effective approaches for partial satisfaction (over-subscription) planning romeo sanchez * menkes...
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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 EngineeringArizona State UniversityTempe, 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 firstEstimating the set of most beneficial goals
Background
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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
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” 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 LENGTH
PSP GOAL
PLAN COST PSP 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: fG Uf (x_addf,n + x_preaddf,n + x_maintainf,n) – lL aA Ca ya,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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AltAltps
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)
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.
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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:
Propagation procedures Max-propagation
Sum-propagation
0
0
0
0
4
0
0
4
5 5
8
5 5
3
l=0 l=1 l=2
hl(p) = Cost of proposition p at level l
0 if p I
hl(p) = min{hl-1(p), cost(a) + Cl(a)} if l > 0
otherwise
Cl(a) = max{hl-1(q) : q prec(a)}
Cl(a) = q prec(a) hl-1(q)
4 4
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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’
Residual cost for gRelaxed plan for G’ (R’p)
Rp for G’ g biased to re-use actions in R’p
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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.
Heuristic value for S equal h(S) = aRpcost(a)
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
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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
SAPAPS: a forward A* approach for PSP
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Heuristic: Variation of SAPA’s ApproachHeuristically extracting the least cost relaxed plan using cost-functionRemove “unbeneficial” goals and related actions
G1
G2
G3
A1
A2
A3
A4
→G1
G2
A1A3
C(A1) + C(A2) > U(G3)
SAPAPS: heuristic
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Empirical results
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Empirical results
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Future work