Symbolic Search and Abstraction Heuristics for Cost-Optimal Planning - PowerPoint PPT Presentation

Optimizing Image Computation Image computation is the main bottleneck in symbolic search How to represent the Transition Relation? ◮ Monolithic relation ⇒ may use exponential memory ◮ Solution in G AMER ⇒ One TR for each action Idea 1: Separate preconditions and effects → avoid using auxiliary variables! move-T (A, B) pre: at-T A eff: at-T B load-P (A) pre: at-T A, at-P A eff: at-P T move-T (B, A) pre: at-T B eff: at-T A . . . ´ Alvaro Torralba PhD Defense June 2, 2015 13 / 54

Optimizing Image Computation Image computation is the main bottleneck in symbolic search How to represent the Transition Relation? ◮ Monolithic relation ⇒ may use exponential memory ◮ Solution in G AMER ⇒ One TR for each action Idea 1: Separate preconditions and effects → avoid using auxiliary variables! Idea 2: Conjunction Tree → check preconditions of all operators simultaneously at-T move-T (A, B) A B ⋆ load-P (A) at-P eff: at-T A . . . move-T (B, A) A B T ⋆ . . . eff: at-P T eff: at-P A eff: at-T B ∅ ´ Alvaro Torralba PhD Defense June 2, 2015 13 / 54

Optimizing Image Computation Image computation is the main bottleneck in symbolic search How to represent the Transition Relation? ◮ Monolithic relation ⇒ may use exponential memory ◮ Solution in G AMER ⇒ One TR for each action Idea 1: Separate preconditions and effects → avoid using auxiliary variables! Idea 2: Conjunction Tree → check preconditions of all operators simultaneously Idea 3: Aggregate TRs → different strategies to group the actions move-T (A, B) move-T (A, B) load-P (A) load-P (A) move-T (B, A) move-T (B, A) . . . . . . ´ Alvaro Torralba PhD Defense June 2, 2015 13 / 54

Empirical Results Compare image computation methods: TR 1 : baseline approach 1 TR 1 + : avoid using x ′ variables 2 CT L 20 : conjunction tree 3 T DT 100 k : aggregate TRs 4 Total coverage of symbolic search algorithms over 1375 instances: TR 1 TR 1 + CT L T DT 20 100 k Forward uniform-cost search 699 676 724 742 Backward uniform-cost search 444 525 529 532 Bidirectional uniform-cost search 729 763 769 793 BDDA ∗ with SPDBs 705 717 724 764 TR 1 ≤ TR 1 + ≤ CT L 20 ≤ T DT 100 k (across all domains) ´ Alvaro Torralba PhD Defense June 2, 2015 14 / 54

Time of Bidirectional Search Solving time of TR 1 (seconds) 10 3 10 2 10 1 10 0 10 0 10 1 10 2 10 3 Solving time of T DT 100 (seconds) ´ Alvaro Torralba PhD Defense June 2, 2015 15 / 54

Outline Introduction 1 Cost-Optimal Planning Symbolic Search 2 (Background) Symbolic Search Image Computation State Invariants Abstraction Heuristics 3 (Background) Abstractions Merge-and-Shrink for Symbolic Search Symbolic Perimeter Merge-and-Shrink Symbolic Bidirectional Heuristic Search 4 5 Conclusions Final Results: IPC14 Conclusions ´ Alvaro Torralba PhD Defense June 2, 2015 16 / 54

Motivation: State Invariants in Symbolic Search Invariant: holds in all states that may belong to a solution path Mutex: pair of facts that cannot be true in the same state 1 → a truck cannot be simultaneously at two locations Invariant group: Set of facts such that exactly one is true 2 → a truck must be somewhere Generated computing h 2 in both directions Useful for: Removing operators from the planning task 1 Pruning invalid states during the search 2 ´ Alvaro Torralba PhD Defense June 2, 2015 17 / 54

