, Formalization & Search PDB-Heuristic Benchmarks Conclusion Solving Non-deterministic Planning Problems with Pattern Database Heuristics Pascal Bercher Robert Mattm¨ uller Institute of Artificial Intelligence Department of Computer Science University of Ulm, Germany University of Freiburg, Germany KI 2009, Paderborn Solving Non-Deterministic Planning Problems with Pattern Database Heuristics 15.-18. Sept 2009, KI ’09 1 / 18
, Formalization & Search PDB-Heuristic Benchmarks Conclusion Formalization A non-deterministic planning problem. Given: (Informally: Initial state, actions, goal states. Nondeterminism: actions may have several outcomes.) A solution to that problem. Desired: (Informally: How to reach a goal state, using the actions?) Solving Non-Deterministic Planning Problems with Pattern Database Heuristics 15.-18. Sept 2009, KI ’09 2 / 18
, Formalization & Search PDB-Heuristic Benchmarks Conclusion Formalization Given: Non-deterministic planning problem P = ( Var , A , s 0 , G ) with: • Var , finite set of state variables . S = 2 Var is the state space. • A , finite set of actions a = � pre ( a ) , eff ( a ) � and: • pre ( a ) ⊆ Var and • eff ( a ) = { � add i , del i � | add i , del i ⊆ Var and i ∈ { 1 , . . . , n } } . • Its application (if pre ( a ) ⊆ s ) leads to: app ( s , a ) = { ( s \ del ) ∪ add | � add , del � ∈ eff ( a ) } • s 0 ∈ S , the initial state . • G ⊆ Var , the goal description . A state s ∈ S is a goal state iff s ⊇ G . Solving Non-Deterministic Planning Problems with Pattern Database Heuristics 15.-18. Sept 2009, KI ’09 3 / 18
, Formalization & Search PDB-Heuristic Benchmarks Conclusion Formalization (Example) Let s = { x , y , z } ∈ S be a state and a ∈ A be an action with: a = � pre ( a ) , eff ( a ) � and pre ( a ) = { x , y } ⊆ s , eff ( a ) = { �{ z } , { x , y }� , �∅ , { t , z }� } . { x , y , z } a add: z add: ε del: x,y del: t,z { z } { x , y } Solving Non-Deterministic Planning Problems with Pattern Database Heuristics 15.-18. Sept 2009, KI ’09 4 / 18
, Formalization & Search PDB-Heuristic Benchmarks Conclusion Formalization Desired: Strong plan. (Success, regardless of non-deterministic outcome.) Solving Non-Deterministic Planning Problems with Pattern Database Heuristics 15.-18. Sept 2009, KI ’09 5 / 18
, Formalization & Search PDB-Heuristic Benchmarks Conclusion Formalization Desired: Strong plan. (Success, regardless of non-deterministic outcome.) Solving Non-Deterministic Planning Problems with Pattern Database Heuristics 15.-18. Sept 2009, KI ’09 5 / 18
, Formalization & Search PDB-Heuristic Benchmarks Conclusion Formalization Desired: Strong plan. (Success, regardless of non-deterministic outcome.) Solving Non-Deterministic Planning Problems with Pattern Database Heuristics 15.-18. Sept 2009, KI ’09 5 / 18
, Formalization & Search PDB-Heuristic Benchmarks Conclusion Formalization Desired: Strong plan. (Success, regardless of non-deterministic outcome.) Solving Non-Deterministic Planning Problems with Pattern Database Heuristics 15.-18. Sept 2009, KI ’09 5 / 18
, Formalization & Search PDB-Heuristic Benchmarks Conclusion Search Algorithm, modification of AO* s 0 , c=4 s 0 , c=5 c 0 c 1 c 0 c 1 Expansion → s 1 s 2 s 3 s 4 s 1 s 2 s 3 s 4 h=2 h=3 h=4 h=3 h=2 c=5 h=4 h=3 c 3 s 5 s 6 s 7 h=2 h=0 h=4 Solving Non-Deterministic Planning Problems with Pattern Database Heuristics 15.-18. Sept 2009, KI ’09 6 / 18
, Formalization & Search PDB-Heuristic Benchmarks Conclusion Idea Use abstraction to simplify the problem: S . . . S 1 S 2 S m Map the search space S to abstract search spaces S i with | S i | ≪ | S | . Compute h ( s ) , s ∈ S , on basis of all h i ( s i ) . Calculation of the h i is done before the search. Solving Non-Deterministic Planning Problems with Pattern Database Heuristics 15.-18. Sept 2009, KI ’09 7 / 18
, Formalization & Search PDB-Heuristic Benchmarks Conclusion Formalization Idea: Disregard some (or rather most of the ) state variables. The abstraction P i = ( Var i , A i , s i 0 , G i ) is the planning problem P , restricted to the pattern P i ⊆ Var : • Var i ≔ Var ∩ P i = P i , • For var ⊆ Var let var i ≔ var ∩ P i . Then: a i ≔ � pre ( a ) i , { � add i , del i � | � add , del � ∈ eff ( a ) }� for a ∈ A . Now, A i ≔ { a i | a ∈ A } . • s i 0 ≔ s 0 ∩ P i • G i ≔ G ∩ P i . Solving Non-Deterministic Planning Problems with Pattern Database Heuristics 15.-18. Sept 2009, KI ’09 8 / 18
