Modelling and SolvingEnglish Peg Solitaire

Chris Jefferson, Angela Miguel,Ian Miguel, Armagan Tarim.

AI Group

Department of Computer Science

University of York

English Peg Solitaire

• The French variant has a slightly larger board, and is considerably more difficult.

0 1 2 3 4 5 6

0123456

0123456

0 1 2 3 4 5 6

Before After

• Horizontal or vertical moves:

Initial: Goal:

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] .• describes the state of the board at time-step t = 0, …, 31.

• M[i,j,t,d] . • denotes whether a move was made from location i, j at

time-step t. d in {N, S, E, W}.

Model A: IP

]bState[],,,[ i,j,tEtjiM ]1bState[],,,[ ,j,tiEtjiM

]2bState[1],,,[ ,j,tiEtjiM

},,,{

],,,[],,bState[]1,,[bStateWESNd

dtjiM-tjitji

],,,2[],,,1[],,,2[],,,1[ WtjiMWtjiMEtjiMEtjiM

],,2,[],,1,[],,2,[],,1,[ NtjiMNtjiMStjiMStjiM

Move Conditions:`1’ means move made.

Connecting board states. Consider all moves affecting a position

Model A: IP

One move at a time:

Bji

WtjiMEtjiMStjiMNtjiM),(

1]),,,[],,,[],,,[],,,[(

hole centre),(

),(

]31,,[

jiBji

jibState

Objective function.Minimise:

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. Trans. No. Trans. No. Trans.

0 2,,0 3 4,,0 6 2,,3

1 2,,2 4 4,2 7 3,,3

2 3,,2 5 2,,1 8… 4,,1

Model B: CSP

• Problem constraints can be stated on moves[] alone.

• Consider transition 0: 2, , 0 at time-step t. The following must hold at time-step t-1.• There must be pegs at 2, 0 and 3, 0.• There must be a hole at 4, 0.

• Ensure by imposing constraints on moves[1..t-1]:

Drawback: many such constraints needed. Some of very large size.

),conflict(),support()pre(}31,...,1{ pppTt

][moves:][moves:][moves hihghggt

01

0 1 2 3 4 5 6

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].

t

bState[t]

moves[]

bState[t+1]

constrains constrains

Model C Channelling Constraints

• These constraints closely resemble pre- and post-conditions of an AI Planning-style operator.

)}),(changes|][moves{(])1,,[bState],,[bState( jittjitji

)}),(pegIn|][moves{()1]1,,[bState0],,[bState( jittjitji

)}),(pegOut|][moves{()0]1,,[bState1],,[bState( jittjitji

Changes(i,j): set of transitions that change the state of i, j

pegIn(i,j): set of transitions that place a peg at i, j

pegOut(i,j): set of transitions that remove a peg from i, j

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:• If pagoda value for an intermediate position is less

than that of final position, backtrack.

a b c a b c

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 re-

established, 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. Trans. No. Trans. No. Trans.

0 2,,0 3 4,,0 6 2,,3

1 2,,2 4 4,2 7 3,,3

2 3,,2 5 2,,1 8… 4,,1

Solitaire Reversals via AI Planning1349--

--

148-

14622

--

25121

->1hr

471--

280.1125

-42

0.15--

16553

->1hr

18>1hr298

-

543---

483544

86---

57-

440.05313

-

171-

>1hr380.627-

3048--

490.0560-

271521

->1hr

219.8154

-

250.7-

1126862

>1hr-

>1hr14

0.0597

>1hr

190.230-

19276

--

210.632-

19-

125-

1427348

>1hr620

>1hr574

>1hr16

1564-

>1hr

BBox4.2FF2.3

HSP2.0Stan4

Bbox, FF most successful, achieve a high percentage of coverage.

Note howsymmetricpositions

differ.

- memory exhausted.

->1hr

--

----

---

>1hr----

Solitaire Reversals via Model C

173.5184.911619.110222,3

>1hr337.8>1hr349.6

72.78

2.71700197

1712199.9

>1hr1891.2>1hr1036

43961.144364.9

BasicPair Sym BreakingPagoda Functions

Pagoda+Sym

Less robust. Bad valueordering?Sym breaking, pagodahelp.

Blank: all >1hr

2903221.5273054.6

164.17.95

Model C + Corner Bias Value Ordering

173.5184.9

11619.110222,3

>1hr337.8>1hr349.6

72.782.7

1700197

1712199.9

>1hr1891.2>1hr1036

43961.144364.9

BasicPair Sym BreakingPagoda Functions

Pagoda+Sym

1.31.2

0.7

Taking symmetry backinto account, can nowcover all but one reversal

Blank: all >1hr

2903221.5273054.6

164.17.95

1.2

0.7

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 Solutions Found

SolutionsPruned

Constraints Added

Time (s)

4 328 32 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.

1],2,[bState],1,[bState],,[bState],,[ tjitjitjitjiC1],2,[bState],1,[bState],,[bState],,[ tjitjitjitjiC

1],,2[bState],,1[bState],,[bState],,[ tjitjitjitjiC1],,2[bState],,1[bState],,[bState],,[ tjitjitjitjiC

New objective function.Minimise:

Bji t

t tjiC),(

31

1

1 ],,[33

C[i,j,t]=1 iff there is a peg at position i,j with a legal move.

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:

deadEnd]1[movesdeadEnd][moves:}30,...,1{ ttt

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.