Infinite Games Recap and Outlook Martin Zimmermann Saarland - PowerPoint PPT Presentation

Reducibility Σ 3 � �� Σ 2 � �� Σ 1 � �� GenReach ( R ) Reach ( R ) coB¨ uchi ( C ) wParity (Ω) wMuller ( F ) Parity (Ω) Muller ( F ) Safety ( S ) B¨ uchi ( F ) ReqRes ( Q j , P j ) � �� Π 1 � �� Π 2 � �� Π 3 Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 16/35

Reducibility Σ 3 � �� Σ 2 � �� Σ 1 � �� GenReach ( R ) Rabin ( Q j , P j ) Reach ( R ) coB¨ uchi ( C ) wParity (Ω) wMuller ( F ) Parity (Ω) Muller ( F ) Safety ( S ) B¨ uchi ( F ) ReqRes ( Q j , P j ) � �� Π 1 � �� Π 2 � �� Π 3 Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 16/35

Reducibility Σ 3 � �� Σ 2 � �� Σ 1 � �� GenReach ( R ) Rabin ( Q j , P j ) Reach ( R ) coB¨ uchi ( C ) wParity (Ω) wMuller ( F ) Parity (Ω) Muller ( F ) Safety ( S ) B¨ uchi ( F ) Streett ( Q j , P j ) ReqRes ( Q j , P j ) � �� Π 1 � �� Π 2 � �� Π 3 Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 16/35

Wadge Games (Wadge) reductions are (Wadge) games! A winning strategy for II in the Wadge game W ( L , L ′ ) is a witness for the existence of a Wadge reduction L ≤ L ′ . A winning strategy for I in the Wadge game W ( L , L ′ ) is a witness for the non-existence of a Wadge reduction L ≤ L ′ . Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 17/35

S2S and Parity Tree Automata S2S: Monadic second-order logic over two successors PTA: Parity tree automata Both formalisms are equivalent: For every A exists ϕ A s.t. t ∈ L ( A ) ⇔ t | = ϕ A For every ϕ exists A ϕ s.t. t | = ϕ ⇔ t ∈ L ( A ϕ ) Consequence: Satisfiability of S2S reduces to PTA emptiness (Parity) games everywhere: Acceptance game G ( A , t ) for complement closure of PTA Emptiness game G ( A ) for emptiness check of PTA “The mother of all decidability results” Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 18/35

Change Log Lecture Notes Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 19/35

Change Log Lecture Notes 1/2 Old definition: New definition: Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 20/35

Change Log Lecture Notes 2/2 Graphical notation for finite-state strategies: We represent the initialization function as labeled initial arrows. Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 21/35

Exam Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 22/35

Organizational Matters End-of-term exam When: August 1st, 2016, 10:15 - 12:15 Where: HS 003, Building E1 3 Mode: Open-book What to bring: Student ID Exam inspection: TBA Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 23/35

Organizational Matters End-of-term exam When: August 1st, 2016, 10:15 - 12:15 Where: HS 003, Building E1 3 Mode: Open-book What to bring: Student ID Exam inspection: TBA End-of-semester exam: September 20th, 2016 (more information after first exam) Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 23/35

Questions Challenge us before we challenge you in the exam. Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 24/35

Outlook Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 25/35

(Simple) Stochastic Games Enter a new player ( ), it flips a coin to pick a successor. 0 wins 1 wins Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 26/35

(Simple) Stochastic Games Enter a new player ( ), it flips a coin to pick a successor. 0 wins 1 wins No (sure) winning strategy... ...but one with probability 1. Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 26/35

(Simple) Stochastic Games Enter a new player ( ), it flips a coin to pick a successor. 0 wins 1 wins No (sure) winning strategy... ...but one with probability 1. Value of the game for Player 0: max min p σ,τ σ τ where p σ,τ is the probability that Player 0 wins when using strategy σ and Player 1 uses strategy τ . Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 26/35

Concurrent Games Both players choose their moves simultaneously Matching pennies: (heads, tails) (heads, heads) (*,*) (tails, tails) (tails, heads) Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 27/35

Concurrent Games Both players choose their moves simultaneously Matching pennies: randomized strategy winning with probability 1. (heads, tails) (heads, heads) (*,*) (tails, tails) (tails, heads) Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 27/35

Concurrent Games Both players choose their moves simultaneously Matching pennies: randomized strategy winning with probability 1. (heads, tails) (heads, heads) (*,*) (tails, tails) (tails, heads) The “Snowball Game”: (hide, wait) (run, wait) (run, throw) (*,*) (*,*) (hide, throw) Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 27/35

Concurrent Games Both players choose their moves simultaneously Matching pennies: randomized strategy winning with probability 1. (heads, tails) (heads, heads) (*,*) (tails, tails) (tails, heads) The “Snowball Game”: for every ε , randomized strategy winning with probability 1 − ε . (hide, wait) (run, wait) (run, throw) (*,*) (*,*) (hide, throw) Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 27/35

Games of Imperfect Information Players do not observe sequence of states, but sequence of non-unique observations (yellow , purple , blue , brown ). Player 0 picks action a / b , Player 1 resolves non-determinism. a,b a v 1 v 3 b a,b v 0 a,b a,b a,b b v 2 v 4 v 5 a Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 28/35

