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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 Outline Background Example Approaches Optiplan Altaltps Sapaps Planning graph heuristics Results Background In one day achieve the following 100 goals: RockData at WP 1, high-res pics at WP 2 & 3, …., SoilData at WP 100 Given: Actions with costs, and goals with utilities, find a plan that has a highest {utility – cost} No way I can achieve that many goals in one day For all your demands, you could’ve bought me a better flash memory stick at least! It’s hard but here is the best I can do: Goal1, Goal5, Goal99 Previous Approaches: Highest utility goal first Estimating the set of most beneficial goals Background Complete satisfaction (traditional) planning Goal state G is a list of conjunctions: G = g1 g2 … gn 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 = {g1, g2 , …, gn} 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 (:goal (and (pointing satellite1 moon) (pointing satellite1 mars) )) There might not be enough resources to achieve all fluents in G (:goal (and (have_rock rover1 waypoint1) (have_rock rover1 waypoint2) )) PSP problems PSP Net benefit: Given a planning problem P = (F, A, I, G), and for each action a “cost” ca 0, and for each goal fluent f G a “utility” uf 0, and a positive number k. Is there a finite sequence of actions = (a1, a2, …, an) that starting from I leads to a state S that has net benefit f(SG) uf – a ca k. PLAN EXISTENCE PLAN LENGTH PSP GOAL PSP GOAL LENGTH PLAN COST PSP UTILITY PSP NET BENEFIT PSP UTILITY COST 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 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 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: fG Uf (x_addf,n + x_preaddf,n + x_maintainf,n) – lL aA Ca ya,l Sum of goal utilities – Sum of action cost Optiplan and partial satisfaction Objective 0 / Minimize #actions Objective Maximize net benefit Goal utility – action cost Constraints Fluent changes Satisfy initial state Satisfy goal Fluent implications Action implications Total satisfaction planning: goal satisfaction is treated as a hard constraint Constraints Fluent changes Satisfy initial state Fluent implications Actions implications Partial satisfaction planning: goal satisfaction is treated as a soft constraint Graphplan based cost propagation AltAltps AltAlt planning system Heuristic state-space search planner (Nguyen, Kambhampati & Sanchez, 2002) Combines Graphplan (Blum & Furst, 1995) with heuristic statespace search techniques (Bonet, Loerincs & Geffner, 1997; Bonet Geffner, 1999; McDermott 1999) AltAltps planning system Total enumeration on 2n 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. AltAltps 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: 0 0 0 5 5 0 4 l=0 4 0 5 5 0 3 8 4 4 l=1 l=2 hl(p) = Cost of proposition p at level l hl(p) = 0 if p I min{hl-1(p), cost(a) + Cl(a)} if l > 0 otherwise Propagation procedures Max-propagation Cl(a) = max{hl-1(q) : q prec(a)} Sum-propagation Cl(a) = q prec(a) hl-1(q) AltAltps 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’ Relaxed plan for G’ (R’p) Residual cost for g Rp for G’ g biased to re-use actions in R’p AltAltps 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. 1. Given a state S, remove the (sub)goal g from S that has highest hl(g) 2. Select the action that supports g with lowest cost (cost(a) + Cl(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 Heuristic value for S equal h(S) = aRpcost(a) Sapaps SAPAPS: a forward A* approach for PSP A5: SampleRock A1: Navigate(X,Y) A2: SampleSoil(Y) A4: Navigate(Y,Z) Anytime A* Algorithm: Search through best beneficial nodes A3: TakePicture g(S) = Util(HasSoilData) – Cost(A1,A2) h(S) = Util(Apply(A3,S)) – Cost(A3) g(S) = U(S) – C(S) h(S) = U(RP(S)) – C(RP(S)) Beneficial Node: A*: f(S) = g(S) + h(S) Nodes evaluation: g(S) > 0 or U(S) > C(S) Termination Node: V S’: g(S) > f(S’) SAPAPS: heuristic Heuristic: Variation of SAPA’s Approach Heuristically extracting the least cost relaxed plan using cost-function Remove “unbeneficial” goals and related actions A1 A3 A4 G1 G2 A2 → A1 A3 G3 C(A1) + C(A2) > U(G3) G1 G2 Empirical results Empirical results Future work