Transcript Modelling and Solving English Peg Solitaire

Modelling and Solving
English Peg Solitaire
Chris Jefferson, Angela Miguel,
Ian Miguel, Armagan Tarim.
AI Group
Department of Computer Science
University of York
English Peg Solitaire
0 1 2 3 4 5 6
0
1
2
Initial: 3
4
5
6
• Horizontal or
vertical moves:
0 1 2 3 4 5 6
0
1
2
Goal: 3
4
5
6
Before
After
• The French variant has a slightly larger
board, and is considerably more difficult.
Solitaire: Interesting Features
• A challenging search problem.
• Highly symmetric.
• Symmetries of the board, symmetries of moves.
• Planning-style problem.
• Not usually tackled directly with constraint
satisfaction/integer programming.
Model A: IP
• 31 moves required to solve a single-peg reversal.
• Exploit this in the modelling.
• bState[i,j,t]  {0, 1}.
• describes the state of the board at time-step t = 0, …, 31.
• M[i,j,t,d]  {0, 1}.
• denotes whether a move was made from location i, j at
time-step t. d in {N, S, E, W}.
Model A: IP
Move Conditions:
`1’ means move made.
M [i, j, t , E]  bState[i,j,t]
M [i, j, t , E ]  bState[i  1,j,t]
M [i, j, t , E ]  1  bState[i  2,j,t]
Connecting board states. Consider all moves affecting a position
bState[i, j, t  1]  bState[i, j, t ]-
 M [i, j, t, d ]
d{ N , S , E ,W }
 M [i  1, j, t , E]  M [i  2, j, t , E]  M [i  1, j, t ,W ]  M [i  2, j, t ,W ]
 M [i, j  1, t , S ]  M [i, j  2, t , S ]  M [i, j  1, t , N ]  M [i, j  2, t , N ]
Model A: IP
One move at a time:
 (M [i, j, t, N ]  M [i, j, t, S ]  M [i, j, t, E]  M [i, j, t,W ])  1
(i , j )B
Objective function.
Minimise:
 bState[i, j,31]
( i , j )B
( i , j )  centre hole
Model B: CSP
• Rather than record the board state, model B
records the sequence of moves required: moves[t]
• Each transition is assigned a unique number:
No.
0
1
2
Trans.
2,04,0
2,02,2
3,03,2
No.
3
4
5
Trans.
4,02,0
4,04,2
2,14,1
No.
6
7
8…
Trans.
2,12,3
3,13,3
4,12,1
Model B: CSP
• Problem constraints can be stated on moves[] alone.
• Consider transition 0: 2, 0  4, 0 at time-step t. The
following must hold at time-step t-1. 0 1 2 3 4 5 6
• There must be pegs at 2, 0 and 3, 0.
• There must be a hole at 4, 0.
0
1
• Ensure by imposing constraints on moves[1..t-1]:
t {1,...,31} Tp  pre( )   support( , p)   conflict( , p)
moves[t ]    g : moves[ g ]     h : g  h  i  moves[h]   
Drawback: many such constraints needed. Some of very large size.
Model C = A + B: CSP
• Combines models A and B to remove some of
the problems of both.
• Maintains: bState[i,j,t], moves[t].
• Discards (A): M[], board state connection constraints.
• Discards (B): Large arity constraints on moves[].
• Channelling constraints are added to maintain
consistency between the two representations.
• These connect bState[i,j,t], moves[t], bState[i,j,t+1].
bState[t]
bState[t+1]
constrains
moves[]
t
constrains
Model C Channelling Constraints
Changes(i,j): set of transitions that change the state of i, j
(bState[i, j, t ]  bState[i, j, t  1])  ({moves[t ]   |   changes(i, j )})
pegIn(i,j): set of transitions that place a peg at i, j
(bState[i, j, t ]  0  bState[i, j, t  1]  1)  ({moves[t ]   |   pegIn(i, j )})
pegOut(i,j): set of transitions that remove a peg from i, j
(bState[i, j, t ]  1  bState[i, j, t  1]  0)  ({moves[t ]   |   pegOut(i, j )}
• These constraints closely resemble pre- and postconditions of an AI Planning-style operator.
Results: Central Solitaire
• Model A (IP): No solution in 12 hours.
• Several alternative formulations also failed.
• Reason: artificial objective function, hence no
tight bounds to exploit.
• Model B (CP): Exhausts memory.
• Model C (A+B, CP solver): 16 seconds.
• So:
• Develop model C further.
• Apply to other variations of Solitaire.
Pagoda Functions
• Used to spot dead-ends early.
• Value assigned to each board position such that:
• Given positions a, b, c in a horizontal/vertical line:
a+b  c.
• Pagoda value of a board state:
• Sum of values at positions where there is a peg.
• Monotonically decreasing as moves made:
• Pagoda condition:
a
b
c
a
b
c
• If pagoda value for an intermediate position is less
than that of final position, backtrack.
Pagoda Functions: Examples
• For a single-peg Solitaire reversal at position i, j,
want pagoda functions with non-zero entries at i, j.
• Otherwise no pruning.
• A rotation of one of these three gives a useful
pagoda function for every board position:
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Board Symmetries
• Rotation.
• Reflection.
• Break rotational symmetry by selecting 1st move:
• Reflection symmetry persists. Remove 5,2  3,2:
Board Symmetries
• Further into the search are both broken and reestablished, depending on the moves made.
• Breaking this symmetry is a possible application for SBDS or SBDD.
Symmetries of Independent Moves
• Many pairs of moves can be performed in any order
without affecting the rest of the solutions.
• Two transitions are independent iff:
• The set of pegs upon which they operate do not intersect.
