infinite games
play

Infinite Games Martin Zimmermann Saarland University October 21st, - PowerPoint PPT Presentation

Jul 10, 2023 •573 likes •1.78k views

Infinite Games Martin Zimmermann Saarland University October 21st, 2014 Research Training Group SCARE, Oldenburg, Germany Martin Zimmermann Saarland University Infinite Games 1/30 Lets Play S T You move at circles and want to reach T

  1. Winning A game G = ( A , Win ) consists of an arena A and a set Win ⊆ V ω of winning plays for Player 0. Set of winning plays for Player 1: V ω \ Win . Strategy σ for Player i is winning strategy from v , if every play that starts in v and is consistent with σ is winning for him. Winning region W i ( G ) : set of vertices from which Player i has a winning strategy. Always: W 0 ( G ) ∩ W 1 ( G ) = ∅ . G determined, if W 0 ( G ) ∪ W 1 ( G ) = V . Solving a game: determine the winning regions and winning strategies. Martin Zimmermann Saarland University Infinite Games 8/30

  2. Three Types of Winning Conditions Win is an (possibly) infinite set of infinite words, and therefore unsuitable as input to an algorithm ⇒ need finite representation. Martin Zimmermann Saarland University Infinite Games 9/30

  3. Three Types of Winning Conditions Win is an (possibly) infinite set of infinite words, and therefore unsuitable as input to an algorithm ⇒ need finite representation. Reachability games: for R ⊆ V define reach ( R ) = { ρ ∈ V ω | ρ visits R at least once } Martin Zimmermann Saarland University Infinite Games 9/30

  4. Three Types of Winning Conditions Win is an (possibly) infinite set of infinite words, and therefore unsuitable as input to an algorithm ⇒ need finite representation. Reachability games: for R ⊆ V define reach ( R ) = { ρ ∈ V ω | ρ visits R at least once } Parity games: for Ω: V → N define parity (Ω) = { ρ ∈ V ω | minimal priority seen infinitely often during ρ is even } Martin Zimmermann Saarland University Infinite Games 9/30

  5. Three Types of Winning Conditions Win is an (possibly) infinite set of infinite words, and therefore unsuitable as input to an algorithm ⇒ need finite representation. Reachability games: for R ⊆ V define reach ( R ) = { ρ ∈ V ω | ρ visits R at least once } Parity games: for Ω: V → N define parity (Ω) = { ρ ∈ V ω | minimal priority seen infinitely often during ρ is even } Muller games: for F ⊆ 2 V define muller ( F ) = { ρ ∈ V ω | set of vertices seen infinitely often during ρ is in F } Martin Zimmermann Saarland University Infinite Games 9/30

  6. Three Types of Winning Conditions Win is an (possibly) infinite set of infinite words, and therefore unsuitable as input to an algorithm ⇒ need finite representation. Reachability games: for R ⊆ V define reach ( R ) = { ρ ∈ V ω | ρ visits R at least once } Parity games: for Ω: V → N define parity (Ω) = { ρ ∈ V ω | minimal priority seen infinitely often during ρ is even } Muller games: for F ⊆ 2 V define muller ( F ) = { ρ ∈ V ω | set of vertices seen infinitely often during ρ is in F } There are many other winning conditions. Martin Zimmermann Saarland University Infinite Games 9/30

  7. What Are We Interested in? Given a type of winning condition (e.g., reachability, parity, Muller),.. .. are games with this condition always determined? .. what kind of strategy do the players need (e.g., positional, finite-state)? .. if finite-state strategies are necessary, how large do they have to be? How hard is it to solve the game? Martin Zimmermann Saarland University Infinite Games 10/30

  8. Outline 1. Definitions 2. Reachability Games 3. Parity Games 4. Muller Games 5. Outlook Martin Zimmermann Saarland University Infinite Games 11/30

  9. Reachability Games Reachability games: for R ⊆ V define reach ( R ) = { ρ ∈ V ω | ρ visits R at least once } v 0 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 Martin Zimmermann Saarland University Infinite Games 12/30

  10. Reachability Games Reachability games: for R ⊆ V define reach ( R ) = { ρ ∈ V ω | ρ visits R at least once } v 0 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 Martin Zimmermann Saarland University Infinite Games 12/30

  11. Reachability Games Reachability games: for R ⊆ V define reach ( R ) = { ρ ∈ V ω | ρ visits R at least once } v 0 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 Martin Zimmermann Saarland University Infinite Games 12/30

  12. Reachability Games Reachability games: for R ⊆ V define reach ( R ) = { ρ ∈ V ω | ρ visits R at least once } v 0 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 Martin Zimmermann Saarland University Infinite Games 12/30

