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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]

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

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  1. 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)

  2. 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)

  3. Constraint Graphs Machine = graph,red & blue edges

  4. Constraint Graphs Machine state = orientation constraint graph

  5. Constraint Logic = 1 1 1 = 2 2 Rule: at least 2 unitsincoming at a vertex Move: reverse an edge, preserving Rule

  6. 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!

  7. SPLIT vertex T T F F F outputs 1 1 1 2 input T F Rule: at least 2 unitsincoming at a vertex

  8. 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!

  9. Decision Problem can you reverse this edge?

  10. 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)

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

  12. Sliding-BlockPuzzles http://www.cs.brandeis.edu/~storer/JimPuzzles/ZPAGES/zzzDadsPuzzler.html

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

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

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

  16. Red-Blue Conversion assume an even number of conversions

  17. Red-Blue Conversion assume an even number of conversions

  18. BooleanFormulas

  19. Quantified Boolean Formulas (QBF) x x y y w w z z

  20. Existential Quantifier

  21. Universal Quantifier

  22. Latch A un locked B

  23. Universal Quantifier

  24. Crossover Gadget

  25. OR fromProtector OR

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

  27. TriangularRush Hour [Hearn & Demaine 2009]

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

  29. Plank Puzzles [Hearn 2004]

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

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

  32. 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)

  33. 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)

  34. Amazons[Hearn 2005] fanout

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