondeterministic domains(kuter et al.2005;kuterandnau2004).inthiswork,itisossibleto laninaweak,strong,orstrongcyclicwayforhnsecifications, but the combination of HNs with declarative temoral secifications like those exressed in EAGLE are not allowed. his is indeed an interesting toic for future investigation, since goals that could combine HNs with temoral formulas could exloit the integration of HNs techniques with the techniques for lanning based on control automata. he idea of abstract task secification using rocedural constructs is widely used in model-based rogramming, see, e.g.,(williams et al. 2004). yically, the model-based rogram is reresented by a control automaton that generates reachability goals according to the rogram source. Each reachability goal is resolved by reactive lanning. In our aroachwedonotconsiderthegoalasarogramtoexecute, therefore the goal interretation and lanning aroach is different. Finally, our work has some similarities in sirit with the workbasedongolog,see,e.g.,(levesqueetal. 1997; McIlraith and Fadel 2002; Baier, Fritz, and McIlraith 2007; Claßenetal. 2007). InGolog,sketchylanswithnondeterministic constructs can be written as logic rograms. Sequences of actions can be inferred by deduction in situation calculus on the basis of user sulied axioms about reconditions and ostconditions. In site of these similarities, both the aroach and the techniques that we roose are very different from Golog. References Bacchus, F., and Kabanza, F Using temoral logic to exress search control knowledge for lanning. Artificial Intelligence 16(1 2): Baier,J.A.;Fritz,C.;andMcIlraith,S.A.2007.Exloiting rocedural domain control knowledge in state-of-theart lanners. In ICAPS. Bertoli, P.; Cimatti, A.; Pistore, M.; Roveri, M.; and raverso,p MBP:aModelBasedPlanner. InIJ- CAI01. Cimatti, A.; Pistore, M.; Roveri, M.; and raverso, P Weak, Strong, and Strong Cyclic Planning via Symbolic Model Checking. Artificial Intelligence 147(1-2): Claßen, J.; Eyerich, P.; Lakemeyer, G.; and Nebel, B owards an integration of golog and lanning. In IJCAI, Dal Lago, U.; Pistore, M.; and raverso, P Planning with a language for exted goals. In AAAI, Edmonton, Alberta, Canada: AAAI-Press/he MI Press. Emerson,E.A emoralandmodallogic. Invan Leeuwen, J., ed., Handbook of heoretical Comuter Science, Volume B: Formal Models and Semantics. Elsevier. chater 16, Hoffmann, J., and Brafman, R Contingent lanning via heuristic forward search with imlicit belief states. In ICAPS, Kabanza, F., and hiébaux, S Search control in lanning for temorally exted goals. In ICAPS, Kabanza, F.; Barbeau, M.; and St-Denis, R Planning control rules for reactive agents. Artificial Intelligence 95(1): Kuter, U., and Nau, D. S Forward-chaining lanning in nondeterministic domains. In AAAI, Kuter,U.;Nau,D.S.;Pistore,M.;andraverso,P A hierarchical task-network lanner based on symbolic model checking. In ICAPS, Levesque, H. J.; Reiter, R.; Leserance, Y.; Lin, F.; and Scherl, R. B GOLOG: A logic rogramming language for dynamic domains. Journal of Logic Programming 31(1-3): McIlraith, S., and Fadel, R Planning with comlex actions. In NMR. Pistore, M., and raverso, P Planning as model checking for exted goals in non-deterministic domains. In Nebel, B., ed., IJCAI-01, Morgan Kaufmann Publisher. Pistore, M.; Bettin, R.; and raverso, P Symbolic techniques for lanning with exted goals in nondeterministic domains. In ECP 01, LNAI. Sringer Verlag. Pistore, M.; Marconi, A.; raverso, P.; and Bertoli, P Automated Comosition of Web Services by Planning at the Knowledge Level. In IJCAI 05. Pistore, M.; raverso, P.; and Bertoli, P Automated Comosition of Web Services by Planning in Asynchronous Domains. In ICAPS 05. Rintanen, J Constructing conditional lans by a theorem-rover. Journal of Artificial Intellegence Research 10: Williams,B.C.;Ingham,M.D.;Chung,S.;Elliott,P.;Hofbaur, M.; and Sullivan, G Model-based rogramming of fault-aware systems. AI Mag. 24(4):