  13. Reachability Games Reachability games: for R ⊆ V define reach ( R ) = { ρ ∈ V ω | ρ visits R at least once } v 0 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 Martin Zimmermann Saarland University Infinite Games 12/30

  14. Attractor Construction Attr A i ( R ) = � n ∈ N A n where A 0 = R and A j + 1 = A j ∪{ v ∈ V i | ∃ ( v , v ′ ) ∈ E s.t. v ′ ∈ A j } ∪{ v ∈ V 1 − i | ∀ ( v , v ′ ) ∈ E we have v ′ ∈ A j } Martin Zimmermann Saarland University Infinite Games 13/30

  15. Attractor Construction Attr A i ( R ) = � n ∈ N A n where A 0 = R and A j + 1 = A j ∪{ v ∈ V i | ∃ ( v , v ′ ) ∈ E s.t. v ′ ∈ A j } ∪{ v ∈ V 1 − i | ∀ ( v , v ′ ) ∈ E we have v ′ ∈ A j } Theorem Reachability games are determined with positional strategies. Martin Zimmermann Saarland University Infinite Games 13/30

  16. Attractor Construction Attr A i ( R ) = � n ∈ N A n where A 0 = R and A j + 1 = A j ∪{ v ∈ V i | ∃ ( v , v ′ ) ∈ E s.t. v ′ ∈ A j } ∪{ v ∈ V 1 − i | ∀ ( v , v ′ ) ∈ E we have v ′ ∈ A j } Theorem Reachability games are determined with positional strategies. Proof. R Martin Zimmermann Saarland University Infinite Games 13/30

  17. Attractor Construction Attr A i ( R ) = � n ∈ N A n where A 0 = R and A j + 1 = A j ∪{ v ∈ V i | ∃ ( v , v ′ ) ∈ E s.t. v ′ ∈ A j } ∪{ v ∈ V 1 − i | ∀ ( v , v ′ ) ∈ E we have v ′ ∈ A j } Theorem Reachability games are determined with positional strategies. Proof. A 1 R Martin Zimmermann Saarland University Infinite Games 13/30

  18. Attractor Construction Attr A i ( R ) = � n ∈ N A n where A 0 = R and A j + 1 = A j ∪{ v ∈ V i | ∃ ( v , v ′ ) ∈ E s.t. v ′ ∈ A j } ∪{ v ∈ V 1 − i | ∀ ( v , v ′ ) ∈ E we have v ′ ∈ A j } Theorem Reachability games are determined with positional strategies. Proof. A 1 A 2 R Martin Zimmermann Saarland University Infinite Games 13/30

  19. Attractor Construction Attr A i ( R ) = � n ∈ N A n where A 0 = R and A j + 1 = A j ∪{ v ∈ V i | ∃ ( v , v ′ ) ∈ E s.t. v ′ ∈ A j } ∪{ v ∈ V 1 − i | ∀ ( v , v ′ ) ∈ E we have v ′ ∈ A j } Theorem Reachability games are determined with positional strategies. Proof. A 1 A 2 · · · A n = A n + 1 R Martin Zimmermann Saarland University Infinite Games 13/30

  20. Attractor Construction Attr A i ( R ) = � n ∈ N A n where A 0 = R and A j + 1 = A j ∪{ v ∈ V i | ∃ ( v , v ′ ) ∈ E s.t. v ′ ∈ A j } ∪{ v ∈ V 1 − i | ∀ ( v , v ′ ) ∈ E we have v ′ ∈ A j } Theorem Reachability games are determined with positional strategies. Proof. R Martin Zimmermann Saarland University Infinite Games 13/30

  21. Attractor Construction Attr A i ( R ) = � n ∈ N A n where A 0 = R and A j + 1 = A j ∪{ v ∈ V i | ∃ ( v , v ′ ) ∈ E s.t. v ′ ∈ A j } ∪{ v ∈ V 1 − i | ∀ ( v , v ′ ) ∈ E we have v ′ ∈ A j } Theorem Reachability games are determined with positional strategies. Proof. R Martin Zimmermann Saarland University Infinite Games 13/30

  22. Attractor Construction Attr A i ( R ) = � n ∈ N A n where A 0 = R and A j + 1 = A j ∪{ v ∈ V i | ∃ ( v , v ′ ) ∈ E s.t. v ′ ∈ A j } ∪{ v ∈ V 1 − i | ∀ ( v , v ′ ) ∈ E we have v ′ ∈ A j } Theorem Reachability games are determined with positional strategies. Proof. R Remark: Attractors can be computed in linear time in | E | . Martin Zimmermann Saarland University Infinite Games 13/30

  23. Outline 1. Definitions 2. Reachability Games 3. Parity Games 4. Muller Games 5. Outlook Martin Zimmermann Saarland University Infinite Games 14/30