Encoding Constraints with cBDD cBDD : BDD that describes invalid states Mutex: f i ∧ f j 1 “At-least-1” invariant: ¬ ( f 1 ∨ f 2 ∨ . . . ∨ f n ) 2 Remove invalid states from S g : S g \ cBDD v 1 v 1 v 1 v 2 v 2 v 2 v 2 v 2 v 3 v 3 v 3 v 3 T F T F T F (c) cBDD (d) S g (e) S g \ cBDD e-deletion : encode invariants in the TRs → no invalid states are generated ´ Alvaro Torralba PhD Defense June 2, 2015 18 / 54

Experimental Results Constraints found in 35 out of 43 domains 10%-74% invalid operators found in 17 out of 43 domains Mutex types: ◮ Baseline (B) ◮ Not pruning invalid states: M ∅ ◮ Pruning invalid states: cBDD or e-deletion ( e - del ) Remove invalid ops B M ∅ cBDD e - del Forward uniform-cost search 699 742 745 750 Backward uniform-cost search 509 532 677 696 Bidirectional uniform-cost search 765 793 836 841 BDDA ∗ with SPDBs 736 764 777 781 ´ Alvaro Torralba PhD Defense June 2, 2015 19 / 54

Time of Bidirectional Uniform-Cost Search 2 , 000 Solving time of O h2 (seconds) 2 , 000 Solving time of O (seconds) 1 , 500 1 , 500 1 , 000 1 , 000 500 500 0 0 0 500 1 , 000 1 , 500 2 , 000 0 500 1 , 000 1 , 500 2 , 000 Solving time of O h2 (seconds) Solving time of e - del (seconds) (f) remove operators (g) prune invalid states Solving time of M 100 k (seconds) 2 , 000 1 , 500 1 , 000 500 0 0 500 1 , 000 1 , 500 2 , 000 Solving time of e - del (seconds) (h) e-deletion vs cBDD ´ Alvaro Torralba PhD Defense June 2, 2015 20 / 54

Comparison with State-of-the-Art Planners 800 600 Coverage 400 200 C G AMER -BD G AMER - BD 0 10 0 10 1 10 2 10 3 Time (seconds) ´ Alvaro Torralba PhD Defense June 2, 2015 21 / 54

Comparison with State-of-the-Art Planners 800 600 Coverage 400 200 C G AMER - FW E XPLICIT -FW 0 10 0 10 1 10 2 10 3 Time (seconds) ´ Alvaro Torralba PhD Defense June 2, 2015 21 / 54

Comparison with State-of-the-Art Planners 800 600 Coverage 400 200 A ∗ + LM- CUT BDDA ∗ + SPDBs 0 10 0 10 1 10 2 10 3 Time (seconds) ´ Alvaro Torralba PhD Defense June 2, 2015 21 / 54

Comparison with State-of-the-Art Planners 800 600 Coverage 400 C G AMER -BD G AMER - BD C G AMER - FW 200 E XPLICIT -FW A ∗ + LM- CUT BDDA ∗ + SPDBs 0 10 0 10 1 10 2 10 3 Time (seconds) ´ Alvaro Torralba PhD Defense June 2, 2015 21 / 54

Summary Image computation 1 ◮ Analyzed different methods for image computation ◮ Best method: aggregate TRs State invariants 2 ◮ Pruning invalid states (specially useful in bw search) ◮ Best encoding for symbolic search: e-edeletion These significantly improved performance of symbolic planning → Symbolic bidirectional blind search is the current state-of-the-art for cost-optimal planning ´ Alvaro Torralba PhD Defense June 2, 2015 22 / 54

Motivation: Heuristics in Symbolic Search Heuristics Delete-relaxation: h max , h + Landmarks: h LA , LM-cut Abstractions: PDBs, M&S, CEGAR, Fork Critical paths: h m Flow max add LP ´ Alvaro Torralba PhD Defense June 2, 2015 24 / 54