Games of Imperfect Information Players do not observe sequence of states, but sequence of non-unique observations (yellow , purple , blue , brown ). Player 0 picks action a / b , Player 1 resolves non-determinism. a,b a v 1 v 3 b a,b v 0 a,b a,b a,b b v 2 v 4 v 5 a No winning strategy for Player 0: every fixed choice of actions to ) ∗ ( pick at ( ) can be countered by going to v 1 or v 2 . Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 28/35

Pushdown Games ⊥ A ⊥ AA ⊥ AAA ⊥ AAAA ⊥ AAAAA ⊥ q I v 0 /0 v 1 /0 v 2 /0 v 3 /0 v 4 /0 v 5 /0 ... q 1 v ′ v ′ v ′ v ′ v ′ v ′ 0 /1 1 /1 2 /1 3 /1 4 /1 5 /1 ... q 2 v ′′ 0 /0 Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 29/35

Pushdown Games ⊥ A ⊥ AA ⊥ AAA ⊥ AAAA ⊥ AAAAA ⊥ q I v 0 /0 v 1 /0 v 2 /0 v 3 /0 v 4 /0 v 5 /0 ... q 1 v ′ v ′ v ′ v ′ v ′ v ′ 0 /1 1 /1 2 /1 3 /1 4 /1 5 /1 ... q 2 v ′′ 0 /0 Pushdown Parity Games can be reduced to parity games in exponentially sized arenas ⇒ Exptime -complete. Both players have positional winning strategies (but these are now infinite objects!). Finite representation of winning strategies: pushdown automata with output. Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 29/35

Playing Infinite Games in a Hurry Parity games in finite time: play until first loop is closed, minimal color in loop determines winner. Positional determinacy ⇒ winning regions preserved Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 30/35

Playing Infinite Games in a Hurry Parity games in finite time: play until first loop is closed, minimal color in loop determines winner. Positional determinacy ⇒ winning regions preserved No longer works for Muller games. Need scoring functions: 0 0 1 1 0 0 1 2 w Sc { 0 } Acc { 0 } Sc { 0 , 1 , 2 } Acc { 0 , 1 , 2 } Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 30/35

Playing Infinite Games in a Hurry Parity games in finite time: play until first loop is closed, minimal color in loop determines winner. Positional determinacy ⇒ winning regions preserved No longer works for Muller games. Need scoring functions: 0 0 1 1 0 0 1 2 w 1 Sc { 0 } ∅ Acc { 0 } Sc { 0 , 1 , 2 } Acc { 0 , 1 , 2 } Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 30/35

Playing Infinite Games in a Hurry Parity games in finite time: play until first loop is closed, minimal color in loop determines winner. Positional determinacy ⇒ winning regions preserved No longer works for Muller games. Need scoring functions: 0 0 1 1 0 0 1 2 w 1 2 Sc { 0 } ∅ ∅ Acc { 0 } Sc { 0 , 1 , 2 } Acc { 0 , 1 , 2 } Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 30/35

Playing Infinite Games in a Hurry Parity games in finite time: play until first loop is closed, minimal color in loop determines winner. Positional determinacy ⇒ winning regions preserved No longer works for Muller games. Need scoring functions: 0 0 1 1 0 0 1 2 w 1 2 0 Sc { 0 } ∅ ∅ ∅ Acc { 0 } Sc { 0 , 1 , 2 } Acc { 0 , 1 , 2 } Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 30/35

Playing Infinite Games in a Hurry Parity games in finite time: play until first loop is closed, minimal color in loop determines winner. Positional determinacy ⇒ winning regions preserved No longer works for Muller games. Need scoring functions: 0 0 1 1 0 0 1 2 w 1 2 0 0 Sc { 0 } ∅ ∅ ∅ ∅ Acc { 0 } Sc { 0 , 1 , 2 } Acc { 0 , 1 , 2 } Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 30/35

Playing Infinite Games in a Hurry Parity games in finite time: play until first loop is closed, minimal color in loop determines winner. Positional determinacy ⇒ winning regions preserved No longer works for Muller games. Need scoring functions: 0 0 1 1 0 0 1 2 w 1 2 0 0 1 Sc { 0 } ∅ ∅ ∅ ∅ ∅ Acc { 0 } Sc { 0 , 1 , 2 } Acc { 0 , 1 , 2 } Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 30/35

Playing Infinite Games in a Hurry Parity games in finite time: play until first loop is closed, minimal color in loop determines winner. Positional determinacy ⇒ winning regions preserved No longer works for Muller games. Need scoring functions: 0 0 1 1 0 0 1 2 w 1 2 0 0 1 2 Sc { 0 } ∅ ∅ ∅ ∅ ∅ ∅ Acc { 0 } Sc { 0 , 1 , 2 } Acc { 0 , 1 , 2 } Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 30/35

