Complexity of Games Puzzles Demaine Hearn many others

Complexity of Games & Puzzles [Demaine, Hearn & many others] un bo un d ed Rengo Kriegspiel? PSPACE EXPTIME Undecidable bridge? bo u nd ed PSPACE P 0 players (simulation) NP 1 player (puzzle) PSPACE 2 players (game) NEXPTIME team, imperfect info

Constraint Logic un bo un d ed [Hearn & Demaine 2009] PSPACE EXPTIME Undecidable bo u nd ed PSPACE P 0 players (simulation) PSPACE NP 1 player (puzzle) 2 players (game) NEXPTIME team, imperfect info

Constraint Graphs Machine = graph, red & blue edges

Constraint Graphs Machine state = orientation constraint graph

Constraint Logic =1 =2 1 1 2 Rule: at least 2 units incoming at a vertex Move: reverse an edge, preserving Rule

AND vertex F T T F 1 1 T T 1 1 2 inputs output F not your usual AND gate! T Rule: at least 2 units incoming at a vertex

SPLIT vertex T F T 1 F F 1 1 2 outputs input T F Rule: at least 2 units incoming at a vertex

OR vertex F T T F 2 2 T T 2 2 2 inputs output T F not your usual OR gate! T Rule: at least 2 units incoming at a vertex

Decision Problem can you reverse this edge?

Constraint Logic un bo un d ed [Hearn & Demaine 2009] PSPACE EXPTIME Undecidable bo u nd ed PSPACE P 0 players (simulation) PSPACE NP 1 player (puzzle) 2 players (game) NEXPTIME team, imperfect info

Decision Problem can you reverse this edge? Theorem: PSPACE-complete

Sliding-Block Puzzles http: //www. cs. brandeis. edu/~storer/Jim. Puzzles/ZPAGES/zzz. Dads. Puzzler. html

Sliding-Block Puzzles [Hearn & Demaine 2002] Corollary: PSPACE-complete

Sliding-Block Puzzles [Hearn & Demaine 2002] Corollary: PSPACE-complete

Wiring Vertices Together AND 1 AND wants red OR wants blue 2 OR

Red-Blue Conversion assume an even number of conversions

Red-Blue Conversion assume an even number of conversions

Boolean Formulas

Quantified Boolean Formulas (QBF) x x y y ww z z

Existential Quantifier

Universal Quantifier

Latch A unlocked B

Universal Quantifier

Crossover Gadget

OR from Protector OR

Rush Hour [Hearn & Demaine 2002] PSPACE-completeness known [Flake & Baum 2002]

Triangular Rush Hour [Hearn & Demaine 2009]

Open: 1× 1 Rush Hour [Tromp & Cilibrasi 2008] · P or PSPACE-complete or …?

Plank Puzzles [Hearn 2004]

Sokoban [Hearn & Demaine 2002] PSPACE-completeness known [Culberson 1998]

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

Constraint Logic un bo un d ed [Hearn & Demaine 2009] PSPACE EXPTIME Undecidable bo u nd ed PSPACE P 0 players (simulation) PSPACE NP 1 player (puzzle) 2 players (game) NEXPTIME team, imperfect info

Constraint Logic un bo un d ed [Hearn & Demaine 2009] PSPACE EXPTIME Undecidable bo u nd ed PSPACE P 0 players (simulation) PSPACE NP 1 player (puzzle) 2 players (game) NEXPTIME team, imperfect info

Amazons [Hearn 2005] fanout