Motivation LTL f in Heuristic Search Finding Information Exploiting Information Experiments Conclusion Finding and Exploiting LTL Trajectory Constraints in Heuristic Search Salom´ e Simon Gabriele R¨ oger University of Basel, Switzerland SoCS 2015
Motivation LTL f in Heuristic Search Finding Information Exploiting Information Experiments Conclusion Motivation Goal framework for describing information about the search space combining information from different sources � creating stronger heuristics decoupling the derivation and exploitation of information � split work between different experts i 3 i 2 i 1 LTL f Framework h 1 h 2
Motivation LTL f in Heuristic Search Finding Information Exploiting Information Experiments Conclusion Linear Temporal Logic on Finite Traces (LTL f ) evaluated over a linear sequence of worlds (= variable assignments) extends propositional logic with: ϕ ϕ ϕ � ϕ Always w 0 w 1 w n ϕ ♦ ϕ Eventually w 0 w i w n ϕ ϕ Next ❡ w 0 w 1 w n ϕ ϕ ψ ϕ U ψ Until w 0 w i w n w i + 1 ψ ψ ∧ ϕ ϕ R ψ Release w 0 w i w n last last Last world w 0 w n
Motivation LTL f in Heuristic Search Finding Information Exploiting Information Experiments Conclusion LTL f Formulas in the Search Space variable ↔ state variable or action world ↔ node in search space (with incoming action) world sequence ↔ path to a goal node LTL f formulas associated to nodes → express conditions all optimal paths to a goal need to fulfill G ϕ
Motivation LTL f in Heuristic Search Finding Information Exploiting Information Experiments Conclusion Feasibility for Nodes Definition (Feasibility for nodes) An LTL f formula ϕ is feasible for n if for all paths ρ such that ρ is applicable in n , the application of ρ leads to a goal state, and g ( n ) + c ( ρ ) = h ∗ it holds that w s ρ | = ϕ. (where w s ρ = �{ a 1 } ∪ s [ a 1 ] , { a 2 } ∪ s [ � a 1 , a 2 � ] , . . . , { a n } ∪ s [ ρ ] , s [ ρ ] � )
Motivation LTL f in Heuristic Search Finding Information Exploiting Information Experiments Conclusion Adding and Propagating Information during the Search 1 new information during the search directly added to the corresponding node with conjunction
Motivation LTL f in Heuristic Search Finding Information Exploiting Information Experiments Conclusion Adding and Propagating Information during the Search 1 new information during the search directly added to the corresponding node with conjunction Example a U d a U b ∧ ♦ d a, c g =5 a a, b g =4 a, d
Motivation LTL f in Heuristic Search Finding Information Exploiting Information Experiments Conclusion Adding and Propagating Information during the Search 1 new information during the search directly added to the corresponding node with conjunction Example a U d a U b ∧ ♦ d ∧ � ¬ c a, c g =5 a a, b g =4 a, d
Motivation LTL f in Heuristic Search Finding Information Exploiting Information Experiments Conclusion Adding and Propagating Information during the Search 2 formulas can be propagated with progression to successor nodes Theorem Let ϕ be a feasible formula for a node n , and let n ′ be the successor node reached from n with action a . Then progress ( ϕ, { a } ∪ s ( n ′ )) is feasible for n ′ .
Motivation LTL f in Heuristic Search Finding Information Exploiting Information Experiments Conclusion Adding and Propagating Information during the Search 2 formulas can be propagated with progression to successor nodes Theorem Let ϕ be a feasible formula for a node n , and let n ′ be the successor node reached from n with action a . Then progress ( ϕ, { a } ∪ s ( n ′ )) is feasible for n ′ . Example a U d a, c a U b ∧ ♦ d ∧ � ¬ c g =5 a g =4 a, b a, d
Motivation LTL f in Heuristic Search Finding Information Exploiting Information Experiments Conclusion Adding and Propagating Information during the Search 2 formulas can be propagated with progression to successor nodes Theorem Let ϕ be a feasible formula for a node n , and let n ′ be the successor node reached from n with action a . Then progress ( ϕ, { a } ∪ s ( n ′ )) is feasible for n ′ . Example a U d a, c a U b ∧ ♦ d ∧ � ¬ c g =5 a g =4 a, b a, d a, d a U b ∧ � ¬ c
Motivation LTL f in Heuristic Search Finding Information Exploiting Information Experiments Conclusion Adding and Propagating Information during the Search 3 duplicate elimination: conjunction of formulas of “cheapest” nodes is feasible Theorem Let n and n ′ be two search nodes such that g ( n ) = g ( n ′ ) and s ( n ) = s ( n ′ ) . Let further ϕ n and ϕ n ′ be feasible for the respective node. Then ϕ n ∧ ϕ n ′ is feasible for both n and n ′ .
