Complexity of Games & Puzzles Rengo Kriegspiel? PSPACE PSPACE EXPTIME Undecidable bridge? P NP PSPACE NEXPTIME 0 players 1 player 2 players team, (simulation) (puzzle) (game) imperfect info Constraint Logic [Hearn & Demaine 2009] PSPACE PSPACE EXPTIME Undecidable PSPACE P NP NEXPTIME 0 players 1 player 2 players team, (simulation) (puzzle) (game) imperfect info Bounded 2CL is PSPACE- complete Bounded 2CL is PSPACE-complete crossover protected OR free edge Amazons [Walter Zamkauskas 1988] Amazons [Walter Zamkauskas 1988] Amazons [Hearn 2005] wire victory shift turn & one way variable Amazons is PSPACE-complete [Hearn 2005] AND OR & SPLIT CHOICE Konane [Hawaii <1778] photo by Thane Plambeck, 2005 Konane is PSPACE-complete [Hearn 2005] OR & variable CHOICE turn AND, SPLIT, parity shift Konane is PSPACE-complete [Hearn 2005] AND, SPLIT, shift input 1 output 1 before input 2 → output 2 → Cross Purposes [Michael Albert 2004] Cross Purposes is PSPACE-complete [Hearn 2005] turn wire free edge variable Cross Purposes is PSPACE-complete [Hearn 2005] protected OR CHOICE Cross Purposes is PSPACE-complete [Hearn 2005] AND, SPLIT, shift input 1 output 1 before input 2 → output 2 → Constraint Logic [Hearn & Demaine 2009] PSPACE PSPACE EXPTIME Undecidable PSPACE P NP NEXPTIME 0 players 1 player 2 players team, (simulation) (puzzle) (game) imperfect info Peek [Stockmeyer & Chandra 1979] BLOCK is EXPTIME- complete [Stockmeyer & Chandra 1979] BLOCK is EXPTIME- complete [Stockmeyer & Chandra 1979] 3 ∧ 5 Checkers is EXPTIME-complete [Robson 1981] reduction from 3 color change Checkers is EXPTIME-complete [Robson 1981] Chess is EXPTIME-complete [Fraenkel & Lichtenstein 1981] reduction from 3 Go is EXPTIME- hard [Robson 1982] 2CL is EXPTIME-complete formula formula fast win fast win slow win slow win slower win slower win variable formula formula fast win fast win slow win slow win slower win slower win 2CL is EXPTIME-complete path equalizer Crossover Gadget No-Repeat Games Private-Information Games Private & Blind Peek [Reif 1984].