  24. Parity Games Parity games: for Ω: V → N define parity (Ω) = { ρ ∈ V ω | minimal priority seen infinitely often during ρ is even } 4 3 2 1 0 1 2 3 0 Martin Zimmermann Saarland University Infinite Games 15/30

  25. Parity Games Parity games: for Ω: V → N define parity (Ω) = { ρ ∈ V ω | minimal priority seen infinitely often during ρ is even } 4 4 3 3 2 2 1 1 0 0 1 1 2 2 3 3 0 0 Martin Zimmermann Saarland University Infinite Games 15/30

  26. Parity Games Applications: Normal form for ω -regular languages: deterministic parity automata. Model-checking game of the modal µ -calculus. Emptiness of parity tree automata equivalent to parity games. Semantics of alternating automata on infinite objects. Martin Zimmermann Saarland University Infinite Games 16/30

  27. Parity Games Applications: Normal form for ω -regular languages: deterministic parity automata. Model-checking game of the modal µ -calculus. Emptiness of parity tree automata equivalent to parity games. Semantics of alternating automata on infinite objects. Theorem Parity games are determined with positional strategies. Martin Zimmermann Saarland University Infinite Games 16/30

  28. Parity Games Applications: Normal form for ω -regular languages: deterministic parity automata. Model-checking game of the modal µ -calculus. Emptiness of parity tree automata equivalent to parity games. Semantics of alternating automata on infinite objects. Theorem Parity games are determined with positional strategies. Proof Sketch: By induction over the number n of vertices. Martin Zimmermann Saarland University Infinite Games 16/30

  29. Parity Games Applications: Normal form for ω -regular languages: deterministic parity automata. Model-checking game of the modal µ -calculus. Emptiness of parity tree automata equivalent to parity games. Semantics of alternating automata on infinite objects. Theorem Parity games are determined with positional strategies. Proof Sketch: By induction over the number n of vertices. n = 1 trivial: c or c Martin Zimmermann Saarland University Infinite Games 16/30

  30. Parity Games Applications: Normal form for ω -regular languages: deterministic parity automata. Model-checking game of the modal µ -calculus. Emptiness of parity tree automata equivalent to parity games. Semantics of alternating automata on infinite objects. Theorem Parity games are determined with positional strategies. Proof Sketch: By induction over the number n of vertices. n = 1 trivial: c or c Player i wins iff Par ( c ) = i Martin Zimmermann Saarland University Infinite Games 16/30

  31. Proof Sketch Now n > 1 and min Ω( V ) = 0. Attr 0 (Ω − 1 ( 0 )) Ω − 1 ( 0 ) Martin Zimmermann Saarland University Infinite Games 17/30

  32. Proof Sketch Martin Zimmermann Saarland University Infinite Games 17/30

  33. Proof Sketch Induction hypothesis applicable.. Martin Zimmermann Saarland University Infinite Games 17/30

  34. Proof Sketch .. yields winning regions W ′ i and positional strategies σ ′ , τ ′ . W ′ 0 W ′ 1 Martin Zimmermann Saarland University Infinite Games 17/30

  35. Proof Sketch W ′ 1 empty: W ′ 0 Martin Zimmermann Saarland University Infinite Games 17/30

  36. Proof Sketch W ′ 1 empty: Player 0 wins from everywhere. Winning strategy: combine σ ′ and attractor strategy, play arbitrarily at Ω − 1 ( 0 ) . W ′ 0 Attr 0 (Ω − 1 ( 0 )) Ω − 1 ( 0 ) Martin Zimmermann Saarland University Infinite Games 17/30

  37. Proof Sketch W ′ 1 non-empty: W ′ 0 Attr 0 (Ω − 1 ( 0 )) W ′ 1 Ω − 1 ( 0 ) Martin Zimmermann Saarland University Infinite Games 17/30

  38. Proof Sketch W ′ 1 non-empty: Attr 1 ( W ′ 1 ) W ′ 1 Martin Zimmermann Saarland University Infinite Games 17/30

  39. Proof Sketch W ′ 1 non-empty: Martin Zimmermann Saarland University Infinite Games 17/30

  40. Proof Sketch W ′ 1 non-empty: Induction hypothesis applicable.. Martin Zimmermann Saarland University Infinite Games 17/30

  41. Proof Sketch W ′ 1 non-empty:.. yields winning regions W ′′ i and positional strategies σ ′′ , τ ′′ . W ′′ 0 W ′′ 1 Martin Zimmermann Saarland University Infinite Games 17/30

  42. Proof Sketch W ′ 1 non-empty:.. yields winning regions W ′′ i and positional strategies σ ′′ , τ ′′ . W ′′ 0 Attr 1 ( W ′ 1 ) W ′′ W ′ 1 1 Martin Zimmermann Saarland University Infinite Games 17/30