Motivation: Heuristics in Symbolic Search Heuristics Delete-relaxation: h max , h + Landmarks: h LA , LM-cut Abstractions: PDBs, M&S, CEGAR, Fork Critical paths: h m Flow max add LP Challenge: How to evaluate h ( s ) on a set of states? ´ Alvaro Torralba PhD Defense June 2, 2015 24 / 54

Abstraction Heuristics Abstraction: Mapping from states to abstract states ◮ Smaller abstract state space → easier to search ◮ Use optimal distances in abstract state space as heuristic ◮ Preserve transitions → admissible estimation 4 5 6 7 start 00 10 20 30 0 3 2 1 01 11 21 31 3 2 1 0 02 12 22 32 ´ Alvaro Torralba PhD Defense June 2, 2015 25 / 54

Abstraction Heuristics Abstraction: Mapping from states to abstract states ◮ Smaller abstract state space → easier to search ◮ Use optimal distances in abstract state space as heuristic ◮ Preserve transitions → admissible estimation 4 5 6 7 start 00 10 20 30 0 3 2 1 01 11 21 31 3 2 1 0 02 12 22 32 h = 3 2 1 0 ´ Alvaro Torralba PhD Defense June 2, 2015 25 / 54

Abstraction Heuristics Abstraction: Mapping from states to abstract states ◮ Smaller abstract state space → easier to search ◮ Use optimal distances in abstract state space as heuristic ◮ Preserve transitions → admissible estimation 4 5 6 7 start 00 10 20 30 0 3 2 1 01 11 21 31 3 2 1 0 02 12 22 32 h = 3 2 1 0 Pattern Databases (PDBs) ◮ Ignore some variables in the problem ◮ Limitation: ignoring a single variable may relax too much ´ Alvaro Torralba PhD Defense June 2, 2015 25 / 54

Abstraction Heuristics Abstraction: Mapping from states to abstract states ◮ Smaller abstract state space → easier to search ◮ Use optimal distances in abstract state space as heuristic ◮ Preserve transitions → admissible estimation 2 2 1 0 start 00 10 20 30 0 1 1 1 01 11 21 31 0 0 0 0 02 12 22 32 h = 0 0 0 0 Pattern Databases (PDBs) ◮ Ignore some variables in the problem ◮ Limitation: ignoring a single variable may relax too much ´ Alvaro Torralba PhD Defense June 2, 2015 25 / 54

Merge-and-Shrink Algorithm (M&S) Algorithm 1: M&S α 1 ← Π υ 1 foreach υ ∈ { υ 2 . . . υ n } : if | α | > N : shrink ( α i − 1 ) ⊗ Π i α i ← α i − 1 ⊗ Π i return α Merge strategy: Linear → variable ordering Shrink strategy → reduce abstraction size ´ Alvaro Torralba PhD Defense June 2, 2015 26 / 54

Merge-and-Shrink Algorithm (M&S) α 1 = T A move A , B Algorithm 1: M&S A B α 1 ← Π υ 1 foreach υ ∈ { υ 2 . . . υ n } : move B , A if | α | > N : shrink ( α i − 1 ) ⊗ Π i α i ← α i − 1 ⊗ Π i return α Merge strategy: Linear → variable ordering Shrink strategy → reduce abstraction size ´ Alvaro Torralba PhD Defense June 2, 2015 26 / 54

Merge-and-Shrink Algorithm (M&S) α 1 = T A move A , B Algorithm 1: M&S A B α 1 ← Π υ 1 foreach υ ∈ { υ 2 . . . υ n } : move B , A if | α | > N : α 3 = T A , P T , P A shrink ( α i − 1 ) ⊗ Π i move A , B α i ← α i − 1 ⊗ Π i return α AA BA move B , A load/unload A Merge strategy: Linear move A , B → variable ordering Ac Bc Shrink strategy move B , A load/unload B → reduce abstraction move A , B size AB BB move B , A ´ Alvaro Torralba PhD Defense June 2, 2015 26 / 54

