Abstraction Heuristics for Rubik’s Cube

(1)

Bachelor Thesis

Abstraction Heuristics for Rubik’s Cube

Clemens B¨ uchner

Departement of Mathematics and Computer Science University of Basel

June 8, 2018

(2)

Abstraction Heuristics for Rubik’s Cube Introduction

Planning: Figure out a sequence of actions that leads to a goal.

(3)

Planning: Figure out a sequence of actions that leads to a goal.

(4)

Planning: Figure out a sequence of actions that leads to a goal.

(5)

Planning: Figure out a sequence of actions that leads to a goal.

(6)

Planning: Figure out a sequence of actions that leads to a goal.

(7)

Planning: Figure out a sequence of actions that leads to a goal.

(8)

Planning: Figure out a sequence of actions that leads to a goal.

(9)

Planning: Figure out a sequence of actions that leads to a goal.

(10)

State Space

. . .

. . . . . . . . . . . . . . . . . . . . .

. . . F

U

R

F U

R

(11)

Abstraction Heuristics for Rubik’s Cube Rubik’s Cube

Terminology

I Cubies: small pieces on Rubik’s Cube

I

corner cubies

I

edge cubies

I

center cubies

I Faces: sets of cubies

I

F, B, L, R, U, D

(12)

Terminology

I Cubies: small pieces on Rubik’s Cube

I

corner cubies

I

edge cubies

I

center cubies

I Faces: sets of cubies

I

F, B, L, R, U, D

(13)

Terminology

I Cubies: small pieces on Rubik’s Cube

I

corner cubies

I

edge cubies

I

center cubies

I Faces: sets of cubies

I

F, B, L, R, U, D

(14)

Terminology

I Cubies: small pieces on Rubik’s Cube

I

corner cubies

I

edge cubies

I

center cubies

I Faces: sets of cubies

I

F, B, L, R, U, D

(15)

Terminology

I Cubies: small pieces on Rubik’s Cube

I

corner cubies

I

edge cubies

I

center cubies

I Faces: sets of cubies

I

F, B, L, R, U, D

(16)

Model

I 20 variables, one for each relevant cubie

I

center cubies and cubie inside never change

I

value describes location and rotation

I size of domain =

#locations · #rotations

I 18 operators, three per face

I

turn 90

^◦

clockwise, turn 90

^◦

counterclockwise and turn 180

^◦

I

no preconditions

I 43, 252 · 10 ¹⁸ reachable states

(17)

Model

I 20 variables, one for each relevant cubie

I

center cubies and cubie inside never change

I

value describes location and rotation

I size of domain =

#locations · #rotations

I 18 operators, three per face

I

turn 90

^◦

clockwise, turn 90

^◦

counterclockwise and turn 180

^◦

I

no preconditions

I 43, 252 · 10 ¹⁸ reachable states

(18)

Model

I 20 variables, one for each relevant cubie

I

center cubies and cubie inside never change

I

value describes location and rotation

I size of domain =

#locations · #rotations

I 18 operators, three per face

I

turn 90

^◦

clockwise, turn 90

^◦

counterclockwise and turn 180

^◦

I

no preconditions

I 43, 252 · 10 ¹⁸ reachable states

(19)

Model

I 20 variables, one for each relevant cubie

I

center cubies and cubie inside never change

I

value describes location and rotation

I size of domain =

#locations · #rotations

I 18 operators, three per face

I

turn 90

^◦

clockwise, turn 90

^◦

counterclockwise and turn 180

^◦

I

no preconditions

I 43, 252 · 10 ¹⁸ reachable states

(20)

Model

I 20 variables, one for each relevant cubie

I

center cubies and cubie inside never change

I

value describes location and rotation

I size of domain =

#locations · #rotations

I 18 operators, three per face

I

turn 90

^◦

clockwise, turn 90

^◦

counterclockwise and turn 180

^◦

I

no preconditions

I 43, 252 · 10 ¹⁸ reachable states

(21)