  43. Proof Sketch W ′ 1 non-empty: Player 0 wins from W ′′ 0 with σ ′′ . W ′′ 0 Attr 1 ( W ′ 1 ) W ′′ W ′ 1 1 Martin Zimmermann Saarland University Infinite Games 17/30

  44. Proof Sketch W ′ 1 non-empty: Player 1 wins from W ′′ 1 ∪ Attr 1 ( W ′ 1 ) . Winning strategy: combine τ ′ , τ ′′ , and attractor strategy. W ′′ 0 Attr 1 ( W ′ 1 ) W ′′ W ′ 1 1 Martin Zimmermann Saarland University Infinite Games 17/30

  45. Algorithms for Parity Games Determinacy proof yields recursive algorithm with exponential running time. c 3 ) . Best deterministic algorithms: O ( m · n Martin Zimmermann Saarland University Infinite Games 18/30

  46. Algorithms for Parity Games Determinacy proof yields recursive algorithm with exponential running time. c 3 ) . Best deterministic algorithms: O ( m · n Intriguing complexity-theoretic status: in NP ∩ Co-NP (even in UP ∩ Co-UP and thus unlikely to be complete for NP or Co-NP ). Martin Zimmermann Saarland University Infinite Games 18/30

  47. Algorithms for Parity Games Determinacy proof yields recursive algorithm with exponential running time. c 3 ) . Best deterministic algorithms: O ( m · n Intriguing complexity-theoretic status: in NP ∩ Co-NP (even in UP ∩ Co-UP and thus unlikely to be complete for NP or Co-NP ). Open problem: is solving parity games in polynomial time? Martin Zimmermann Saarland University Infinite Games 18/30

  48. Outline 1. Definitions 2. Reachability Games 3. Parity Games 4. Muller Games 5. Outlook Martin Zimmermann Saarland University Infinite Games 19/30

  49. Muller Games Muller games: for F ⊆ 2 V define muller ( F ) = { ρ ∈ V ω | set of vertices seen infinitely often during ρ is in F } Martin Zimmermann Saarland University Infinite Games 20/30

  50. Muller Games Muller games: for F ⊆ 2 V define muller ( F ) = { ρ ∈ V ω | set of vertices seen infinitely often during ρ is in F } A 1 B 2 C 3 D 4 F ∈ F iff | F ∩ { A , B , C , D }| = max ( F ∩ { 1 , 2 , 3 , 4 } ) Martin Zimmermann Saarland University Infinite Games 20/30

  51. Muller Games Muller games: for F ⊆ 2 V define muller ( F ) = { ρ ∈ V ω | set of vertices seen infinitely often during ρ is in F } A 1 B 2 in general: DJW n here: DJW 4 C 3 D 4 F ∈ F iff | F ∩ { A , B , C , D }| = max ( F ∩ { 1 , 2 , 3 , 4 } ) Martin Zimmermann Saarland University Infinite Games 20/30

  52. Latest Appearance Records Need to estimate set of vertices in { A , B , C , D } visited infinitely often during the play: Track order of last appearance of vertices in { A , B , C , D } Martin Zimmermann Saarland University Infinite Games 21/30

  53. Latest Appearance Records Need to estimate set of vertices in { A , B , C , D } visited infinitely often during the play: Track order of last appearance of vertices in { A , B , C , D } C A B C D # Martin Zimmermann Saarland University Infinite Games 21/30

  54. Latest Appearance Records Need to estimate set of vertices in { A , B , C , D } visited infinitely often during the play: Track order of last appearance of vertices in { A , B , C , D } C A B C D # 4 Martin Zimmermann Saarland University Infinite Games 21/30

  55. Latest Appearance Records Need to estimate set of vertices in { A , B , C , D } visited infinitely often during the play: Track order of last appearance of vertices in { A , B , C , D } C A B C D # 4 B B A # C D Martin Zimmermann Saarland University Infinite Games 21/30

  56. Latest Appearance Records Need to estimate set of vertices in { A , B , C , D } visited infinitely often during the play: Track order of last appearance of vertices in { A , B , C , D } C A B C D # 4 B B A # C D 2 Martin Zimmermann Saarland University Infinite Games 21/30

  57. Latest Appearance Records Need to estimate set of vertices in { A , B , C , D } visited infinitely often during the play: Track order of last appearance of vertices in { A , B , C , D } C A B C D # 4 B B A # C D 2 D D B A C # Martin Zimmermann Saarland University Infinite Games 21/30

  58. Latest Appearance Records Need to estimate set of vertices in { A , B , C , D } visited infinitely often during the play: Track order of last appearance of vertices in { A , B , C , D } C A B C D # 4 B B A # C D 2 D D B A C # 4 Martin Zimmermann Saarland University Infinite Games 21/30