Merge-and-Shrink for Symbolic Search Hypothesis: BDDA ∗ lacks good heuristics → BDDA ∗ + M&S can improve results How to use M&S in symbolic search: M&S algorithm M&S heuristic to ADD ADD to BDDs BDDs to use in symbolic search ´ Alvaro Torralba PhD Defense June 2, 2015 28 / 54

Merge-and-Shrink as ADDs α 1 = T A move A , B A B move B , A α 3 = T A , P T , P A move A , B AA BA move B , A load/unload A move A , B Ac Bc move B , A load/unload B move A , B AB BB move B , A ´ Alvaro Torralba PhD Defense June 2, 2015 29 / 54

Merge-and-Shrink as ADDs α 1 = T A move A , B A B move B , A T A α 3 = T A , P T , P A P T move A , B AA BA P A move B , A load/unload A move A , B Ac Bc move B , A load/unload B move A , B AB BB move B , A ´ Alvaro Torralba PhD Defense June 2, 2015 29 / 54

Merge-and-Shrink as ADDs α 1 = T A move A , B A B move B , A T A α 3 = T A , P T , P A P T move A , B AA BA P A move B , A load/unload A move A , B AA Ac AB ✗ BA Bc BB ✗ Ac Bc move B , A load/unload B move A , B AB BB move B , A ´ Alvaro Torralba PhD Defense June 2, 2015 29 / 54

Merge-and-Shrink as ADDs α 1 = T A move A , B A B move B , A T A α 3 = T A , P T , P A P T move A , B AA BA P A move B , A load/unload A move A , B AA Ac AB ✗ BA Bc Ac Bc move B , A load/unload B move A , B AB BB move B , A ´ Alvaro Torralba PhD Defense June 2, 2015 29 / 54

Theoretical Results M&S to ADDs/BDDs in polynomial time Related empirical results: ◮ ADD representation of heuristics reduces memory ◮ Variable ordering has a huge impact ADD/BDD reduction rules may achieve exponential gain in memory with respect to shrinking perfect strategies → shows potential of improvement for M&S strategies ´ Alvaro Torralba PhD Defense June 2, 2015 30 / 54

Empirical Results Used M&S in symbolic search → Worse than symbolic PDBs 657 M&S g 749 638 M&S b 739 568 Blind ( ∅ ) 756 550 600 650 700 750 Coverage Symbolic-A ∗ Explicit-A ∗ ◮ Contradicts our hypothesis ´ Alvaro Torralba PhD Defense June 2, 2015 31 / 54

Motivation: Combine Symbolic Search and M&S Symbolic PDBs: larger abstract state spaces 1 M&S: flexible abstractions 2 Can we get the best of both worlds? ´ Alvaro Torralba PhD Defense June 2, 2015 33 / 54

Motivation: Combine Symbolic Search and M&S Symbolic PDBs: larger abstract state spaces 1 M&S: flexible abstractions 2 Can we get the best of both worlds? → Use symbolic search to search M&S abstractions! Symbolic Perimeter M&S: Symbolic M&S abstractions: larger M&S abstract state spaces 1 Perimeter abstractions 2 ´ Alvaro Torralba PhD Defense June 2, 2015 33 / 54

SM&S Hierarchy Enlarged M&S abstractions: to perform symbolic search v 1 v 2 v 3 v 4 v 5 α SM&S (original problem) 0 α SM&S α 1 1 α SM&S α 2 2 α SM&S α 3 3 α SM&S α 4 (M&S abstraction) 4 ´ Alvaro Torralba PhD Defense June 2, 2015 34 / 54

Perimeter Abstractions Challenges addressed with symbolic search Regression 1 Expensive operations: 2 ⋆ membership in perimeter ⋆ frontier mapping S ⋆ Set perimeter radius 3 Exp ( α 0 ) Contributions Exp ( α 1 ) Multiple abstraction levels 1 Improved initialization of 2 abstract searches ´ Alvaro Torralba PhD Defense June 2, 2015 35 / 54

