bun

Complexity of Games & PuzzlesComplexity of Games & Puzzles[Demaine, Hearn & many others][Demaine, Hearn & many others]

ound

ed

boun

ded

NP

PSPACE EXPTIME

P

Undecidable

NEXPTIMEPSPACE

PSPACE

Rengo Kriegspiel?

bridge?

Image courtesy of Jason Whittaker.

Courtesy of BigJ_Smack

Image courtesy of Nguyen Dai.

Image courtesy of Herman Hiddema

Image courtesy of Marie-Lan Nguyen.

Image courtesy of PartsnPieces.

Image is in the public domain.

Courtesy of Sam Cancilla. Used with permission.

Courtesy of Glenn Peters. Used with permission.

0 players 1 player 2 players team, (simulation) (puzzle) (game) imperfect info

Images by MIT OpenCourseWare.

1

Constraint LogicConstraint Logic[Hearn & Demaine 2009][Hearn & Demaine 2009]

boun

ded

unbo

unde

d

PSPACE EXPTIME

P

Undecidable

NEXPTIME

PSPACE

NP PSPACE

0 players 1 player 2 players team, (simulation) (puzzle) (game) imperfect info

Copyright (2009) From Games, Puzzles, and Computation by Robert A. Hearn and Erik D. Demaine.Reproduced by permission of Taylor and Francis Group, LLC, a division of Informa plc.

2

Constraint GraphsConstraint Graphs

Machine = graph,red & blue edges

Copyright (2009) From Games, Puzzles, and Computation by Robert A. Hearn and Erik D. Demaine.Reproduced by permission of Taylor and Francis Group, LLC, a division of Informa plc. 3

Constraint GraphsConstraint Graphs

Machine state= orientation

Copyright (2009) From Games, Puzzles, and Computation by Robert A. Hearn and Erik D. Demaine.Reproduced by permission of Taylor and Francis Group, LLC, a division of Informa plc.

constraint graph

4

Constraint LogicConstraint Logic

= 11 1

= 2 2

Rule: at least 2 unitsincoming at a vertex

Move: reverse an edge, preserving RuleCopyright (2009) From Games, Puzzles, and Computation by Robert A. Hearn and Erik D. Demaine.Reproduced by permission of Taylor and Francis Group, LLC, a division of Informa plc.

5

AND vertexAND vertex

FT F TT T inputs1 1

1 1

2 output

F T

not your usual Rule: at least 2 units AND gate! incoming at a vertex

Copyright (2009) From Games, Puzzles, and Computation by Robert A. Hearn and Erik D. Demaine.Reproduced by permission of Taylor and Francis Group, LLC, a division of Informa plc.

6

SPLIT vertexSPLIT vertex

T T FF F outputs1

1 1

2 input

T F

Rule: at least 2 units incoming at a vertex

Copyright (2009) From Games, Puzzles, and Computation by Robert A. Hearn and Erik D. Demaine.Reproduced by permission of Taylor and Francis Group, LLC, a division of Informa plc.

7

OR vertexOR vertex

FT F TT T inputs2 2

2 2

2 output

TF T

not your usual Rule: at least 2 unitsOR gate! incoming at a vertex

Copyright (2009) From Games, Puzzles, and Computation by Robert A. Hearn and Erik D. Demaine.Reproduced by permission of Taylor and Francis Group, LLC, a division of Informa plc.

8

Decision ProblemDecision Problem

Copyright (2009) From Games, Puzzles, and Computation by Robert A. Hearn and Erik D. Demaine.Reproduced by permission of Taylor and Francis Group, LLC, a division of Informa plc.

can you reverse this edge?

9

Constraint LogicConstraint Logic[Hearn & Demaine 2009][Hearn & Demaine 2009]

boun

ded

unbo

unde

d

PSPACE EXPTIME

P

Undecidable

NEXPTIME

PSPACE

NP PSPACE

0 players 1 player 2 players team, (simulation) (puzzle) (game) imperfect info

Copyright (2009) From Games, Puzzles, and Computation by Robert A. Hearn and Erik D. Demaine.Reproduced by permission of Taylor and Francis Group, LLC, a division of Informa plc. 10

Decision ProblemDecision Problem

Copyright (2009) From Games, Puzzles, and Computation by Robert A. Hearn and Erik D. Demaine.Reproduced by permission of Taylor and Francis Group, LLC, a division of Informa plc.

can you reverse this edge?

Theorem: PSPACE-complete

11

SlidingSliding--BlockBlockPuzzlesPuzzles

Courtesy of Dr. Jim Storer. Used with permission.

12

SlidingSliding--Block PuzzlesBlock PuzzlesCorollary:[Hearn & Demaine 2002][Hearn & Demaine 2002] PSPACE-complete

Copyright (2009) From Games, Puzzles, and Computation by Robert A. Hearn and Erik D. Demaine.Reproduced by permission of Taylor and Francis Group, LLC, a division of Informa plc. 13

SlidingSliding--Block PuzzlesBlock PuzzlesCorollary:[Hearn & Demaine 2002][Hearn & Demaine 2002] PSPACE-complete

Copyright (2009) From Games, Puzzles, and Computation by Robert A. Hearn and Erik D. Demaine.Reproduced by permission of Taylor and Francis Group, LLC, a division of Informa plc. 14

Wiring Vertices TogetherWiring Vertices Together

AND wants red OR wants blue

2 OR

AND 1

Copyright (2009) From Games, Puzzles, and Computation by Robert A. Hearn and Erik D. Demaine.Reproduced by permission of Taylor and Francis Group, LLC, a division of Informa plc. 15

RedRed--Blue ConversionBlue Conversion

assume an even number of conversionsCopyright (2009) From Games, Puzzles, and Computation by Robert A. Hearn and Erik D. Demaine.Reproduced by permission of Taylor and Francis Group, LLC, a division of Informa plc.

