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 Geißer 2 Robert Mattmüller 2 1 University of Basel 2 University 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 New challenge: abstraction admissible uninformative combination Solution: multiple Solution: partial costs abstractions in abstractions; then take sum

✪ Challenge: single New challenge: abstraction admissible uninformative combination Solution: multiple Solution: partial costs abstractions in abstractions; then take sum Requirement for admissibility: Assume c ( a ) = 6 . c 1 ( a ) = 3 c 2 ( a ) = 2 ✧ ∑ = 5

Challenge: single New challenge: abstraction admissible uninformative combination Solution: multiple Solution: partial costs abstractions in abstractions; then take sum Requirement for admissibility: Assume c ( a ) = 6 . c 1 ( a ) = 3 c 1 ( a ) = 4 c 2 ( a ) = 2 c 2 ( a ) = 3 ✧ ✪ ∑ = 5 ∑ = 7

State-dependent cost partitioning (uncharted territory!) State-independent cost partitioning (you are here!)

Running Example: Logistics Task Cost of optimal plan: 5 2016-09-28 Keller, Pommerening, Seipp, Geißer, Mattmüller – State-Dependent Cost Partitioning 7 / 16

Running Example: Logistics Task Abstraction 1 1 Cost of abstract plan: 2 2016-09-28 Keller, Pommerening, Seipp, Geißer, Mattmüller – State-Dependent Cost Partitioning 8 / 16

Cost Partitioning A cost partitioning is a tuple P = � c 1 ,..., c n � where each c i : A → R is a cost function, sum of partial action costs bounded by original action cost ∑ c i ( a ) ≤ c ( a ) for all a ∈ A . i Theorem [Katz and Domshlak, 2010; Pommerening et al., 2015] For admissible heuristics h 1 ,..., h n and cost partitioning P , the cost partitioning heuristic h P ( s ) = ∑ i h i ( s , c i ) is admissible. 2016-09-28 Keller, Pommerening, Seipp, Geißer, Mattmüller – State-Dependent Cost Partitioning 9 / 16

Optimal Cost Partitioning An optimal cost partitioning for a state s is one which maximizes the heuristic estimate, h ocp ( s ) = max h P ( s ) . P 2016-09-28 Keller, Pommerening, Seipp, Geißer, Mattmüller – State-Dependent Cost Partitioning 10 / 16

Running Example: Logistics Task Optimal Cost Partitioning h 1 ( s 0 ) = c 1 ( )+ c 1 ( ) h 2 ( s 0 ) = c 2 ( )+ c 2 ( ) 2016-09-28 Keller, Pommerening, Seipp, Geißer, Mattmüller – State-Dependent Cost Partitioning 11 / 16

Running Example: Logistics Task Optimal Cost Partitioning 0 h 1 ( s 0 ) = c 1 ( )+ c 1 ( ) h 2 ( s 0 ) = c 2 ( )+ c 2 ( ) x x 0 1 h ocp ( s 0 ) = 1 + c 1 ( )+ c 2 ( )+ 1 = 2 + x + y 1 y y 0 0 2016-09-28 Keller, Pommerening, Seipp, Geißer, Mattmüller – State-Dependent Cost Partitioning 11 / 16

Running Example: Logistics Task Optimal Cost Partitioning 0 h 1 ( s 0 ) = c 1 ( )+ c 1 ( ) h 2 ( s 0 ) = c 2 ( )+ c 2 ( ) x x 0 1 h ocp ( s 0 ) = 1 + c 1 ( )+ c 2 ( )+ 1 = 2 + x + y maximize this subject to x + y ≤ 1 1 y y 0 0 2016-09-28 Keller, Pommerening, Seipp, Geißer, Mattmüller – State-Dependent Cost Partitioning 11 / 16

Running Example: Logistics Task Optimal Cost Partitioning 0 h 1 ( s 0 ) = c 1 ( )+ c 1 ( ) h 2 ( s 0 ) = c 2 ( )+ c 2 ( ) x x 0 1 h ocp ( s 0 ) = 1 + c 1 ( )+ c 2 ( )+ 1 = 2 + x + y maximize this subject to x + y ≤ 1 1 y y 0 h ocp ( s 0 ) = 3 ⇒ 0 2016-09-28 Keller, Pommerening, Seipp, Geißer, Mattmüller – State-Dependent Cost Partitioning 11 / 16

Running Example: Logistics Task Optimal Cost Partitioning 0 Problem: Action counted only once, x x 0 but could be counted twice. 1 1 y y 0 0 2016-09-28 Keller, Pommerening, Seipp, Geißer, Mattmüller – State-Dependent Cost Partitioning 11 / 16

Running Example: Logistics Task Optimal Cost Partitioning 0 Problem: Action counted only once, x x 0 but could be counted twice. 1 Idea: Distinguish states in which action is applied. (= Assign partial costs state-wise.) 1 If different, allow counting it y y 0 both times. 0 2016-09-28 Keller, Pommerening, Seipp, Geißer, Mattmüller – State-Dependent Cost Partitioning 11 / 16

