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?

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

Constraint GraphsConstraint Graphs

Machine = graph,red & blue edges

Constraint GraphsConstraint Graphs

Machine state= orientation

constraint graph

Constraint LogicConstraint Logic

= 1

= 2

Rule: at least 2 unitsincoming at a vertex

1 1

2

Move: reverse an edge, preserving Rule

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

SPLIT vertexSPLIT vertex

Rule: at least 2 unitsincoming at a vertex

1 11

2

F F

FT

T TF

outputs

input

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

Decision ProblemDecision Problemcan you reverse this edge?

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

Decision ProblemDecision Problemcan you reverse this edge?

Theorem:PSPACE-complete

Sliding-BlockSliding-BlockPuzzlesPuzzles

http://www.cs.brandeis.edu/~storer/JimPuzzles/ZPAGES/zzzDadsPuzzler.html

Sliding-Block PuzzlesSliding-Block Puzzles[Hearn & Demaine 2002][Hearn & Demaine 2002]

Corollary:PSPACE-complete

Sliding-Block PuzzlesSliding-Block Puzzles[Hearn & Demaine 2002][Hearn & Demaine 2002]

Corollary:PSPACE-complete

Wiring Vertices TogetherWiring Vertices Together

AND

OR

1

2

AND wants redOR wants blue

Red-Blue ConversionRed-Blue Conversion

assume an even number of conversions

Red-Blue ConversionRed-Blue Conversion

assume an even number of conversions

BooleanBooleanFormulasFormulas

Quantified Boolean Formulas Quantified Boolean Formulas (QBF)(QBF)

x x y y w w z z

Existential QuantifierExistential Quantifier

Universal QuantifierUniversal Quantifier

LatchLatch

A

Blockedun

Universal QuantifierUniversal Quantifier

Crossover GadgetCrossover Gadget

OR fromOR fromProtector ORProtector OR

Rush HourRush Hour[Hearn & Demaine 2002][Hearn & Demaine 2002]

PSPACE-completeness known [Flake & Baum 2002]

TriangularTriangularRush HourRush Hour

[Hearn & Demaine 2009]

Open: 1×1 Rush HourOpen: 1×1 Rush Hour[Tromp & Cilibrasi 2008][Tromp & Cilibrasi 2008]

P or PSPACE-complete or …?

Plank Puzzles Plank Puzzles [Hearn 2004][Hearn 2004]

SokobanSokoban[Hearn & Demaine 2002][Hearn & Demaine 2002]

PSPACE-completeness known [Culberson 1998]

Push-2FPush-2F[Demaine, Hearn, Hoffmann 2002][Demaine, Hearn, Hoffmann 2002]

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

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

AmazonsAmazons[Hearn 2005][Hearn 2005]

fanout