Complexity of Games & Puzzles - courses.csail.mit.educourses.csail.mit.edu/6.890/fall14/lectures/L18_images.pdf · Figures 25 & 19 & 30 of “Winning Ways for Your Mathematical Plays”
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Complexity of Games & Puzzles 0 players (simulation) 1 player (puzzle) 2 players (game) team, imperfect info NP PSPACE EXPTIME P Undecidable NEXPTIME PSPACE PSPACE Rengo Kriegspiel? bridge?
Slide from Lecture 1. http://erikdemaine.org/papers/GPC/
Constraint Logic[Hearn & Demaine 2009]
PSPACE EXPTIME
P
Undecidable
NEXPTIME
PSPACE
NPPSPACE
0 players(simulation)
1 player(puzzle)
2 players(game)
team,imperfect info
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Presentation Notes
Slide from Lecture 1. http://erikdemaine.org/papers/GPC/
Conway’s Game of Life [1970]
Glider
Pulsar
Spaceship
Gosper’s Glider GunStill Life
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Images are in public domain: http://commons.wikimedia.org/wiki/File:Game_of_life_pulsar.gif http://commons.wikimedia.org/wiki/File:Game_of_life_block_with_border.svg http://en.wikipedia.org/wiki/File:Game_of_life_beehive.svg http://en.wikipedia.org/wiki/File:Game_of_life_loaf.svg http://commons.wikimedia.org/wiki/File:Game_of_life_animated_glider.gif http://commons.wikimedia.org/wiki/File:Game_of_life_animated_LWSS.gif Glider Gun image created by Erik Demaine using Golly software: http://golly.sourceforge.net/
Life is PSPACE-hard [Paul Rendell 2000]
Turing machine simulation
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http://rendell-attic.org/gol/tm.htm
Life is PSPACE-hard [Paul Rendell 2000]
universal Turing machine simulation
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http://rendell-attic.org/gol/utm/index.htm
Life is Undecidable [Paul Rendell 2001]
universal Turing machine simulation
149 milliongenerations
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http://rendell-attic.org/gol/fullutm/index.htm
Life is Undecidable[Berlekamp, Conway, Guy 1982]
wire
terminator
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Figures 17 & 24 of “Winning Ways for Your Mathematical Plays” by Elwyn R. Berlakmp, John H. Conway, Richard K. Guy, 2nd edition, volume 4
Life is Undecidable[Berlekamp, Conway, Guy 1982]
turn
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Presentation Notes
Figures 19 & 20 of “Winning Ways for Your Mathematical Plays” by Elwyn R. Berlakmp, John H. Conway, Richard K. Guy, 2nd edition, volume 4
Life is Undecidable[Berlekamp, Conway, Guy 1982]
multiple turns/delay
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Figure 22 of “Winning Ways for Your Mathematical Plays” by Elwyn R. Berlakmp, John H. Conway, Richard K. Guy, 2nd edition, volume 4
Life is Undecidable[Berlekamp, Conway, Guy 1982]
shift
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Figure 21 of “Winning Ways for Your Mathematical Plays” by Elwyn R. Berlakmp, John H. Conway, Richard K. Guy, 2nd edition, volume 4
OR
Life is Undecidable[Berlekamp, Conway, Guy 1982]
AND OR
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Figure 29 of “Winning Ways for Your Mathematical Plays” by Elwyn R. Berlakmp, John H. Conway, Richard K. Guy, 2nd edition, volume 4
Life is Undecidable[Berlekamp, Conway, Guy 1982]
kick back
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Presentation Notes
Figure 27 of “Winning Ways for Your Mathematical Plays” by Elwyn R. Berlakmp, John H. Conway, Richard K. Guy, 2nd edition, volume 4
Life is Undecidable[Berlekamp, Conway, Guy 1982]
thinning/crossover
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Presentation Notes
Figure 28 of “Winning Ways for Your Mathematical Plays” by Elwyn R. Berlakmp, John H. Conway, Richard K. Guy, 2nd edition, volume 4
Life is Undecidable[Berlekamp, Conway, Guy 1982]
split/NOT
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Figure 30 of “Winning Ways for Your Mathematical Plays” by Elwyn R. Berlakmp, John H. Conway, Richard K. Guy, 2nd edition, volume 4
Life is Undecidable[Berlekamp, Conway, Guy 1982]
incr
emen
t
integer registerstest
create
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Figures 25 & 19 & 30 of “Winning Ways for Your Mathematical Plays” by Elwyn R. Berlakmp, John H. Conway, Richard K. Guy, 2nd edition, volume 4