State-Dependent Cost Partitioning A state-dependent cost partitioning is a tuple P = � c 1 ,..., c n � where each c i : A × S → R is a state-dependent cost function sum of partial action costs bounded by original action cost ∑ c i ( a , s ) ≤ c ( a ) for all a ∈ A , s ∈ S . i Theorem For admissible heuristics h 1 ,..., h n and state-dependent cost partitioning P , the cost partitioning heuristic h P ( s ) = ∑ i h i ( s , c i ) is admissible. 2016-09-28 Keller, Pommerening, Seipp, Geißer, Mattmüller – State-Dependent Cost Partitioning 12 / 16

Running Example: Logistics Task Optimal State-Dependent Cost Partitioning 0 c 1 ( , s 0 ) = 1 ( s ′ � = s 0 ) , s ′ ) = 0 c 1 ( 1 0 0 c 2 ( , s ) = 1 − c 1 ( , s ) 1 for all s 1 0 0 1 0 2016-09-28 Keller, Pommerening, Seipp, Geißer, Mattmüller – State-Dependent Cost Partitioning 13 / 16

Running Example: Logistics Task Optimal State-Dependent Cost Partitioning 0 c 1 ( , s 0 ) = 1 ( s ′ � = s 0 ) , s ′ ) = 0 c 1 ( 1 0 0 c 2 ( , s ) = 1 − c 1 ( , s ) 1 for all s h ocp-dep ( s 0 ) = 1 + 1 + 1 + 1 = 4 ⇒ 1 0 0 1 0 2016-09-28 Keller, Pommerening, Seipp, Geißer, Mattmüller – State-Dependent Cost Partitioning 13 / 16

Running Example: Logistics Task Optimal State-Dependent Cost Partitioning 0 c 1 ( , s 0 ) = 1 ( s ′ � = s 0 ) , s ′ ) = 0 c 1 ( 1 0 0 c 2 ( , s ) = 1 − c 1 ( , s ) 1 for all s h ocp-dep ( s 0 ) = 1 + 1 + 1 + 1 = 4 ⇒ > 3 = h ocp ( s 0 ) . 1 0 0 1 0 2016-09-28 Keller, Pommerening, Seipp, Geißer, Mattmüller – State-Dependent Cost Partitioning 13 / 16

Saturated Cost Partitioning 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]. 2016-09-28 Keller, Pommerening, Seipp, Geißer, Mattmüller – State-Dependent Cost Partitioning 14 / 16

Theoretical Results Dominance and Incomparability Opt-Dep Sat-Dep Opt-Ind Sat-Ind 2016-09-28 Keller, Pommerening, Seipp, Geißer, Mattmüller – State-Dependent Cost Partitioning 15 / 16

Theoretical Results Dominance and Incomparability optimal Opt-Dep Sat-Dep Opt-Ind Sat-Ind 2016-09-28 Keller, Pommerening, Seipp, Geißer, Mattmüller – State-Dependent Cost Partitioning 15 / 16

Theoretical Results Dominance and Incomparability optimal Opt-Dep Sat-Dep Opt-Ind Sat-Ind saturated 2016-09-28 Keller, Pommerening, Seipp, Geißer, Mattmüller – State-Dependent Cost Partitioning 15 / 16

Theoretical Results Dominance and Incomparability optimal Opt-Dep Sat-Dep Opt-Ind Sat-Ind saturated 2016-09-28 Keller, Pommerening, Seipp, Geißer, Mattmüller – State-Dependent Cost Partitioning 15 / 16

Theoretical Results Dominance and Incomparability Opt-Dep Sat-Dep Opt-Ind Sat-Ind 2016-09-28 Keller, Pommerening, Seipp, Geißer, Mattmüller – State-Dependent Cost Partitioning 15 / 16

Theoretical Results Dominance and Incomparability state-dependent Opt-Dep Sat-Dep Opt-Ind Sat-Ind 2016-09-28 Keller, Pommerening, Seipp, Geißer, Mattmüller – State-Dependent Cost Partitioning 15 / 16

Theoretical Results Dominance and Incomparability state-dependent Opt-Dep Sat-Dep Opt-Ind Sat-Ind state-independent 2016-09-28 Keller, Pommerening, Seipp, Geißer, Mattmüller – State-Dependent Cost Partitioning 15 / 16

Theoretical Results Dominance and Incomparability state-dependent Opt-Dep Sat-Dep Opt-Ind Sat-Ind state-independent 2016-09-28 Keller, Pommerening, Seipp, Geißer, Mattmüller – State-Dependent Cost Partitioning 15 / 16

Theoretical Results Dominance and Incomparability state-dependent Opt-Dep Sat-Dep Opt-Ind Sat-Ind state-independent 2016-09-28 Keller, Pommerening, Seipp, Geißer, Mattmüller – State-Dependent Cost Partitioning 15 / 16

Theoretical Results Dominance and Incomparability Opt-Dep Sat-Dep Opt-Ind Sat-Ind 2016-09-28 Keller, Pommerening, Seipp, Geißer, Mattmüller – State-Dependent Cost Partitioning 15 / 16

Theoretical Results Dominance and Incomparability Opt-Dep Sat-Dep Opt-Ind Sat-Ind 2016-09-28 Keller, Pommerening, Seipp, Geißer, Mattmüller – State-Dependent Cost Partitioning 15 / 16

Conclusion State-dependent more fine-grained than state-independent cost partitioning. Complete classification of dominance among { Opt , Sat }×{ Dep , Ind } . Only preliminary empirical results. 2016-09-28 Keller, Pommerening, Seipp, Geißer, Mattmüller – State-Dependent Cost Partitioning 16 / 16