Model

I 20 variables, one for each relevant cubie

I

center cubies and cubie inside never change

I

value describes location and rotation

I size of domain =

#locations · #rotations

I 18 operators, three per face

I

turn 90

^◦

clockwise, turn 90

^◦

counterclockwise and turn 180

^◦

I

no preconditions

I 43, 252 · 10 ¹⁸ reachable states

(22)

Model

I 20 variables, one for each relevant cubie

I

center cubies and cubie inside never change

I

value describes location and rotation

I size of domain =

#locations · #rotations

I 18 operators, three per face

I

turn 90

^◦

clockwise, turn 90

^◦

counterclockwise and turn 180

^◦

I

no preconditions

I 43, 252 · 10 ¹⁸ reachable states

(23)

Model

I 20 variables, one for each relevant cubie

I

center cubies and cubie inside never change

I

value describes location and rotation

I size of domain =

#locations · #rotations

I 18 operators, three per face

I

turn 90

^◦

clockwise, turn 90

^◦

counterclockwise and turn 180

^◦

I

no preconditions

I 43, 252 · 10 ¹⁸ reachable states

(24)

Abstraction Heuristics for Rubik’s Cube Abstraction Heuristics

Abstraction Heuristics

(25)

Abstractions

I disregard information on purpose

I combine states with similarities to abstract states

I always admissible → A ^∗ yields optimal plan

(26)

Example: Boat and Sluice

fill empty enter

leave

fill empty

leave enter

fill

empty

(27)

Projection

fill, empty

enter leave

fill, empty

leave enter

fill, empty

(28)

Cartesian Abstraction

fill, empty

enter, leave

(29)

Cartesian Abstraction

fill, empty enter, leave

leave enter

fill, empty

(30)

Cartesian Abstraction

fill, empty

enter leave

fill, empty

leave enter

fill, empty

(31)

Cartesian Abstraction

fill, empty

enter leave

fill empty

leave enter

fill, empty

(32)

Merge-and-Shrink Abstraction

fill, empty

enter leave

fill empty

leave enter

fill, empty

(33)

Abstraction Heuristics for Rubik’s Cube Cartesian Abstraction Refinement

Cartesian Abstraction Refinement

(34)

Regression

I regression of Cartesian Set not Cartesian

I no support for conditional effects

I Rubik’s Cube has only conditional effects

(35)

Factored Effect Task

Definition

Operator o is called a factored effect operator if eff (o ) has the following form:

X 7→ x ₁ B X 7→ x ₂ ∧ X 7→ x 3 B X 7→ x 4 ∧

. . . ∧

Z 7→ z 1 B Z 7→ z 2

I factored effect tasks only have factored effect operators

(36)

Transition Check (before)

1: function CheckTransition (a, o , b) 2: for all v ∈ V do

3: if v ∈ vars (pre (o )) and pre (o) [v ] 6∈ dom (v, a) then 4: return false

5: if v ∈ vars (post (o )) and post(o)[v ] 6∈ dom (v, b) then 6: return false

7: if v 6∈ vars (post (o )) and dom (v, a) ∩ dom (v , b) = ∅ then 8: return false

9: return true

(37)

Transition Check (before)

1: function CheckTransition (a, o , b) 2: for all v ∈ V do

3: if v ∈ vars (pre (o )) and pre (o) [v ] 6∈ dom (v, a) then 4: return false

5: if v ∈ vars (post(o) ) and post (o)[v] 6∈ dom (v , b) then 6: return false

7: if v 6∈ vars (post(o) ) and dom (v , a) ∩ dom (v, b) = ∅ then 8: return false

9: return true

(38)

Functions

possible (a, o, X ) = [

x∈dom(X ,a)

{resulting fact (X 7→ x, o )}

(39)

Transition Check (after)