Motivation LTL f in Heuristic Search Finding Information Exploiting Information Experiments Conclusion Adding and Propagating Information during the Search 3 duplicate elimination: conjunction of formulas of “cheapest” nodes is feasible Theorem Let n and n ′ be two search nodes such that g ( n ) = g ( n ′ ) and s ( n ) = s ( n ′ ) . Let further ϕ n and ϕ n ′ be feasible for the respective node. Then ϕ n ∧ ϕ n ′ is feasible for both n and n ′ . Example a U d a, c a U b ∧ ♦ d ∧ � ¬ c g =5 a g =4 a, b a, d a U b ∧ � ¬ c
Motivation LTL f in Heuristic Search Finding Information Exploiting Information Experiments Conclusion Adding and Propagating Information during the Search 3 duplicate elimination: conjunction of formulas of “cheapest” nodes is feasible Theorem Let n and n ′ be two search nodes such that g ( n ) = g ( n ′ ) and s ( n ) = s ( n ′ ) . Let further ϕ n and ϕ n ′ be feasible for the respective node. Then ϕ n ∧ ϕ n ′ is feasible for both n and n ′ . Example a U d a, c a U b ∧ ♦ d ∧ � ¬ c g =5 a g =4 a, b a, d a U b ∧ � ¬ c
Motivation LTL f in Heuristic Search Finding Information Exploiting Information Experiments Conclusion Adding and Propagating Information during the Search 3 duplicate elimination: conjunction of formulas of “cheapest” nodes is feasible Theorem Let n and n ′ be two search nodes such that g ( n ) = g ( n ′ ) and s ( n ) = s ( n ′ ) . Let further ϕ n and ϕ n ′ be feasible for the respective node. Then ϕ n ∧ ϕ n ′ is feasible for both n and n ′ . Example a U d a, c a U b ∧ ♦ d ∧ � ¬ c g =5 a U d a g =4 a, b a, d a U b ∧ � ¬ c
Motivation LTL f in Heuristic Search Finding Information Exploiting Information Experiments Conclusion Adding and Propagating Information during the Search 3 duplicate elimination: conjunction of formulas of “cheapest” nodes is feasible Theorem Let n and n ′ be two search nodes such that g ( n ) = g ( n ′ ) and s ( n ) = s ( n ′ ) . Let further ϕ n and ϕ n ′ be feasible for the respective node. Then ϕ n ∧ ϕ n ′ is feasible for both n and n ′ . Example a U d a, c a U b ∧ ♦ d ∧ � ¬ c g =5 a U d ∧ � ¬ c a g =4 a, b a, d a U b ∧ � ¬ c
Motivation LTL f in Heuristic Search Finding Information Exploiting Information Experiments Conclusion Encoding Information in LTL f Formulas Possible sources of information: domain-specific knowledge temporally extended goals here: information used in specialized heuristics Landmarks and their orderings (Hoffmann et al. 2004, Richter et al. 2008) Unjustified Action Applications (Karpas and Domshlak 2012)
Motivation LTL f in Heuristic Search Finding Information Exploiting Information Experiments Conclusion Landmarks Fact Landmark : A fact that must be true at some point in every solution (Hoffmann et al. 2004) → In LTL f : ♦ l Further information about landmarks: First achievers : l ∨ � a ∈ FA l ♦ a Required again : ( ♦ l ) U l ′ � g ′ ∈ G g ′ � Goal : � ( ♦ g ) U � g ∈ G
Motivation LTL f in Heuristic Search Finding Information Exploiting Information Experiments Conclusion Unjustified Action Applications If an action is applied, its effects must be of some use (Karpas and Domshlak 2012) one of its effects is necessary for applying another action 1 one of its effects is a goal variable (that is not made false again) 2 � � ¬ a ′ ) U � a ′ � � ϕ a = ( e ∧ ∨ e ∈ add ( a ) \ G a ′ ∈ A with a ′ ∈ A with e ∈ add ( a ′ ) e ∈ pre ( a ′ ) � � � ¬ a ′ ) U a ′ �� � � ( e ∧ last ∨ e ∈ add ( a ) ∩ G a ′ ∈ A with a ′ ∈ A with e ∈ add ( a ′ ) e ∈ pre ( a ′ )
Motivation LTL f in Heuristic Search Finding Information Exploiting Information Experiments Conclusion Heuristics Very rich temporal information possible → heuristics can use different levels of relaxation Proof of concept heuristic extracts landmarks from node-associated formulas → looses temporal information between landmarks
Motivation LTL f in Heuristic Search Finding Information Exploiting Information Experiments Conclusion Experiment Setup Configurations: 1 h LA : standard admissible landmark heuristic (Karpas and Domshlak 2009) h LM AL : LTL landmark extraction heuristic with sources: 2 Landmarks (First achievers, Required again, Goal) h LM+UAA : LTL landmark extraction heuristic with sources: 3 AL Landmarks (First achievers, Required again, Goal) Unjustified Action Applications all heuristics use BJOLP landmark extraction and optimal cost partitioning search algorithm: h LA uses LM-A ∗ , the others a slight variant we call LTL-A ∗
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