Planning with graded nondeterministic actions: A possibilistic approach

ion hierarchy acceptability hierarchy Precondition propositions are partitioned into abstraction levels. Within a given abstraction level only a subset of the subgoals are considered. All precondition propositions are considered at each iteration. For a given subgoal, the set of potential establishers is independent of the current abstraction level. Elementary consequences are partitioned into acceptability levels. Within a given acceptability level, only a subset of the potential establishers of a given subgoal are considered. Abstraction hierarchies implement a speci c search control heuristic concerning the choice of the aw to consider. Acceptability hierarchies implement a speci c search control heuristic concerning both the choice of the aw and the choice of the way this aw will be removed (lines 1 and 2(a) in the re nement function). Figure 8: Comparaison of hierarchy principles plan from being present in the search tree. The following result establishes the completeness of POSPLAN*. Proposition 6 (Completeness of POSPLAN*) Let = h init; Goals;Ai be a possibilistic planning problem, and max be the greater i such that there exists an essential solution plan haiiN 1 i=0 with N [Goalsj init; haiiN 1 i=0 ] = i. Then POSPLAN* will generate a partial solution plan P with N [Goalsj init;P] = max. Proof: The following lemma insures that for each i, if there exists a safe plan for 1 i, then POSPLAN* will nd a solution at the i level. Then from the equivalence proposition (2), the proposition (6) is proved. Lemma 2 8i, if haiiN 1 i=0 is an essential safe plan for 1 i, then POSPLAN* will generate a partial plan P essential and safe for 1 i, such that haiiN 1 i=0 is a completion of P. proof: By induction on i. For i = n and i = n, this is a consequence of POSPLAN algorithm's completeness (proposition (5)). Assume the proposition true till i+ 1. If haiiN 1 i=0 is an essential safe plan for 1 i , it is also a safe plan for 1 i+1 , but not necessarily essential. Let haijip 1 j=0 , p N be a subplan of haiiN 1 i=0 , essential and safe for 1 i+1 . By induction we know that there exists a partial solution plan P essential and safe for 1 i+1 , such that haijip 1 j=0 is a completion of P . Then from the lemma 1 we deduce that 22 POSPLAN* can generate a partial safe plan P 0 for 1 i , such that haiiN 1 i=0 is a completion of P . 2 Example: We still consider the possibilistic planning problem = h init = f(1; f ^ g ^ s ^ p ^ y)g; Goals = y;Ai. A= fsow-normal, sow-better, treat, harvestg is an 8-level action set, with i 2 f0:2; 0:3; 0:6; 0:7; 0:8; 0:9; 1:0g. At 6 = 0:2, A0:8 is deterministic and POSPLAN*( ) generates the safe plan hsow-normal, harvesti. At 5 = 0:3, hsow-normal, harvesti is still a safe plan but at 4 = 0:6, due to the fact that sow-normal0:4 has two possible e ects for its second discriminant POSPLAN* must generate a new safe plan which is hsow-better; treat; harvesti. This plan remains safe for 6 and 5 but does not for 3 = 0:7 because the second discriminant of sow-better0:3 is then associated with two possile e ects. Since no other plan can be generated, max = 0:6 and the returned plan is hsow-better; treat; harvesti. 5 Conclusions 5.1 Contributions The main goal of this paper was to formalize a possibilistic approach of planning under uncertainty in domain model characterized by incomplete knowledge of the initial state and by actions having both context-dependent and graded nondeterministic e ects. It was inspired by the work done on the BURIDAN[Kus95] planner that relies on a probabilistic representation of uncertainty. In practice, it seems more natural and easier to see actions in terms of normal and more or less exceptional e ects rather than probable ones. Thus, the ordinal nature of a possibilistic representation of uncertainty is quite appealing for this purpose. Beside its representational adequacy, the possibilistic approach has interesting computational properties since the search for -acceptable or optimal plans amounts to solve induced planning problems that have only crisp nondeterministic actions (i.e. each action having then only normal e ects). Moreover, in the case of optimal plan generation, the proposed sound and complete POSPLAN* algorithm is an anytime least-commitment planner; it possesses the additional feature of iteratively solving derived planning problems that are progressively more complex and exploits at each iteration the partial plans developed in the previous ones. The POSPLAN* planner and, consequently, the NDP and POSPLAN algorithms have been implemented in Common Lisp reusing part of BURIDAN's code (in particular its SNLP basis). The generation