  59. Latest Appearance Records Need to estimate set of vertices in { A , B , C , D } visited infinitely often during the play: Track order of last appearance of vertices in { A , B , C , D } C A B C D # A A D B # C 4 B B A # C D 2 D D B A C # 4 Martin Zimmermann Saarland University Infinite Games 21/30

  60. Latest Appearance Records Need to estimate set of vertices in { A , B , C , D } visited infinitely often during the play: Track order of last appearance of vertices in { A , B , C , D } C A B C D # A A D B # C 4 3 B B A # C D 2 D D B A C # 4 Martin Zimmermann Saarland University Infinite Games 21/30

  61. Latest Appearance Records Need to estimate set of vertices in { A , B , C , D } visited infinitely often during the play: Track order of last appearance of vertices in { A , B , C , D } C A B C D # A A D B # C 4 3 B B A # C D C C A D B # 2 D D B A C # 4 Martin Zimmermann Saarland University Infinite Games 21/30

  62. Latest Appearance Records Need to estimate set of vertices in { A , B , C , D } visited infinitely often during the play: Track order of last appearance of vertices in { A , B , C , D } C A B C D # A A D B # C 4 3 B B A # C D C C A D B # 2 4 D D B A C # 4 Martin Zimmermann Saarland University Infinite Games 21/30

  63. Latest Appearance Records Need to estimate set of vertices in { A , B , C , D } visited infinitely often during the play: Track order of last appearance of vertices in { A , B , C , D } C A B C D # A A D B # C 4 3 B B A # C D C C A D B # 2 4 D C D B A C # C # A D B 4 Martin Zimmermann Saarland University Infinite Games 21/30

  64. Latest Appearance Records Need to estimate set of vertices in { A , B , C , D } visited infinitely often during the play: Track order of last appearance of vertices in { A , B , C , D } C A B C D # A A D B # C 4 3 B B A # C D C C A D B # 2 4 D C D B A C # C # A D B 4 1 Martin Zimmermann Saarland University Infinite Games 21/30

  65. Latest Appearance Records Need to estimate set of vertices in { A , B , C , D } visited infinitely often during the play: Track order of last appearance of vertices in { A , B , C , D } C A B C D # A A D B # C A A C # D B 4 3 B B A # C D C C A D B # 2 4 D C D B A C # C # A D B 4 1 Martin Zimmermann Saarland University Infinite Games 21/30

  66. Latest Appearance Records Need to estimate set of vertices in { A , B , C , D } visited infinitely often during the play: Track order of last appearance of vertices in { A , B , C , D } C A B C D # A A D B # C A A C # D B 4 3 2 B B A # C D C C A D B # 2 4 D C D B A C # C # A D B 4 1 Martin Zimmermann Saarland University Infinite Games 21/30

  67. Latest Appearance Records Need to estimate set of vertices in { A , B , C , D } visited infinitely often during the play: Track order of last appearance of vertices in { A , B , C , D } C A B C D # A A D B # C A A C # D B 4 3 2 B B A # C D C C A D B # C C A # D B 2 4 D C D B A C # C # A D B 4 1 Martin Zimmermann Saarland University Infinite Games 21/30

  68. Latest Appearance Records Need to estimate set of vertices in { A , B , C , D } visited infinitely often during the play: Track order of last appearance of vertices in { A , B , C , D } C A B C D # A A D B # C A A C # D B 4 3 2 B B A # C D C C A D B # C C A # D B 2 4 2 D C D B A C # C # A D B 4 1 Martin Zimmermann Saarland University Infinite Games 21/30

  69. Latest Appearance Records Need to estimate set of vertices in { A , B , C , D } visited infinitely often during the play: Track order of last appearance of vertices in { A , B , C , D } C A B C D # A A D B # C A A C # D B 4 3 2 B B A # C D C C A D B # C C A # D B 2 4 2 D C A D B A C # C # A D B A C # D B 4 1 Martin Zimmermann Saarland University Infinite Games 21/30

  70. Latest Appearance Records Need to estimate set of vertices in { A , B , C , D } visited infinitely often during the play: Track order of last appearance of vertices in { A , B , C , D } C A B C D # A A D B # C A A C # D B 4 3 2 B B A # C D C C A D B # C C A # D B 2 4 2 D C A D B A C # C # A D B A C # D B 4 1 2 Martin Zimmermann Saarland University Infinite Games 21/30

  71. Latest Appearance Records Need to estimate set of vertices in { A , B , C , D } visited infinitely often during the play: Track order of last appearance of vertices in { A , B , C , D } C A B C D # A A D B # C A A C # D B 4 3 2 B B A # C D C C A D B # C C A # D B 2 4 2 D C A D B A C # C # A D B A C # D B 4 1 2 From some point onwards only vertices that are visited infinitely often are in front of # , and infinitely often exactly the set of vertices that are visited infinitely often is in front of # . Martin Zimmermann Saarland University Infinite Games 21/30