1: function CheckTransition (a, o , b) 2: for all v ∈ V do

3: if v ∈ vars (pre (o )) and pre (o) [v ] 6∈ dom (v, a) then 4: return false

5: if v ∈ vars (o ) and possible (a, o , v ) ∩ dom (v, b) = ∅ then 6: return false

7: if v 6∈ vars (o ) and dom (v, a) ∩ dom (v, b) = ∅ then 8: return false

9: return true

(40)

Abstraction Heuristics for Rubik’s Cube Evaluation

Evaluation

(41)

Setup

I 200 problem instances of different difficulties

I seven heuristics

I

Blind Search Heuristic as a baseline

I

different Pattern Database heuristics

I

Cartesian Abstraction with conditional effects

I

Merge-and-Shrink

(42)

Coverage

Summary blind man syst corner edge cegar m&s

coverage 68 128 122 108 105 111 95

out-of-memory

^a

132 72 0 92 95 89 105

timeout

^b

0 0 78 0 0 0 0

total time

^c

0.32 16.98 141.36 2.64 2.21 0.61 19.94

a

memory limit: 3.5 GiB

b

time limit: 30 minutes

c

geometric mean of instances solved by all configurations

(43)

Coverage

Summary blind man syst corner edge cegar m&s

coverage 68 128 122 108 105 111 95

out-of-memory

^a

132 72 0 92 95 89 105

timeout

^b

0 0 78 0 0 0 0

total time

^c

0.32 16.98 141.36 2.64 2.21 0.61 19.94

a

memory limit: 3.5 GiB

b

time limit: 30 minutes

c

geometric mean of instances solved by all configurations

(44)

Coverage

Summary blind man syst corner edge cegar m&s

coverage 68 128 122 108 105 111 95

out-of-memory

^a

132 72 0 92 95 89 105

timeout

^b

0 0 78 0 0 0 0

total time

^c

0.32 16.98 141.36 2.64 2.21 0.61 19.94

a

memory limit: 3.5 GiB

b

time limit: 30 minutes

c

geometric mean of instances solved by all configurations

(45)

Coverage

Summary blind man syst corner edge cegar m&s

coverage 68 128 122 108 105 111 95

out-of-memory

^a

132 72 0 92 95 89 105

timeout

^b

0 0 78 0 0 0 0

total time

^c

0.32 16.98 141.36 2.64 2.21 0.61 19.94

a

memory limit: 3.5 GiB

b

time limit: 30 minutes

c

geometric mean of instances solved by all configurations

(46)

Expansions Until the Last f -layer

10

⁻¹

10

⁰

10

¹

10

²

10

³

10

⁴

10

⁵

10

⁶

10

⁻¹

10

⁰

10

¹

10

²

10

³

10

⁴

10

⁵

10

⁶

uns.

blind search

single co rner pattern

(47)

Expansions Until the Last f -layer

10

⁻¹

10

⁰

10

¹

10

²

10

³

10

⁴

10

⁵

10

⁶

10

⁻¹

10

⁰

10

¹

10

²

10

³

10

⁴

10

⁵

10

⁶

uns.

single corner pattern

max over manual PDBs

(48)

Expansions Until the Last f -layer

10

⁻¹

10

⁰

10

¹

10

²

10

³

10

⁴

10

⁵

10

⁶

10

⁻¹

10

⁰

10

¹

10

²

10

³

10

⁴

10

⁵

10

⁶

uns.

max over manual PDBs

CEGAR

(49)

Expansions Until the Last f -layer

10

⁻¹

10

⁰

10

¹

10

²

10

³

10

⁴

10

⁵

10

⁶

10

⁻¹

10

⁰

10

¹

10

²

10

³

10

⁴

10

⁵

10

⁶

uns.

CEGAR

merge-and-shrink

(50)

Initial h-values

0 2 4 6 8 10 12

problem instances

differencetoh?

blind search manual patterns CEGAR merge-and-shrink

(51)

Abstraction Heuristics for Rubik’s Cube Conclusion

Summary

I factored effect tasks can be abstracted with CEGAR

I PDBs find most solutions with least expansions

I CEGAR is fast and yields perfect heuristic values

(52)

Abstraction Heuristics for Rubik’s Cube Conclusion

Thank you for your attention.

