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Complexity of Games & Puzzles [Demaine, Hearn & many others] Rengo Kriegspiel? unbounded EXPTIME Undecidable PSPACE PSPACE bridge? bounded NP PSPACE P NEXPTIME team, imperfect info 0 players (simulation) 1 player (puzzle) 2 players (game) Constraint Logic [Hearn & Demaine 2009]

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complexity of games puzzles demaine hearn many others
Complexity of Games & Puzzles[Demaine, Hearn & many others]

Rengo Kriegspiel?

unbounded

EXPTIME

Undecidable

PSPACE

PSPACE

bridge?

bounded

NP

PSPACE

P

NEXPTIME

team,imperfect info

0 players

(simulation)

1 player

(puzzle)

2 players

(game)

constraint logic hearn demaine 2009
Constraint Logic[Hearn & Demaine 2009]

unbounded

EXPTIME

Undecidable

PSPACE

PSPACE

bounded

PSPACE

NP

P

NEXPTIME

team,imperfect info

0 players

(simulation)

1 player

(puzzle)

2 players

(game)

constraint graphs
Constraint Graphs

Machine = graph,red & blue edges

constraint graphs4
Constraint Graphs

Machine state

= orientation

constraint graph

constraint logic
Constraint Logic

= 1

1

1

= 2

2

Rule: at least 2 unitsincoming at a vertex

Move: reverse an edge, preserving Rule

and vertex
AND vertex

T

F

T

F

T

T

inputs

1

1

1

1

2

output

F

T

Rule: at least 2 unitsincoming at a vertex

not your usualAND gate!

split vertex
SPLIT vertex

T

T

F

F

F

outputs

1

1

1

2

input

T

F

Rule: at least 2 unitsincoming at a vertex

or vertex
OR vertex

T

F

F

T

T

T

inputs

2

2

2

2

2

output

T

F

T

Rule: at least 2 unitsincoming at a vertex

not your usualOR gate!

decision problem
Decision Problem

can you reverse this edge?

constraint logic hearn demaine 200910
Constraint Logic[Hearn & Demaine 2009]

unbounded

EXPTIME

Undecidable

PSPACE

PSPACE

bounded

PSPACE

NP

P

NEXPTIME

team,imperfect info

0 players

(simulation)

1 player

(puzzle)

2 players

(game)

decision problem11
Decision Problem

can you reverse this edge?

Theorem:

PSPACE-complete

sliding block puzzles
Sliding-BlockPuzzles

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

wiring vertices together
Wiring Vertices Together

AND

1

AND wants red

OR wants blue

2

OR

red blue conversion
Red-Blue Conversion

assume an even number of conversions

red blue conversion17
Red-Blue Conversion

assume an even number of conversions

latch
Latch

A

un

locked

B

rush hour hearn demaine 2002
Rush Hour[Hearn & Demaine 2002]

PSPACE-completeness known [Flake & Baum 2002]

triangular rush hour
TriangularRush Hour

[Hearn & Demaine 2009]

sokoban hearn demaine 2002
Sokoban[Hearn & Demaine 2002]

PSPACE-completeness known [Culberson 1998]

constraint logic hearn demaine 200932
Constraint Logic[Hearn & Demaine 2009]

unbounded

EXPTIME

Undecidable

PSPACE

PSPACE

bounded

PSPACE

NP

P

NEXPTIME

team,imperfect info

0 players

(simulation)

1 player

(puzzle)

2 players

(game)

constraint logic hearn demaine 200933
Constraint Logic[Hearn & Demaine 2009]

unbounded

EXPTIME

Undecidable

PSPACE

PSPACE

bounded

PSPACE

NP

P

NEXPTIME

team,imperfect info

0 players

(simulation)

1 player

(puzzle)

2 players

(game)