  72. Muller Games Bookkeeping works in general (use permutations over V ). Product of arena and LAR-structure can be turned into equivalent parity game from which finite-state strategies can be derived (“Muller games are reducible to parity games”). Martin Zimmermann Saarland University Infinite Games 22/30

  73. Muller Games Bookkeeping works in general (use permutations over V ). Product of arena and LAR-structure can be turned into equivalent parity game from which finite-state strategies can be derived (“Muller games are reducible to parity games”). Theorem Muller games are determined with finite-state strategies of size n · n ! . Martin Zimmermann Saarland University Infinite Games 22/30

  74. Muller Games Bookkeeping works in general (use permutations over V ). Product of arena and LAR-structure can be turned into equivalent parity game from which finite-state strategies can be derived (“Muller games are reducible to parity games”). Theorem Muller games are determined with finite-state strategies of size n · n ! . Matching lower bounds via DJW n games. Complexity depends on encoding of F : P , if F is given as list of sets. NP ∩ Co-NP , if F is encoded by a tree. Pspace -complete, if F is encoded by circuit or boolean formula (with variables V ). Martin Zimmermann Saarland University Infinite Games 22/30

  75. Outline 1. Definitions 2. Reachability Games 3. Parity Games 4. Muller Games 5. Outlook Martin Zimmermann Saarland University Infinite Games 23/30

  76. Concurrent Games Both players choose their moves simultaneously Matching pennies: (heads, tails) (heads, heads) (*,*) (tails, heads) (tails, tails) Martin Zimmermann Saarland University Infinite Games 24/30

  77. Concurrent Games Both players choose their moves simultaneously Matching pennies: randomized strategy winning with probability 1. (heads, tails) (heads, heads) (*,*) (tails, heads) (tails, tails) Martin Zimmermann Saarland University Infinite Games 24/30

  78. Concurrent Games Both players choose their moves simultaneously Matching pennies: randomized strategy winning with probability 1. (heads, tails) (heads, heads) (*,*) (tails, heads) (tails, tails) The “Snowball Game”: (hide, wait) (run, wait) (run, throw) (*,*) (*,*) (hide, throw) Martin Zimmermann Saarland University Infinite Games 24/30

Recommend

Admissibility in Infinite Games Dietmar Berwanger EPFL, Lausanne STACS , Aachen

Admissibility in Infinite Games Dietmar Berwanger EPFL, Lausanne STACS , Aachen

Admissibility in Infinite Games Dietmar Berwanger EPFL, Lausanne STACS , Aachen Dietmar Berwanger (EPFL) Admissibility in Infinite Games STACS07 1 / 18 Strategic Interaction Models of reactive systems: zero-sum two-player

333 views • 29 slides
Tradeoffs in Infinite Games Martin Zimmermann Saarland University May 15th, 2018 Scientific

Tradeoffs in Infinite Games Martin Zimmermann Saarland University May 15th, 2018 Scientific

Tradeoffs in Infinite Games Martin Zimmermann Saarland University May 15th, 2018 Scientific Talk in the Habilitation Process Martin Zimmermann Saarland University Tradeoffs in Infinite Games 1/36 Churchs Synthesis Problem o 1 i 1 . ?

1.8k views • 162 slides
Infinite Games Recap and Outlook Martin Zimmermann Saarland University July 26th, 2016 Martin

Infinite Games Recap and Outlook Martin Zimmermann Saarland University July 26th, 2016 Martin

Infinite Games Recap and Outlook Martin Zimmermann Saarland University July 26th, 2016 Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 1/35 Plan for Today Review Change Log Lecture Notes Exam Organizational

1.35k views • 101 slides
Infinite Automata, Logics and Games Angeliki Chalki NTUA March 28, 2017 . . . . . . . .

Infinite Automata, Logics and Games Angeliki Chalki NTUA March 28, 2017 . . . . . . . .

Outline -Automata Nondeterministic Tree Automata Ehrenfeucht-Frass Games Infinite Automata, Logics and Games Angeliki Chalki NTUA March 28, 2017 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

293 views • 26 slides
Automata, Logics, and Infinite Games S. Pinchinat IRISA, Rennes, France Master2 RI 2007 S.

Automata, Logics, and Infinite Games S. Pinchinat IRISA, Rennes, France Master2 RI 2007 S.