(53)

Abstraction Heuristics for Rubik’s Cube Appendix

Planning Tasks

Definition

A planning task is a 4-tuple Π = hV , I , O, γi where

I V is a finite set of finite-domain state variables,

I I is a state over V called the initial state,

I O is a finite set of finite-domain operators over V , and

I γ is a formula over V called the goal.

I operators o ∈ O with precondition pre(o), effect eff (o), and

cost cost(o)

(54)

Locations

I coordinate system

I distinguish categories

I

corner cubies and edge cubies

0

1

2

3 4 6

0 7 1

2 3

4 6

7 9

11

(55)

Rotations

I number of facelets corresponds to possible rotations

I triple representation hx, y, z i

I

use blanks (#) where less than 3 facelets

I

rotation is first non-blank triple element

x

y

z

(56)

Rotations

I number of facelets corresponds to possible rotations

I triple representation hx, y, z i

I

use blanks (#) where less than 3 facelets

I

rotation is first non-blank triple element

x

y

z

(57)

Regression (1)

Definition

Let X 7→ x

₁

Abstraction Heuristics for Rubik’s Cube

Bachelor Thesis

Abstraction Heuristics for Rubik’s Cube

Clemens B¨ uchner

Departement of Mathematics and Computer Science University of Basel

June 8, 2018

Planning: Figure out a sequence of actions that leads to a goal.

Planning: Figure out a sequence of actions that leads to a goal.

Planning: Figure out a sequence of actions that leads to a goal.

Planning: Figure out a sequence of actions that leads to a goal.

Planning: Figure out a sequence of actions that leads to a goal.

Planning: Figure out a sequence of actions that leads to a goal.

Planning: Figure out a sequence of actions that leads to a goal.

Planning: Figure out a sequence of actions that leads to a goal.

State Space

. . .

. . . . . . . . . . . . . . . . . . . . .

. . . F

U

R

F U

R

Terminology

I Cubies: small pieces on Rubik’s Cube

corner cubies

edge cubies

center cubies

I Faces: sets of cubies

F, B, L, R, U, D

Terminology

I Cubies: small pieces on Rubik’s Cube

corner cubies

edge cubies

center cubies

I Faces: sets of cubies

F, B, L, R, U, D

Terminology

I Cubies: small pieces on Rubik’s Cube

corner cubies

edge cubies

center cubies

I Faces: sets of cubies

F, B, L, R, U, D

Terminology

I Cubies: small pieces on Rubik’s Cube

corner cubies

edge cubies

center cubies

I Faces: sets of cubies

F, B, L, R, U, D

Terminology

I Cubies: small pieces on Rubik’s Cube

corner cubies

edge cubies

center cubies

I Faces: sets of cubies

F, B, L, R, U, D

Model

I 20 variables, one for each relevant cubie

center cubies and cubie inside never change

value describes location and rotation

I size of domain =

#locations · #rotations

I 18 operators, three per face

turn 90

clockwise, turn 90

counterclockwise and turn 180

no preconditions

I 43, 252 · 10 18 reachable states

Model

I 20 variables, one for each relevant cubie

center cubies and cubie inside never change

value describes location and rotation

I size of domain =

#locations · #rotations

I 18 operators, three per face

turn 90

clockwise, turn 90

counterclockwise and turn 180

no preconditions

I 43, 252 · 10 ¹⁸ reachable states

I 43, 252 · 10 ¹⁸ reachable states

I 43, 252 · 10 ¹⁸ reachable states

I 43, 252 · 10 ¹⁸ reachable states

I 43, 252 · 10 ¹⁸ reachable states

I 43, 252 · 10 ¹⁸ reachable states

I 43, 252 · 10 ¹⁸ reachable states