, Formalization & Search PDB-Heuristic Benchmarks Conclusion Heuristic Computation Recall: • A pattern is a set of state variables P i ⊆ Var . Then, a pattern collection P is a set of patterns. • Compute h ( s ) , s ∈ S , on basis of all h i ( s i ) , P i ∈ P , P finite pattern collection, i.e. set of patterns. How to calculate those h i ( s i ) , s i ∈ S i ? h i ( s i ) is the true cost value cost * of the planning problem P i . Calcuation is done by a complete exhaustive search. (Thus, S i and therefore P i have to be small!) ( True means: prefer shallow solution graphs.) Solving Non-Deterministic Planning Problems with Pattern Database Heuristics 15.-18. Sept 2009, KI ’09 9 / 18
, Formalization & Search PDB-Heuristic Benchmarks Conclusion Additivity (Theorem) How to calculate h ( s ) , s ∈ S ? By using additivity! A pattern collection P is called additive , if for all states s ∈ S : h i ( s i ) ≤ cost * ( s ) , i.e. if this sum is still admissible. � P i ∈ P Known from classical planning: Theorem ( textual description ) If there is no action a ∈ A that affects variables in more than one pattern from P, then P is additive. Solving Non-Deterministic Planning Problems with Pattern Database Heuristics 15.-18. Sept 2009, KI ’09 10 / 18
, Formalization & Search PDB-Heuristic Benchmarks Conclusion Additivity (Theorem) How to calculate h ( s ) , s ∈ S ? By using additivity! A pattern collection P is called additive , if for all states s ∈ S : h i ( s i ) ≤ cost * ( s ) , i.e. if this sum is still admissible. � P i ∈ P Known from classical planning: Theorem ( mathematical description ) If for all a ∈ A and for all patterns P i ∈ P holds: If P i ∩ effvar ( a ) � ∅ , then P j ∩ effvar ( a ) = ∅ for all P j ∈ P with P j � P i , where effvar ( a ) = � � add , del �∈ eff ( a ) add ∪ del. Then P is additive. Solving Non-Deterministic Planning Problems with Pattern Database Heuristics 15.-18. Sept 2009, KI ’09 10 / 18
, Formalization & Search PDB-Heuristic Benchmarks Conclusion Additivity (Example) P = ( { a , b , c , d , e } , A , { a } , { b , c , d , e } ) with A = { a 1 , . . . , a 9 } and: a 1 = �{ a } , {�{ b } , { a }� , �{ c } , { a }�}� a 6 = �{ b , e } , {�{ c } , ∅�}� a 2 = �{ b } , {�{ e } , ∅� , �{ d } , ∅�}� a 7 = �{ c , e } , {�{ b } , ∅�}� a 3 = �{ c } , {�{ e } , ∅� , �{ d } , ∅�}� a 8 = �{ b , c , d } , {�{ e } , ∅�}� a 4 = �{ b , d } , {�{ c } , ∅�}� a 9 = �{ b , c , e } , {�{ d } , ∅�}� a 5 = �{ c , d } , {�{ b } , ∅�}� Solving Non-Deterministic Planning Problems with Pattern Database Heuristics 15.-18. Sept 2009, KI ’09 11 / 18
, Formalization & Search PDB-Heuristic Benchmarks Conclusion Additivity (Example) P = ( { a , b , c , d , e } , A , { a } , { b , c , d , e } ) with A = { a 1 , . . . , a 9 } and: a 1 = �{ a } , {�{ b } , { a }� , �{ c } , { a }�}� a 6 = �{ b , e } , {�{ c } , ∅�}� a 2 = �{ b } , {�{ e } , ∅� , �{ d } , ∅�}� a 7 = �{ c , e } , {�{ b } , ∅�}� a 3 = �{ c } , {�{ e } , ∅� , �{ d } , ∅�}� a 8 = �{ b , c , d } , {�{ e } , ∅�}� a 4 = �{ b , d } , {�{ c } , ∅�}� a 9 = �{ b , c , e } , {�{ d } , ∅�}� a 5 = �{ c , d } , {�{ b } , ∅�}� Now, consider the pattern collection P = {{ a , b , c } , { d , e }} . Solving Non-Deterministic Planning Problems with Pattern Database Heuristics 15.-18. Sept 2009, KI ’09 11 / 18
, Formalization & Search PDB-Heuristic Benchmarks Conclusion Additivity (Example) P = ( { a , b , c , d , e } , A , { a } , { b , c , d , e } ) with A = { a 1 , . . . , a 9 } and: a 1 = �{ a } , {�{ b } , { a }� , �{ c } , { a }�}� a 6 = �{ b , e } , {�{ c } , ∅�}� a 2 = �{ b } , {�{ e } , ∅� , �{ d } , ∅�}� a 7 = �{ c , e } , {�{ b } , ∅�}� a 3 = �{ c } , {�{ e } , ∅� , �{ d } , ∅�}� a 8 = �{ b , c , d } , {�{ e } , ∅�}� a 4 = �{ b , d } , {�{ c } , ∅�}� a 9 = �{ b , c , e } , {�{ d } , ∅�}� a 5 = �{ c , d } , {�{ b } , ∅�}� Now, consider the pattern collection P = {{ a , b , c } , { d , e }} . Solving Non-Deterministic Planning Problems with Pattern Database Heuristics 15.-18. Sept 2009, KI ’09 11 / 18
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