Automata, Logics, and Infinite Games S. Pinchinat IRISA, Rennes, France Master2 RI 2007 S. Pinchinat (IRISA) Automata, Logics, and Infinite Games Master2 RI 2007 1 / 79 Temporal Logics and Model-checking 1 Introductory Example Kripke

1.01k views • 83 slides
Procedural Generation of Infinite 3D Worlds for Games Lus Miguel Coelho e Magalhes Mestrado

Procedural Generation of Infinite 3D Worlds for Games Lus Miguel Coelho e Magalhes Mestrado

F ACULDADE DE E NGENHARIA DA U NIVERSIDADE DO P ORTO Procedural Generation of Infinite 3D Worlds for Games Lus Miguel Coelho e Magalhes Mestrado Integrado em Engenharia Informtica e Computao Supervisor: Joo Tiago Pinheiro Neto Jacob

1.1k views • 94 slides
Determinacy strength of infinite games in -languages recognized by variations of pushdown

Determinacy strength of infinite games in -languages recognized by variations of pushdown

Determinacy strength of infinite games in -languages recognized by variations of pushdown automata Wenjuan Li jointly with Prof. Kazuyuki Tanaka Mathematical Institute, Tohoku University Sep. 16, 2016 Workshop on Mathematical Logic and its

580 views • 39 slides
Games Miheer Dewaskar Chennai Mathematical Institute April 27, 2016 1 / 19 Outline Finite

Games Miheer Dewaskar Chennai Mathematical Institute April 27, 2016 1 / 19 Outline Finite

Games Miheer Dewaskar Chennai Mathematical Institute April 27, 2016 1 / 19 Outline Finite Duration Games Win-Lose Games Payoff Games Infinite Duration Games Parity Games Mean Payoff Games Simple Stochastic Games 2 / 19 Outline Finite

2.02k views • 166 slides
Time-optimal Strategies for Infinite Games Martin Zimmermann RWTH Aachen University March 10th,

Time-optimal Strategies for Infinite Games Martin Zimmermann RWTH Aachen University March 10th,

Time-optimal Strategies for Infinite Games Martin Zimmermann RWTH Aachen University March 10th, 2010 DIMAP Seminar Warwick University, United Kingdom Martin Zimmermann RWTH Aachen University Time-optimal Strategies for Infinite Games 1/32

1.02k views • 88 slides
How Much Lookahead is Needed to Win Infinite Games? Joint work with Felix Klein (Saarland

How Much Lookahead is Needed to Win Infinite Games? Joint work with Felix Klein (Saarland

How Much Lookahead is Needed to Win Infinite Games? Joint work with Felix Klein (Saarland University) Martin Zimmermann Saarland University February 13th, 2015 Workshop Automata, Concurrency and Timed Systems Chennai Mathematical Institute,

1.68k views • 126 slides
Optimizing Winning Strategies in Regular Infinite Games SOFSEM 2008, January 2008 Wolfgang

Optimizing Winning Strategies in Regular Infinite Games SOFSEM 2008, January 2008 Wolfgang

Optimizing Winning Strategies in Regular Infinite Games SOFSEM 2008, January 2008 Wolfgang Thomas A Quotation of 50 Years Ago Alonzo Church at the Summer Institute of Symbolic Logic Cornell University, 1957: Given a requirement

665 views • 54 slides
Degrees of Lookahead in Context-free Infinite Games Joint work with Wladimir Fridman and Christof

Degrees of Lookahead in Context-free Infinite Games Joint work with Wladimir Fridman and Christof

Degrees of Lookahead in Context-free Infinite Games Joint work with Wladimir Fridman and Christof L oding Martin Zimmermann RWTH Aachen University August 31st, 2011 Games Workshop 2011 Paris, France Martin Zimmermann RWTH Aachen

1.1k views • 68 slides
Time-optimal Winning Strategies in Infinite Games Martin Zimmermann RWTH Aachen University

Time-optimal Winning Strategies in Infinite Games Martin Zimmermann RWTH Aachen University

Time-optimal Winning Strategies in Infinite Games Martin Zimmermann RWTH Aachen University zimmermann@automata.rwth-aachen.de Gasics meeting Brussels, March 5-6, 2009 Martin Zimmermann RWTH Aachen Time-optimal Winning Strategies in Infinite

713 views • 43 slides
Winning Infinite Games in Finite Time Wolfgang Thomas Francqui Lecture, Mons, April 2013

Winning Infinite Games in Finite Time Wolfgang Thomas Francqui Lecture, Mons, April 2013

Winning Infinite Games in Finite Time Wolfgang Thomas Francqui Lecture, Mons, April 2013 Wolfgang Thomas R. McNaughton Wolfgang Thomas The Problem Given a Muller game with the collection F of winning loops for Player 2 play this like a

661 views • 34 slides
Infinite graphs P eter Komj ath LC12 P eter Komj ath Infinite graphs Infinite