Perimeter Abstractions Challenges addressed with symbolic search Regression 1 Expensive operations: 2 ⋆ membership in perimeter ⋆ frontier mapping S ⋆ Set perimeter radius 3 Exp ( α 0 ) Contributions Exp ( α 1 ) Multiple abstraction levels 1 Exp ( α 2 ) Improved initialization of 2 abstract searches ´ Alvaro Torralba PhD Defense June 2, 2015 35 / 54

Symbolic Perimeter Merge-and-Shrink V α 0 = ∅ Exp ( α 0 ) υ 0 υ 1 υ 2 S 1 S 2 ∅ S ⋆ υ 3 . . . h 0 1 2 truncated h Exp ( α 0 ) h SPM&S heuristic is admissible and consistent ´ Alvaro Torralba PhD Defense June 2, 2015 36 / 54

Symbolic Perimeter Merge-and-Shrink V α 1 υ 0 V α 0 = ∅ υ 1 M&S e 0 e 1 e 2 Exp ( α 0 ) υ 0 υ 1 υ 2 ∅ S ⋆ S 1 S 2 υ 3 . . . h 0 1 2 truncated h Exp ( α 0 ) h SPM&S heuristic is admissible and consistent ´ Alvaro Torralba PhD Defense June 2, 2015 36 / 54

Symbolic Perimeter Merge-and-Shrink V α 1 υ 0 V α 0 = ∅ υ 1 M&S e 0 e 1 e 2 Exp ( α 0 ) Exp ( α 1 ) υ 0 υ 1 α 1 α 1 υ 2 S 3 ∅ S ⋆ S 1 S 2 S 2 υ 3 . . . h 0 1 2 2 3 truncated truncated h Exp ( α 0 ) h Exp ( α 1 ) h SPM&S heuristic is admissible and consistent ´ Alvaro Torralba PhD Defense June 2, 2015 36 / 54

Symbolic Perimeter Merge-and-Shrink V α 1 υ 0 υ 0 V α 0 = ∅ V α 2 υ 1 υ 1 M&S M&S υ 2 e 0 e 1 e 2 e 0 e 1 e 2 Exp ( α 0 ) Exp ( α 1 ) Exp ( α 2 ) υ 0 υ 1 α 1 α 1 α 2 α 2 υ 2 S 3 S 3 ∅ S ⋆ S 1 S 2 S 2 S 4 υ 3 . . . h 0 1 2 2 3 3 4 truncated truncated h Exp ( α 0 ) h Exp ( α 1 ) h Exp ( α 2 ) h SPM&S heuristic is admissible and consistent ´ Alvaro Torralba PhD Defense June 2, 2015 36 / 54

Empirical Results LM- CUT 796 M&S 650 809 SPM&S SPPDB 822 SP 800 650 700 750 800 850 Coverage ´ Alvaro Torralba PhD Defense June 2, 2015 37 / 54

Empirical Results: Expanded Nodes 10 8 10 8 Expanded nodes M&S bop 10k Expanded nodes LM- CUT 10 6 10 6 10 4 10 4 10 2 10 2 10 0 10 0 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7 Expanded nodes SPM&S bop 10k Expanded nodes SPM&S bop 10k ´ Alvaro Torralba PhD Defense June 2, 2015 38 / 54

Empirical Results: Expanded Nodes 10 8 Expanded nodes SPPDB 10 6 Expanded nodes SP 10 6 10 4 10 4 10 2 10 2 10 0 10 0 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 Expanded nodes SPM&S bop 10k Expanded nodes SPM&S bop 10k ´ Alvaro Torralba PhD Defense June 2, 2015 39 / 54