of -acceptable and optimal plans relies on the NDP algorithm which is a planner operating on pure nondeterministic actions. Thus, our approach gives practical usefulness to such a kind of planner that so far were only of theoretical interest [Ped88] [Ped91]. Several recent planning approaches are based on Bayesian decision theory (probabilistic uncertainty and additive utility functions)[Bou96]. Now, possibility theory o ers a well-suited base for a more qualitative version of decision theory [Dub95]; thus, our approach can be seen as a preliminary step toward a more qualitative 23 approach to decision-theoretic planning.5.2 Limitations and future workAlthough our approach enables an extension of the STRIPS representation by sup-porting graded nondeterministic actions and partially known intial state, it is stillfar from being able to cope with the complexity of most practical problems. One ofthe main limitations comes from the synthactic restrictions that the propositionalnature of the action representation language imposes. The other limitations con-cern the capabilities of hierachical planning (the possibility to specify and searchplans at various levels of detail), of dealing with time constraints (duration) and ofhandling constraints on resources. Important progresses have been accomplished onthese last two issues over the past ten years. It would be interesting to reconsiderthem in the context of graded nondeterministic actions considered in this paper.Especially appealing is the development within the framework of possibility theoryof an homogeneous treatment of uncertainty on initial state and e ects of actionstogether with imprecision and uncertainty on durations and resource consumptionsand productions.As pointed out in Section 2.4, the possibilistic modeling of the e ects of an ac-tion may be generated automatically from an initial set of hard (secure) and defaultrules expressing the (qualitative) knowledge about the general behavior of the actionunder consideration. Each rule may encodes the change from an initial situation toone or several possible resulting situations. Hard rules express sure transitions andcan most of the time only specify the general context in which the action is usefuland what its e ects might be. Default rules apply in particular situations that areconsistent but more speci c than the condition part of any hard rule; they expressthe normal though not sure e ects of the action. The idea of automatic genera-tion of a possibilistic action exploits the default rule ranking procedure initiated byPearl[Pea91] and extended in[Ben92] to the case of graded defaults. This procedureproduces a partition of the default rules according to the relative speci city; thehigher the speci city,the higher the rank. This ranking can then be used to inducea ranking on the possible resulting states depending on the situation prior to theexecution of the action; the rank of a state is the rank of the most speci c ruleviolated in this state. Each possible situation prior to the execution of the actionwill give rise to a discriminant in the correponding possibilistic action and the latterranking provides the basis for deriving a coherent possibility distribution on the ele-mentary e ects of each discriminant. The complete formalization of this knowledgeengineering construction procedure is the subject of an ongoing work.Our approach of planning under uncertainty assumes a complete non-observabilityof states during execution. In the opposite case of complete observability, one mayswitch to dynamic programming approaches that have been extended to possibilis-tic transition functions (see, for instance,[Far96]). For the intermediate case ofpartial observability, we hope to extend our system so that information-gatheringactions can also be incorporated in the planning process as has been done with24 C-Buridan[Dra94] for probabilistic planning.In this paper, a goal is simply a conjunction of literals that de nes a set ofequally good goal states (i.e. reaching any of them would be ne). We are thinkingof extending our framework so as to take account of preferences over the goal states,using a qualitative utility function or equivalently a fuzzy set of goal states 5. Moregenerally, the approach could be reconsidered to cope with a more elaborate notionof goals in the spirit of utility functions in decision theoretic planning approachessuch that a preference would be associated to each state that might be gone throughin the execution of a sequence of actions. The objective would then be to nd thebest sequence with respect to a criteria embedding a compromise between uncer-tainty and preference. 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