Infinite graphs P eter Komj ath LC12 P eter Komj ath Infinite graphs Infinite

Infinite graphs Infinite graphs P eter Komj ath LC12 P eter Komj ath Infinite graphs Infinite graphs Introduction Graph : ( V , X ), where X [ V ] 2 , V : vertices, X : edges ( W , Y ) is a subgraph of ( V , X ) if W V

403 views • 26 slides
CSC2556 Lecture 11 Noncooperative Games 2: Zero-Sum Games, Stackelberg Games CSC2556 - Nisarg

CSC2556 Lecture 11 Noncooperative Games 2: Zero-Sum Games, Stackelberg Games CSC2556 - Nisarg

CSC2556 Lecture 11 Noncooperative Games 2: Zero-Sum Games, Stackelberg Games CSC2556 - Nisarg Shah 1 Zero-Sum Games Total reward is constant in all outcomes (w.l.o.g. 0 ) Focus on two-player zero-sum games (2p-zs) The more I

539 views • 27 slides
Digital Games An Introduction What are Digital Games? Commonly referred to as video games

Digital Games An Introduction What are Digital Games? Commonly referred to as video games

Digital Games An Introduction What are Digital Games? Commonly referred to as video games People who play video games are called gamers Rapidly growing industry Generated close to USD 100 billion in revenue in 2015

706 views • 45 slides
Parametric LTL Games Martin Zimmermann RWTH Aachen University February 25th, 2010 AlMoTh 2010

Parametric LTL Games Martin Zimmermann RWTH Aachen University February 25th, 2010 AlMoTh 2010

Parametric LTL Games Martin Zimmermann RWTH Aachen University February 25th, 2010 AlMoTh 2010 Frankfurt am Main, Germany Martin Zimmermann RWTH Aachen University Parametric LTL Games 1/18 Motivation We consider infinite games with winning

628 views • 38 slides
Foundations of Artificial Intelligence 6. Board Games Search Strategies for Games, Games with

Foundations of Artificial Intelligence 6. Board Games Search Strategies for Games, Games with

Foundations of Artificial Intelligence 6. Board Games Search Strategies for Games, Games with Chance, State of the Art Joschka Boedecker and Wolfram Burgard and Bernhard Nebel Albert-Ludwigs-Universit at Freiburg May 12, 2017 Contents

396 views • 36 slides
Modal -Calculus and Alternating Tree Automata Seminar Automata, Logics, and Infinite

Modal -Calculus and Alternating Tree Automata Seminar Automata, Logics, and Infinite

Department of Computer Science Institute for Theoretical Computer Science Modal -Calculus and Alternating Tree Automata Seminar Automata, Logics, and Infinite Games Patrick Bahr s0404888@inf.tu-dresden.de Dresden, February 1, 2008

461 views • 45 slides
HOW TO USE INFINITE CAMPUS August 24, 2016 Seminar Information WHAT IS INFINITE CAMPUS?

HOW TO USE INFINITE CAMPUS August 24, 2016 Seminar Information WHAT IS INFINITE CAMPUS?

HOW TO USE INFINITE CAMPUS August 24, 2016 Seminar Information WHAT IS INFINITE CAMPUS? NEW STUDENT INFORMATION SYSTEM REPLACING JMC YOU CAN USE IT TO . . . CHECK YOUR GRADES MANAGE YOUR SCHEDULE SEE CLASSES,

442 views • 16 slides
HOW TO USE INFINITE CAMPUS August 24, 2017 Seminar Information WHAT IS INFINITE CAMPUS?

HOW TO USE INFINITE CAMPUS August 24, 2017 Seminar Information WHAT IS INFINITE CAMPUS?

HOW TO USE INFINITE CAMPUS August 24, 2017 Seminar Information WHAT IS INFINITE CAMPUS? NEW STUDENT INFORMATION SYSTEM REPLACING JMC YOU CAN USE IT TO . . . CHECK YOUR GRADES MANAGE YOUR SCHEDULE SEE CLASSES,

401 views • 15 slides
Happy 103rd birthday, Richard Guy Karl Dilcher Infinite products Infinite products involving

Happy 103rd birthday, Richard Guy Karl Dilcher Infinite products Infinite products involving

Happy 103rd birthday, Richard Guy Karl Dilcher Infinite products Infinite products involving Dirichlet characters and cyclotomic polynomials Karl Dilcher Dalhousie Number Theory Seminar, Sept. 30, 2019 Karl Dilcher Infinite products Joint

965 views • 69 slides
Exploring Trading Strategy Spaces Michael Wellman University of Michigan Trading Games

Exploring Trading Strategy Spaces Michael Wellman University of Michigan Trading Games

Exploring Trading Strategy Spaces Michael Wellman University of Michigan Trading Games Market interaction defines a game Complications: infinite action set incomplete information dynamics, information revelation Large

472 views • 22 slides

More recommend