State-Dependent Cost Partitionings for Cartesian Abstractions in - - PowerPoint PPT Presentation
State-Dependent Cost Partitionings for Cartesian Abstractions in Classical Planning Thomas Keller 1 Florian Pommerening 1 Jendrik Seipp 1 Florian Geier 2 Robert Mattmller 2 1 University of Basel 2 University of Freiburg September 28, 2016 KI
Thomas Keller1 Florian Pommerening1 Jendrik Seipp1 Florian Geißer2 Robert Mattmüller2
1University of Basel 2University of Freiburg
September 28, 2016 KI 2016, Klagenfurt
Given: Initial state Goal Actions Find: Plan from initial state to goal
Approach: State-space search for a plan Guided by a distance heuristic ... ...that is automatically derived
Heuristic derivation: Abstract problem Shortest goal distances there Use as heuristic values
Challenge: single abstraction uninformative Solution: multiple abstractions
Challenge: single abstraction uninformative Solution: multiple abstractions New challenge: admissible combination Solution: partial costs in abstractions; then take sum
Challenge: single abstraction uninformative Solution: multiple abstractions New challenge: admissible combination Solution: partial costs in abstractions; then take sum Requirement for admissibility: Assume c(a) = 6.
Challenge: single abstraction uninformative Solution: multiple abstractions New challenge: admissible combination Solution: partial costs in abstractions; then take sum Requirement for admissibility: Assume c(a) = 6.
Cost of optimal plan: 5
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Abstraction
Cost of abstract plan: 2
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A cost partitioning is a tuple P = c1,...,cn where each ci : A → R is a cost function, sum of partial action costs bounded by original action cost
i
ci(a) ≤ c(a)
for all a ∈ A.
Theorem [Katz and Domshlak, 2010; Pommerening et al., 2015]
For admissible heuristics h1,...,hn and cost partitioning P, the cost partitioning heuristic hP(s) = ∑i hi(s,ci) is admissible.
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An optimal cost partitioning for a state s is one which maximizes the heuristic estimate,
hocp(s) = max
P
hP(s).
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Optimal Cost Partitioning
h1(s0) = c1( )+c1( ) h2(s0) = c2( )+c2( )
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Optimal Cost Partitioning
x 1 x y 1 y
h1(s0) = c1( )+c1( ) h2(s0) = c2( )+c2( ) hocp(s0) = 1+c1( )+c2( )+1 = 2+x+y
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Optimal Cost Partitioning
x 1 x y 1 y
h1(s0) = c1( )+c1( ) h2(s0) = c2( )+c2( ) hocp(s0) = 1+c1( )+c2( )+1 = 2+x+y
maximize this subject to x+y ≤ 1
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Optimal Cost Partitioning
x 1 x y 1 y
h1(s0) = c1( )+c1( ) h2(s0) = c2( )+c2( ) hocp(s0) = 1+c1( )+c2( )+1 = 2+x+y
maximize this subject to x+y ≤ 1
⇒ hocp(s0) = 3
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Optimal Cost Partitioning
x 1 x y 1 y
Problem: Action counted only once, but could be counted twice.
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Optimal Cost Partitioning
x 1 x y 1 y
Problem: Action counted only once, but could be counted twice. Idea: Distinguish states in which action is applied. (= Assign partial costs state-wise.) If different, allow counting it both times.
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A state-dependent cost partitioning is a tuple P = c1,...,cn where each ci : A×S → R is a state-dependent cost function sum of partial action costs bounded by original action cost
i
ci(a,s) ≤ c(a)
for all a ∈ A,s ∈ S.
Theorem
For admissible heuristics h1,...,hn and state-dependent cost partitioning P, the cost partitioning heuristic hP(s) = ∑i hi(s,ci) is admissible.
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Optimal State-Dependent Cost Partitioning
1 1 1 1
c1( ,s0) = 1 c1( ,s′) = 0 (s′ = s0) c2( ,s) = 1−c1( ,s)
for all s
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Optimal State-Dependent Cost Partitioning
1 1 1 1
c1( ,s0) = 1 c1( ,s′) = 0 (s′ = s0) c2( ,s) = 1−c1( ,s)
for all s
⇒ hocp-dep(s0) = 1+1+1+1 = 4
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Optimal State-Dependent Cost Partitioning
1 1 1 1
c1( ,s0) = 1 c1( ,s′) = 0 (s′ = s0) c2( ,s) = 1−c1( ,s)
for all s
⇒ hocp-dep(s0) = 1+1+1+1 = 4 > 3 = hocp(s0).
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State-Independent or State-Dependent
Problem: Optimal state-dependent cost partitioning too expensive to compute (exponential). Remedy: Consider saturated cost-partitioning as an alternative. Idea: In current abstraction, leave as much remaining costs for subsequent abstractions as possible without making current abstraction less informative. Details: See IJCAI 2016 paper [KPSGM16].
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Dominance and Incomparability
Opt-Dep Sat-Dep Opt-Ind Sat-Ind
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Dominance and Incomparability
Opt-Dep Sat-Dep Opt-Ind Sat-Ind
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Dominance and Incomparability
saturated
Opt-Dep Sat-Dep Opt-Ind Sat-Ind
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Dominance and Incomparability
saturated
Opt-Dep Sat-Dep Opt-Ind Sat-Ind
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Dominance and Incomparability
Opt-Dep Sat-Dep Opt-Ind Sat-Ind
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Dominance and Incomparability
state-dependent
Opt-Dep Sat-Dep Opt-Ind Sat-Ind
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Dominance and Incomparability
state-dependent state-independent
Opt-Dep Sat-Dep Opt-Ind Sat-Ind
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Dominance and Incomparability
state-dependent state-independent
Opt-Dep Sat-Dep Opt-Ind Sat-Ind
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Dominance and Incomparability
state-dependent state-independent
Opt-Dep Sat-Dep Opt-Ind Sat-Ind
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Dominance and Incomparability
Opt-Dep Sat-Dep Opt-Ind Sat-Ind
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Dominance and Incomparability
Opt-Dep Sat-Dep Opt-Ind Sat-Ind
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State-dependent more fine-grained than state-independent cost partitioning. Complete classification of dominance among
{Opt,Sat}×{Dep,Ind}.
Only preliminary empirical results.
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Preliminary but promising. Opt-Dep really too expensive. Though theoretically incomparable, Sat-Dep often more accurate than Sat-Ind. Conjecture: Sat-Dep and Opt-Ind have potential to complement each other.
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