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Effective Approaches for Partial Satisfaction (Over-subscription) Planning Romeo Sanchez * Menkes van den Briel ** Subbarao Kambhampati * * Department of Computer Science and Engineering ** Department of Industrial Engineering Arizona State University Tempe, Arizona

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Outline Background Example Approaches Optiplan Altaltps Sapaps Planning graph heuristics Results

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For all your demands, you could’ve bought me a better flash memory stick at least! In one day achieve the following 100 goals: RockData at WP 1, high-res pics at WP 2 & 3, …., SoilData at WP 100 No way I can achieve that many goals in one day It’s hard but here is the best I can do: Goal1, Goal5, Goal99 Given: Actions with costs, and goals with utilities, find a plan that has a highest {utility – cost} Previous Approaches: Highest utility goal first Estimating the set of most beneficial goals Background

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Complete satisfaction (traditional) planning Goal state G is a list of conjunctions: G = g 1 g 2 … g n A plan that achieves n – 1 goal fluents is as good as a plan that achieves 0 goal fluents Partial satisfaction planning (PSP) Goal state G is a list of fluents: G = {g 1, g 2, …, g n } Goal fluents might have utilities, actions might have costs, therefore achieving a partial plan might be more beneficial than the “null” plan. Achieving all goal fluents might be impossible… The goal state G may contain logically conflicting fluents There might not be enough resources to achieve all fluents in G (:goal (and (pointing satellite1 moon) (pointing satellite1 mars) )) (:goal (and (have_rock rover1 waypoint1) (have_rock rover1 waypoint2) ))

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PSP problems PSP Net benefit: Given a planning problem P = (F, A, I, G), and for each action a “cost” c a 0, and for each goal fluent f G a “utility” u f 0, and a positive number k. Is there a finite sequence of actions = (a 1, a 2, …, a n ) that starting from I leads to a state S that has net benefit f (S G) u f – a c a k. PLAN EXISTENCE PLAN LENGTH PSP GOAL LENGTH PSP GOAL PLAN COSTPSP UTILITY PSP UTILITY COST PSP NET BENEFIT

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Example Getting from Las Vegas (LV) to San Jose (SJ) C: action cost U(G): utility of goal G G1,G2,G3,G4: goals P = {travel(LV,DL), travel(DL,SJ), travel(SJ,SF)} achieves G1, G2, G3

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Approaches Optiplan Integer programming based STRIPS planner Solves the PSP problem by encoding it as an integer program Altaltps Heuristic regression planner Solves the PSP problem through a goal selection heuristic Sapaps Heuristic forward state space planner Solves the PSP problem using an anytime A* algorithm

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Optiplan Optiplan planning system: Combines Graphplan (Blum & Furst, 1995) with State Change Encoding (Vossen et al., 1999) As in the Blackbox planning system, Graphplan reduces the encoding size generated by Optiplan Computes optimal plans for a given parallel length Objective: f G U f (x_add f,n + x_preadd f,n + x_maintain f,n ) – l L a A C a y a,l Sum of goal utilities – Sum of action cost

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Optiplan and partial satisfaction Objective 0 / Minimize #actions Constraints Fluent changes Satisfy initial state Satisfy goal Fluent implications Action implications Total satisfaction planning: goal satisfaction is treated as a hard constraint Objective Maximize net benefit Goal utility – action cost Constraints Fluent changes Satisfy initial state Fluent implications Actions implications Partial satisfaction planning: goal satisfaction is treated as a soft constraint

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Graphplan based cost propagation

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AltAlt ps AltAlt planning system Heuristic state-space search planner (Nguyen, Kambhampati & Sanchez, 2002) Combines Graphplan (Blum & Furst, 1995) with heuristic state- space search techniques (Bonet, Loerincs & Geffner, 1997; Bonet Geffner, 1999; McDermott 1999) AltAlt ps planning system Total enumeration on 2 n goal subsets is too costly Selects a promising subset of the top-level goals upfront Searches for a plan using a regression state space search combined with cost-sensitive planning graph heuristics.

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AltAlt ps cost propagation Using a planning graph structure Propositions in the initial state come for free (they have zero cost) Other propositions have costs computed as follows: Propagation procedures Max-propagation Sum-propagation l=0l=1l=2 h l (p) = Cost of proposition p at level l 0 if p I h l (p) =min{h l-1 (p), cost(a) + C l (a)} if l > 0 otherwise C l (a) = max{h l-1 (q) : q prec(a)} C l (a) = q prec(a) h l-1 (q) 44

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AltAlt ps goal set selection Main idea Start with the original goal set G and an empty goal set G’ Iteratively add goals to G’ as long as the estimated NET BENEFIT increases The cost of adding another goal g to G’ depends on the goals that are already in G’ G’ G’ g Cost for achieving G’ Residual cost for g Relaxed plan for G’ (R’p) Rp for G’ g biased to re-use actions in R’p

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AltAlt ps cost-sensitive relaxed plan heuristic General procedure States are ranked during search using the relaxed plan heuristic and the propagated costs The idea is to compute the cost of a relaxed plan Rp in terms of the costs of the actions composing it. Heuristic value for S equal h(S) = a Rp cost(a) 1. Given a state S, remove the (sub)goal g from S that has highest h l (g) 2. Select the action that supports g with lowest cost ( cost(a) + C l (a) ) 3. Regress S over a to get S’ = S prec(a) \ eff(a) 4. Stop when each proposition q S is present in the initial state

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Sapaps

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Nodes evaluation: g(S) = U(S) – C(S) h(S) = U(RP(S)) – C(RP(S)) Beneficial Node: g(S) > 0 or U(S) > C(S) Termination Node: V S’: g(S) > f(S’) A*: f(S) = g(S) + h(S) A1: Navigate(X,Y)A2: SampleSoil(Y) A3: TakePicture A4: Navigate(Y,Z) A5: SampleRock g(S) = Util(HasSoilData) – Cost(A1,A2) h(S) = Util(Apply(A3,S)) – Cost(A3) Anytime A* Algorithm: Search through best beneficial nodes SAPA PS : a forward A* approach for PSP

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Heuristic: Variation of SAPA’s Approach Heuristically extracting the least cost relaxed plan using cost-function Remove “unbeneficial” goals and related actions G1 G2 G3 A1 A2 A3 A4 → G1 G2 A1 A3 C(A1) + C(A2) > U(G3) SAPA PS : heuristic

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Empirical results

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Future work

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