16

RedRed--Blue ConversionBlue Conversion

assume an even number of conversionsCopyright (2009) From Games, Puzzles, and Computation by Robert A. Hearn and Erik D. Demaine.Reproduced by permission of Taylor and Francis Group, LLC, a division of Informa plc. 17

BooleanBooleanFormulasFormulas

Copyright (2009) From Games, Puzzles, and Computation by Robert A. Hearn and Erik D. DeReproduced by permission of Taylor and Francis Group, LLC, a division of Informa plc.

maine.18

Quantified Boolean FormulasQuantified Boolean Formulas (QBF)(QBF)

x x y y w w z z

Copyright (2009) From Games, Puzzles, and Computation by Robert A. Hearn and Erik D. Demaine.Reproduced by permission of Taylor and Francis Group, LLC, a division of Informa plc.

19

Existential QuantifierExistential Quantifier

Copyright (2009) From Games, Puzzles, and Computation by Robert A. Hearn and Erik D. Demaine.Reproduced by permission of Taylor and Francis Group, LLC, a division of Informa plc. 20

Copyright (2009) From Games, Puzzles, and Computation by Robert A. Hearn and Erik D. Demaine.Reproduced by permission of Taylor and Francis Group, LLC, a division of Informa plc.

Universal QuantifierUniversal Quantifier

21

LatchLatch

A

Blockedun

Copyright (2009) From Games, Puzzles, and Computation by Robert A. Hearn and Erik D. Demaine.Reproduced by permission of Taylor and Francis Group, LLC, a division of Informa plc.

22

Universal QuantifierUniversal Quantifier

Copyright (2009) From Games, Puzzles, and Computation by Robert A. Hearn and Erik D. Demaine.Reproduced by permission of Taylor and Francis Group, LLC, a division of Informa plc. 23

Crossover GadgetCrossover Gadget

Copyright (2009) From Games, Puzzles, and Computation by Robert A. Hearn and Erik D. Demaine.Reproduced by permission of Taylor and Francis Group, LLC, a division of Informa plc. 24

OR fromOR fromProtector ORProtector OR

Copyright (2009) From Games, Puzzles, and Computation by Robert A. Hearn and Erik D. Demaine.Reproduced by permission of Taylor and Francis Group, LLC, a division of Informa plc. 25

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

PSPACE-completeness known [Flake & Baum 2002]Copyright (2009) From Games, Puzzles, and Computation by Robert A. Hearn and Erik D. Demaine.Reproduced by permission of Taylor and Francis Group, LLC, a division of Informa plc.

26

TriangularTriangularRush HourRush Hour[Hearn & Demaine 2009]

Copyright (2009) From Games, Puzzles, and Computation by Robert A. Hearn and Erik D. Demaine.Reproduced by permission of Taylor and Francis Group, LLC, a division of Informa plc. 27

Open: 1Open: 1××1 Rush Hour1 Rush Hour[Tromp & Cilibrasi 2008][Tromp & Cilibrasi 2008]

• P or PSPACE-complete or …?

Image courtesy of John Tromp. Used with permission.

28

Plank PuzzlesPlank Puzzles [Hearn 2004][Hearn 2004]

Copyright (2009) From Games, Puzzles, and Computation by Robert A. Hearn and Erik D. Demaine.Reproduced by permission of Taylor and Francis Group, LLC, a division of Informa plc.

29

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

PSPACE-completeness known [Culberson 1998]Copyright (2009) From Games, Puzzles, and Computation by Robert A. Hearn and Erik D. Demaine.Reproduced by permission of Taylor and Francis Group, LLC, a division of Informa plc.

30

PushPush--2F2F[Demaine, Hearn, Hoffmann 2002][Demaine, Hearn, Hoffmann 2002]

Copyright (2009) From Games, Puzzles, and Computation by Robert A. Hearn and Erik D. Demaine.Reproduced by permission of Taylor and Francis Group, LLC, a division of Informa plc. 31

Constraint LogicConstraint Logic[Hearn & Demaine 2009][Hearn & Demaine 2009]

boun

ded

unbo

unde

d

PSPACE EXPTIME

P

Undecidable

NEXPTIME

PSPACE

NP PSPACE

0 players 1 player 2 players team, (simulation) (puzzle) (game) imperfect info

Copyright (2009) From Games, Puzzles, and Computation by Robert A. Hearn and Erik D. Demaine.Reproduced by permission of Taylor and Francis Group, LLC, a division of Informa plc. 32

Constraint LogicConstraint Logic[Hearn & Demaine 2009][Hearn & Demaine 2009]

boun

ded

unbo

unde

d

PSPACE EXPTIME

P

Undecidable

NEXPTIME

PSPACE

NP PSPACE

0 players 1 player 2 players team, (simulation) (puzzle) (game) imperfect info

Copyright (2009) From Games, Puzzles, and Computation by Robert A. Hearn and Erik D. Demaine.Reproduced by permission of Taylor and Francis Group, LLC, a division of Informa plc. 33

AmazonsAmazons[Hearn 2005][Hearn 2005]

fanout

Copyright (2009) From Games, Puzzles, and Computation by Robert A. Hearn and Erik D. Demaine.Reproduced by permission of Taylor and Francis Group, LLC, a division of Informa plc. 34

MIT OpenCourseWarehttp://ocw.mit.edu

ES.268 The Mathematics in Toys and GamesSpring 2010

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