Page 1
Complexity of Games & Complexity of Games & PuzzlesPuzzles[Demaine, Hearn & many others][Demaine, Hearn & many others]
boun
ded
unbo
unde
d
0 players(simulation)
1 player(puzzle)
2 players(game)
team,imperfect info
NP
PSPACE EXPTIME
P
Undecidable
NEXPTIMEPSPACE
PSPACE
Rengo Kriegspiel?
bridge?
Page 2
Constraint LogicConstraint Logic[Hearn & Demaine 2009][Hearn & Demaine 2009]
boun
ded
unbo
unde
d
0 players(simulation)
1 player(puzzle)
2 players(game)
team,imperfect info
PSPACE EXPTIME
P
Undecidable
NEXPTIME
PSPACE
NP
PSPACE
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Constraint GraphsConstraint Graphs
Machine = graph,red & blue edges
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Constraint GraphsConstraint Graphs
Machine state= orientation
constraint graph
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Constraint LogicConstraint Logic
= 1
= 2
Rule: at least 2 unitsincoming at a vertex
1 1
2
Move: reverse an edge, preserving Rule
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AND vertexAND vertex
Rule: at least 2 unitsincoming at a vertex
1 11 1
2
T T
TF
F FTT
not your usualAND gate!
inputs
output
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SPLIT vertexSPLIT vertex
Rule: at least 2 unitsincoming at a vertex
1 11
2
F F
FT
T TF
outputs
input
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OR vertexOR vertex
Rule: at least 2 unitsincoming at a vertex
2 22 2
2
T T
TF
F FTT
not your usualOR gate!
inputs
output
T
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Decision ProblemDecision Problemcan you reverse this edge?
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Constraint LogicConstraint Logic[Hearn & Demaine 2009][Hearn & Demaine 2009]
boun
ded
unbo
unde
d
0 players(simulation)
1 player(puzzle)
2 players(game)
team,imperfect info
PSPACE EXPTIME
P
Undecidable
NEXPTIME
PSPACE
NP
PSPACE
Page 11
Decision ProblemDecision Problemcan you reverse this edge?
Theorem:PSPACE-complete
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Sliding-BlockSliding-BlockPuzzlesPuzzles
http://www.cs.brandeis.edu/~storer/JimPuzzles/ZPAGES/zzzDadsPuzzler.html
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Sliding-Block PuzzlesSliding-Block Puzzles[Hearn & Demaine 2002][Hearn & Demaine 2002]
Corollary:PSPACE-complete
Page 14
Sliding-Block PuzzlesSliding-Block Puzzles[Hearn & Demaine 2002][Hearn & Demaine 2002]
Corollary:PSPACE-complete
Page 15
Wiring Vertices TogetherWiring Vertices Together
AND
OR
1
2
AND wants redOR wants blue
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Red-Blue ConversionRed-Blue Conversion
assume an even number of conversions
Page 17
Red-Blue ConversionRed-Blue Conversion
assume an even number of conversions
Page 18
BooleanBooleanFormulasFormulas
Page 19
Quantified Boolean Formulas Quantified Boolean Formulas (QBF)(QBF)
x x y y w w z z
Page 20
Existential QuantifierExistential Quantifier
Page 21
Universal QuantifierUniversal Quantifier
Page 22
LatchLatch
A
Blockedun
Page 23
Universal QuantifierUniversal Quantifier
Page 24
Crossover GadgetCrossover Gadget
Page 25
OR fromOR fromProtector ORProtector OR
Page 26
Rush HourRush Hour[Hearn & Demaine 2002][Hearn & Demaine 2002]
PSPACE-completeness known [Flake & Baum 2002]
Page 27
TriangularTriangularRush HourRush Hour
[Hearn & Demaine 2009]
Page 28
Open: 1×1 Rush HourOpen: 1×1 Rush Hour[Tromp & Cilibrasi 2008][Tromp & Cilibrasi 2008]
P or PSPACE-complete or …?
Page 29
Plank Puzzles Plank Puzzles [Hearn 2004][Hearn 2004]
Page 30
SokobanSokoban[Hearn & Demaine 2002][Hearn & Demaine 2002]
PSPACE-completeness known [Culberson 1998]
Page 31
Push-2FPush-2F[Demaine, Hearn, Hoffmann 2002][Demaine, Hearn, Hoffmann 2002]
Page 32
Constraint LogicConstraint Logic[Hearn & Demaine 2009][Hearn & Demaine 2009]
boun
ded
unbo
unde
d
0 players(simulation)
1 player(puzzle)
2 players(game)
team,imperfect info
PSPACE EXPTIME
P
Undecidable
NEXPTIME
PSPACE
NP
PSPACE
Page 33
Constraint LogicConstraint Logic[Hearn & Demaine 2009][Hearn & Demaine 2009]
boun
ded
unbo
unde
d
0 players(simulation)
1 player(puzzle)
2 players(game)
team,imperfect info
PSPACE EXPTIME
P
Undecidable
NEXPTIME
PSPACE
NP
PSPACE
Page 34
AmazonsAmazons[Hearn 2005][Hearn 2005]
fanout