Searching for Macro-operators with Automatically Generated Heuristics

1
Searching for Macro-operators with Automatically
Generated Heuristics

• István T. Hernádvölgyi
• University of Ottawa
• istvan_at_site.uottawa.ca

2
Motivation
Time
1 week
Korf 97
us
Korf 85
Sims 70, Osterlund 95
1 sec
85
50
455
18
Quality
3
Subgoals and Macros
1
2
3
4
5
4
1
2
3
4
5
5
1
2
3
4
5
1
2
1
3
1
2
4
1
2
3
6
1
2
3
4
5
1
1
1
1
1
2
1
2
1
2
1
2
1
3
1
2
3
1
2
3
1
2
4
1
2
3
4
1
2
3
7
1
2
3
4
5

1
o1
1
o2
1
o3
1
o4
1

2
1
o2,o1,o2
2
1
o3,o2,o3
2
1
o4,o3,o4
2
1

3
1
2
o3,o1,o3
3
1
2
o4,o2,o4
3
1
2

4
1
2
3
o4,o1,o4
4
1
2
3
8
4
1
2
5
3

1
o1
1
o2
1
o3
1
o4
1

2
1
o2,o1,o2
2
1
o3,o2,o3
2
1
o4,o3,o4
2
1

3
1
2
o3,o1,o3
3
1
2
o4,o2,o4
3
1
2

4
1
2
3
o4,o1,o4
4
1
2
3
9
4
1
2
5
3

1
o1
1
o2
1
o3
1
o4
1

2
1
o2,o1,o2
2
1
o3,o2,o3
2
1
o4,o3,o4
2
1

3
1
2
o3,o1,o3
3
1
2
o4,o2,o4
3
1
2

4
1
2
3
o4,o1,o4
4
1
2
3
o1
10
1
4
2
5
3

1
o1
1
o2
1
o3
1
o4
1

2
1
o2,o1,o2
2
1
o3,o2,o3
2
1
o4,o3,o4
2
1

3
1
2
o3,o1,o3
3
1
2
o4,o2,o4
3
1
2

4
1
2
3
o4,o1,o4
4
1
2
3
o1,o2,o1,o2
11
1
2
4
5
3

1
o1
1
o2
1
o3
1
o4
1

2
1
o2,o1,o2
2
1
o3,o2,o3
2
1
o4,o3,o4
2
1

3
1
2
o3,o1,o3
3
1
2
o4,o2,o4
3
1
2

4
1
2
3
o4,o1,o4
4
1
2
3
o1,o2,o1,o2,o4,o2,o4
12
1
2
3
5
4

1
o1
1
o2
1
o3
1
o4
1

2
1
o2,o1,o2
2
1
o3,o2,o3
2
1
o4,o3,o4
2
1

3
1
2
o3,o1,o3
3
1
2
o4,o2,o4
3
1
2

4
1
2
3
o4,o1,o4
4
1
2
3
o1,o2,o1,o2,o4,o2,o1,o4
o1,o2,o1,o2,o4,o2,o4,o4,o1,o4
o2,o4,o3,o2
13

2
1
o2,o1,o2
2
1
o3,o2,o3
2
1
o4,o3,o4
2
1

3
1
2
o3,o1,o3
3
1
2
o4,o2,o4
3
1
2
14

2
3
1
o3,o1,o3
2
3
1
o4,o2,o4
2
3
1
o1,o2,o1
2
3
1
o1,o3,o2
2
3
1
o1,o4,o1,o3
2
3
1
o2,o3,o1
2
3
1
o3,o2,o3
2
3
1
o2,o4,o2
2
3
1
o3,o4,o1
2
3
1
o2,o3,o4,o2,o3
2
3
1
o4,o3,o4
2
3
1
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Results
17
The cost of merging subgoals
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Solution use heuristics to speed up search for
each macro

• For each subgoal, create a heuristic specifically
  for that subgoal
• Can combine this with the heuristic for the
  previous subgoal (because that subgoal is a
  subset of the current subgoal)
• Need to create these automatically, they are
  unintuitive spaces and the heuristics are throw
  away

19
Previous WorkAutomatically Generated Heuristics

• Absolver Prieditis90
• Pattern Database CulbersonSchaeffer96
• 15 Puzzle, dont care tiles, Rubiks Cube
  Korf97
• Domain Abstraction HolteHernadvolgyi99
• Planning (Strips) Edelkamp2000

20
Pattern Database
181,440
15,120
21
Pattern Database
6
3
4
6
3
8
7
5
1
2
1
2
22
Domain Abstraction
181,440
5,040
23
Rubiks Cube
ULB
UB
URB
UL
UR
U
ULF
UF
URF
ULB
UL
UR
URB
UB
ULB
LF
RF
F
LB
L
RB
R
LB
B
DLF
DF
DRF
DLB
DL
DR
DRB
DB
DLB
DL
DR
D
DLB
DB
DRB
24
State Representation
Goal state (subscripts indicate orientation)
Typical start state
25
Macro Search Space
Goal state for the subgoal fix RF and RB
Typical start state for this subgoal
26
Domain Abstraction

• dont care about a cubies orientation
• Make 2 (or more) cubies indistinguishable from
  each other
• Continue to add abstractions until the abstract
  space is sufficiently small (2 million entries)

27
Abstraction Example
Abstracted goal state (x orientation doesnt
matter)
Abstracted start state (typical)
28
The cost of merging subgoals
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Contributions

• Automatically Generated Heuristics for Finding
  Macro-operators
• Merging Subgoals to Obtain Shorter Solutions
• 44 Improvement for Rubiks Cube
• Method is General and Mostly Automatic

30
Using Symmetries
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Using Symmetries
URF, URB,ULB,ULF,UF,UL,UR,UB
LF,LB,DL
LF,RF,DR
RF,RB,DR
LB,RB,DL
LF, RF
RF,RB
LF,RB
LF,LB
DF,DB (gt DR)
DRF,DRB
DLF, DLB
DLF (gtDLB)
DRF (gt DRB)
32
Kociembas Two-phase Algorithm
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Open Questions

• Do the improvements carry over to other spaces?
• Does the order of the subgoals matter?

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Motivation
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Motivation
Time
1 week
Korf 97
human
Kloosterman 90
Thistlethwaite 80
Korf 85
Sims 70, Osterlund 95
1 sec
Kociemba 92
85
50
455
18
Quality
46
Merging Consecutive Subgoals
Serial Decomposability Korf83
Idea
47
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Previous WorkMacro-Operators

• Stabilizer Chain Sims70, Osterlund95
• Bi-directional Partial-Match Korf85

51
Domain Abstraction

• select two invariant cubies c1 and c2
• toss coin
• if (heads) // mask orientations
• if (both orientations masked)
• goto 1
• mask orientations of c1 and c2
• else // mask cubie identity
• if (mask(c1) mask(c2) goto 1
• select new mask A
• if (!masked(c1) !masked(c2))
• c1 A, c2 A
• else if (masked(c1) !masked(c2))
• all cubies with mask(c1) and c2 A
• else if (!masked(c1) masked(c2))
• all cubies with mask(c2) and c1 A
• else // both has a mask
• cubies with mask(c1) and mask(c2) A
• H expand(N)
• if (H 0) goto 1

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Search

• IDA Korf85
• Maximum of 3 pattern databases (5M entries)
• 18 subgoals
• 6 subgoals
• 10,000 random instances

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Statistics