Life is Undecidable[Berlekamp, Conway, Guy 1982]
pull blockback 3
push blockforward 1
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Figure 30 of “Winning Ways for Your Mathematical Plays” by Elwyn R. Berlakmp, John H. Conway, Richard K. Guy, 2nd edition, volume 4
Life is Undecidable[Berlekamp, Conway, Guy 1982]
precise glider
positioning
boomerang
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Figures 34 & 35 of “Winning Ways for Your Mathematical Plays” by Elwyn R. Berlakmp, John H. Conway, Richard K. Guy, 2nd edition, volume 4
Life is Undecidable[Berlekamp, Conway, Guy 1982]
destroy terminator
destroy gun
self-destruction
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Figures 37 & 38 & 25 of “Winning Ways for Your Mathematical Plays” by Elwyn R. Berlakmp, John H. Conway, Richard K. Guy, 2nd edition, volume 4
DCLSwitch
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Figure C.1 of http://erikdemaine.org/papers/GPC/
Reduction from Q-CNF-SAT
𝑥𝑥 𝑥𝑥 𝑦𝑦 𝑦𝑦 𝑤𝑤 𝑤𝑤 𝑧𝑧 𝑧𝑧
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Figure 5.4 of http://erikdemaine.org/papers/GPC/
DCL is PSPACE-complete
existentialquantifier
universalquantifier
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Figure 4.3 of http://erikdemaine.org/papers/GPC/
CNFFormula
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Slide from Lecture 17
DCL is PSPACE-complete
OR′
AND′
…before…
split
connectionbetween CNF& quantifiers
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Figures 4.4 & 4.5 of http://erikdemaine.org/papers/GPC/
Removing Degree-2 Vertices
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Mostly new figures with parts from Figures 2.4 & 2.5 of http://erikdemaine.org/papers/GPC/
Planar DCL
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Figure 4.7 of http://erikdemaine.org/papers/GPC/
PlanarDCL
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Figures C.11 & C.12 of http://erikdemaine.org/papers/GPC/
Reversi/Othello[<1883]
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Images from http://gamescrafters.berkeley.edu/games.php?game=Othello & http://www.iggamecenter.com/info/en/reversi.html
Reversi/Othello is PSPACE-complete[Iwata & Kasai 1994]
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http://dx.doi.org/10.1016/0304-3975(94)90131-7
Reversi/Othello is PSPACE-complete[Iwata & Kasai 1994]
shift
turn/degree 2
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Presentation Notes
http://dx.doi.org/10.1016/0304-3975(94)90131-7
Reversi/Othello is PSPACE-complete[Iwata & Kasai 1994]
double visit⇒ black wins
double visit⇒ white wins
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Presentation Notes
http://dx.doi.org/10.1016/0304-3975(94)90131-7
Reversi/Othello is PSPACE-complete[Iwata & Kasai 1994]
white chooses black chooses
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Presentation Notes
http://dx.doi.org/10.1016/0304-3975(94)90131-7
Constraint Logic[Hearn & Demaine 2009]
PSPACE EXPTIME
P
Undecidable
NEXPTIME
PSPACE
NPPSPACE
0 players(simulation)
1 player(puzzle)
2 players(game)
team,imperfect info
Presenter
Presentation Notes
Slide from Lecture 1. http://erikdemaine.org/papers/GPC/
Constraint GraphSatisfaction isNP-complete
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Slide from Lecture 17. Based on Theorem 5.4. Figures 5.1 & 2.5 of http://erikdemaine.org/papers/GPC/
Bounded NCL isNP-complete
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Figures 5.1 & 2.5 of http://erikdemaine.org/papers/GPC/
Bounded Crossover Gadget
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Figures 5.2 & 5.3 of http://erikdemaine.org/papers/GPC/
Bounded 2CLis PSPACE-complete
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Figures 6.1 & 6.2 of http://erikdemaine.org/papers/GPC/
Bounded Crossover Gadget
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Figures 6.2 & 5.3 of http://erikdemaine.org/papers/GPC/
Protected OR
free edge
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Figures 6.2 & 10.12 of http://erikdemaine.org/papers/GPC/
Amazons[Walter Zamkauskas 1988]
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Presentation Notes
Images from http://en.wikipedia.org/wiki/Game_of_the_Amazons licensed under BSD license by en:User:Cburnett