Summary Symbolic Perimeter M&S Combines M&S, perimeter abstractions and symbolic search Improvements to perimeter abstractions Synergy between symbolic search and perimeter abstractions More accurate heuristic than both! But... Results still worse than symbolic bidirectional uniform-cost search ´ Alvaro Torralba PhD Defense June 2, 2015 40 / 54

Motivation: Heuristics in Symbolic Bidirectional Search Observations Bidirectional brute-force search is a state-of-the-art technique 1 Good symbolic abstraction heuristics 2 ´ Alvaro Torralba PhD Defense June 2, 2015 42 / 54

Motivation: Heuristics in Symbolic Bidirectional Search Observations Bidirectional brute-force search is a state-of-the-art technique 1 Good symbolic abstraction heuristics 2 Use abstraction heuristics in symbolic bidirectional search! ´ Alvaro Torralba PhD Defense June 2, 2015 42 / 54

Motivation: Heuristics in Symbolic Bidirectional Search Observations Bidirectional brute-force search is a state-of-the-art technique 1 Good symbolic abstraction heuristics 2 Use abstraction heuristics in symbolic bidirectional search! However, bidirectional heuristic search is not so easy: ◮ Very promising since years ago ◮ Never really able to outperform A ∗ or bidirectional uniform-cost search ´ Alvaro Torralba PhD Defense June 2, 2015 42 / 54

Algorithm Main idea: Start symbolic bidirectional uniform-cost search 1 ⋆ If it succeeds → done! Detect when it is going to fail and activate heuristics 2 Abstraction heuristics: Bidirectional, Partial, Perimeter s 0 s ⋆ ´ Alvaro Torralba PhD Defense June 2, 2015 43 / 54

Algorithm Main idea: Start symbolic bidirectional uniform-cost search 1 ⋆ If it succeeds → done! Detect when it is going to fail and activate heuristics 2 Abstraction heuristics: Bidirectional, Partial, Perimeter Decide which search advance: useful and feasible s 0 s ⋆ ´ Alvaro Torralba PhD Defense June 2, 2015 43 / 54

Empirical Results Best 873 SM&S 840 PDB ran Full SymBA ∗ 844 PDB cgl 842 BD ( ∅ ) 842 820 840 860 880 Coverage ´ Alvaro Torralba PhD Defense June 2, 2015 44 / 54

Empirical Results 870 Best 873 842 SM&S 840 Full SymBA ∗ 836 PDB ran No perimeter 844 abstraction 844 PDB cgl 842 842 BD ( ∅ ) 842 820 840 860 880 Coverage ´ Alvaro Torralba PhD Defense June 2, 2015 44 / 54

Empirical Results 868 Best 870 873 842 842 SM&S Full SymBA ∗ 840 No perimeter 836 PDB ran 836 abstraction 844 No bidir 837 abstraction PDB cgl 844 842 842 BD ( ∅ ) 842 842 820 840 860 880 Coverage ´ Alvaro Torralba PhD Defense June 2, 2015 44 / 54

Summary Contributions: ◮ SymBA ∗ : a symbolic bidirectional heuristic search algorithm ◮ Bidirectional search in abstract state spaces ◮ Synergy: Symbolic search + Bidirectional search + Perimeter abstractions Symbolic Bidirectional A ∗ is possible ◮ Future work: domain-independent abstraction strategies (better than a random selection) ´ Alvaro Torralba PhD Defense June 2, 2015 45 / 54

Final Results 800 600 Coverage 400 200 Sym-BD G AMER -BD 0 10 0 10 1 10 2 10 3 Time (seconds) ´ Alvaro Torralba PhD Defense June 2, 2015 47 / 54

Symbolic Search and Abstraction Heuristics for Cost-Optimal Planning - PowerPoint PPT Presentation

Symbolic Search and Abstraction Heuristics for Cost-Optimal Planning Alvaro Torralba Advisors: Daniel Borrajo and Carlos Linares L opez Universidad Carlos III de Madrid June 2, 2015 Alvaro Torralba PhD Defense June 2, 2015 1 /

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