• Break this symmetry by ordering adjacent entries in
moves[]:
• independent(moves[i], moves[i+1]) 
moves[i]  moves[i+1]
• This problem extends to larger sets of transitions.
• If 2 is independent of {3, 1}, can have 2, 3, 1 and 3, 1, 2.
Results: Solitaire Reversals
• Compared Model C against state of the art AI
planning systems:
• Blackbox 4.2, FastForward 2.3, HSP 2.0, and Stan 4.
• Experiments on the full set of single-peg reversals.
• Although many board positions symmetrical, these
positions are distinguished by the transition ordering.
• Transitions chosen in ascending order.
No.
0
1
2
Trans.
2,04,0
2,02,2
3,03,2
No.
3
4
5
Trans.
4,02,0
4,04,2
2,14,1
No.
6
7
8…
Trans.
2,12,3
3,13,3
4,12,1
Solitaire Reversals via AI Planning
BBox4.2
FF2.3
HSP2.0
Stan4
- memory exhausted.
19
0.2
30
19
276
-
Bbox, FF most
successful, achieve a
high percentage of
coverage.
25
0.7
1126
862
>1hr
>1hr
14
0.05
97
>1hr
13
49
25
121
>1hr
47
1
28
0.1
125
42
0.15
16
553
>1hr
18
>1hr
298
-
148
543
48
3544
86
57
44
0.05
313
>1hr
-
14
622
17
1
>1hr
38
0.6
27
30
48
49
0.05
60
27
1521
>1hr
21
9.8
154
-
14
273
48
>1hr
620
>1hr
574
>1hr
16
1564
>1hr
21
0.6
32
>1hr
19
125
-
Note how
symmetric
positions
differ.
Solitaire Reversals via Model C
Basic
Pair Sym Breaking
Pagoda Functions
Pagoda+Sym
2903
221.5
2730
54.6
Blank: all >1hr
439
61.1
443
64.9
>1hr
1891.2
>1hr
1036
Less robust. Bad value
ordering?
Sym breaking, pagoda
help.
17
3.5
18
4.9
116
19.1
102
22,3
16
4.1
7.9
5
>1hr
337.8
>1hr
349.6
7
2.7
8
2.7
1700
197
1712
199.9
Model C + Corner Bias Value Ordering
Basic
Pair Sym Breaking
Pagoda Functions
Pagoda+Sym
2903
221.5
2730
54.6
Blank: all >1hr
439
61.1
443
64.9
>1hr
1891.2
>1hr
1036
Taking symmetry back
into account, can now
cover all but one reversal
17
3.5
18 1.2
4.9
116
19.1
102
22,3
16
4.1
7.9 1.3
5
>1hr
337.8
>1hr
349.6
7
2.7
0.7
8
2.7
1700
197
1712
199.9
0.7
1.2
4.6
Symmetric Paths
• There are often multiple ways of arriving at the
same board state.
• Some are due to independent moves. Others are not:
Symmetric Paths
• Find all solutions to a given depth.
• Group the transition sequences that lead to identical
positions.
• Insert constraints that allow one representative per group.
Depth
4
Solutions
Found
328
Solutions
Pruned
32
Constraints Time (s)
Added
32
<1
5
1572
234
205
1.5
6
7152
1504
1256
10
7
29953
8111
6167
116
Total
50600
28818
7660
Fool’s Solitaire
• An optimisation variant.
• Reach a position where no further moves are
possible in the shortest sequence of moves.
• Not easily stated as an AI planning problem.
• Shows the flexibility of the CP and IP approaches.
Fool’s Solitaire: IP Model
• Moves, connection of board states same as model A.
C[i,j,t]=1 iff there is a peg at position i,j with a legal move.
C[i,
C[i,
C[i,
C[i,
j, t ]  bState[i,
j, t ]  bState[i,
j, t ]  bState[i,
j, t ]  bState[i,
j, t ]  bState[i  1, j, t ]  bState[i  2, j, t ]  1
j, t ]  bState[i  1, j, t ]  bState[i  2, j, t ]  1
j, t ]  bState[i, j  1, t ]  bState[i, j  2, t ]  1
j, t ]  bState[i, j  1, t ]  bState[i, j  2, t ]  1
New objective function.
Minimise:
31
  33
( i , j )B t 1
t 1
C[i, j, t ]
Fool’s Solitaire: CP Model
• Modified version of model C.
• An extra transition, deadEnd, is added to the
domain of the moves[] variables.
• Assigned when no other move is possible.
• deadEnd transition is only allowed when no
other transitions are possible.
• Preconditions based on bState[].
• If deadEnd at moves[t], then also at all following
time-steps:
t {1,...,30} : moves[t ]  deadEnd  moves[t  1]  deadEnd
Fool’s Solitaire: Results
• CP, reverse instantiation order: 20s
• IP, iterative approach: 27s
Conclusions
• Basic, and ineffective CP and IP models
combined into a superior CP model.
• Another instance of the utility of channelling
between two complementary models.
• Each allows easy statement of different aspects of
the problem:
• Model A: preconditions on state changes without
considering entire move history.
• Model B: one move at once, combines 3 state changes
into a single token.
Conclusions
• Encouraging results versus dedicated AI
planning systems.
• Lessons learned should generalise to other
sequential planning-style problems.
• Channelling constraints specify action pre- and
post-conditions.
• Breaking symmetry of independent
actions/paths.
Future Work
• Further configurations of English Solitaire.
• Other optimisation variants:
• Minimise number of draughts-like multiple
moves using a single peg.
• Proving unsolvability.
• Large search space to explore.
• Will need improved symmetry breaking.
• French Solitaire.