Playing Infinite Games in a Hurry Parity games in finite time: play until first loop is closed, minimal color in loop determines winner. Positional determinacy ⇒ winning regions preserved No longer works for Muller games. Need scoring functions: 0 0 1 1 0 0 1 2 w 1 2 0 0 1 2 0 Sc { 0 } ∅ ∅ ∅ ∅ ∅ ∅ ∅ Acc { 0 } Sc { 0 , 1 , 2 } Acc { 0 , 1 , 2 } Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 30/35

Playing Infinite Games in a Hurry Parity games in finite time: play until first loop is closed, minimal color in loop determines winner. Positional determinacy ⇒ winning regions preserved No longer works for Muller games. Need scoring functions: 0 0 1 1 0 0 1 2 w 1 2 0 0 1 2 0 0 Sc { 0 } ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ Acc { 0 } Sc { 0 , 1 , 2 } Acc { 0 , 1 , 2 } Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 30/35

Playing Infinite Games in a Hurry Parity games in finite time: play until first loop is closed, minimal color in loop determines winner. Positional determinacy ⇒ winning regions preserved No longer works for Muller games. Need scoring functions: 0 0 1 1 0 0 1 2 w 1 2 0 0 1 2 0 0 Sc { 0 } ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ Acc { 0 } Sc { 0 , 1 , 2 } 0 Acc { 0 , 1 , 2 } { 0 } Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 30/35

Playing Infinite Games in a Hurry Parity games in finite time: play until first loop is closed, minimal color in loop determines winner. Positional determinacy ⇒ winning regions preserved No longer works for Muller games. Need scoring functions: 0 0 1 1 0 0 1 2 w 1 2 0 0 1 2 0 0 Sc { 0 } ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ Acc { 0 } Sc { 0 , 1 , 2 } 0 0 Acc { 0 , 1 , 2 } { 0 } { 0 } Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 30/35

Playing Infinite Games in a Hurry Parity games in finite time: play until first loop is closed, minimal color in loop determines winner. Positional determinacy ⇒ winning regions preserved No longer works for Muller games. Need scoring functions: 0 0 1 1 0 0 1 2 w 1 2 0 0 1 2 0 0 Sc { 0 } ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ Acc { 0 } Sc { 0 , 1 , 2 } 0 0 0 Acc { 0 , 1 , 2 } { 0 } { 0 } { 0 , 1 } Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 30/35

Playing Infinite Games in a Hurry Parity games in finite time: play until first loop is closed, minimal color in loop determines winner. Positional determinacy ⇒ winning regions preserved No longer works for Muller games. Need scoring functions: 0 0 1 1 0 0 1 2 w 1 2 0 0 1 2 0 0 Sc { 0 } ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ Acc { 0 } Sc { 0 , 1 , 2 } 0 0 0 0 Acc { 0 , 1 , 2 } { 0 } { 0 } { 0 , 1 } { 0 , 1 } Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 30/35

Playing Infinite Games in a Hurry Parity games in finite time: play until first loop is closed, minimal color in loop determines winner. Positional determinacy ⇒ winning regions preserved No longer works for Muller games. Need scoring functions: 0 0 1 1 0 0 1 2 w 1 2 0 0 1 2 0 0 Sc { 0 } ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ Acc { 0 } Sc { 0 , 1 , 2 } 0 0 0 0 0 Acc { 0 , 1 , 2 } { 0 } { 0 } { 0 , 1 } { 0 , 1 } { 0 , 1 } Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 30/35

Playing Infinite Games in a Hurry Parity games in finite time: play until first loop is closed, minimal color in loop determines winner. Positional determinacy ⇒ winning regions preserved No longer works for Muller games. Need scoring functions: 0 0 1 1 0 0 1 2 w 1 2 0 0 1 2 0 0 Sc { 0 } ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ Acc { 0 } Sc { 0 , 1 , 2 } 0 0 0 0 0 0 Acc { 0 , 1 , 2 } { 0 } { 0 } { 0 , 1 } { 0 , 1 } { 0 , 1 } { 0 , 1 } Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 30/35

Playing Infinite Games in a Hurry Parity games in finite time: play until first loop is closed, minimal color in loop determines winner. Positional determinacy ⇒ winning regions preserved No longer works for Muller games. Need scoring functions: 0 0 1 1 0 0 1 2 w 1 2 0 0 1 2 0 0 Sc { 0 } ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ Acc { 0 } Sc { 0 , 1 , 2 } 0 0 0 0 0 0 0 Acc { 0 , 1 , 2 } { 0 } { 0 } { 0 , 1 } { 0 , 1 } { 0 , 1 } { 0 , 1 } { 0 , 1 } Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 30/35

Playing Infinite Games in a Hurry Parity games in finite time: play until first loop is closed, minimal color in loop determines winner. Positional determinacy ⇒ winning regions preserved No longer works for Muller games. Need scoring functions: 0 0 1 1 0 0 1 2 w 1 2 0 0 1 2 0 0 Sc { 0 } ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ Acc { 0 } Sc { 0 , 1 , 2 } 0 0 0 0 0 0 0 1 Acc { 0 , 1 , 2 } { 0 } { 0 } { 0 , 1 } { 0 , 1 } { 0 , 1 } { 0 , 1 } { 0 , 1 } ∅ Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 30/35

Playing Infinite Games in a Hurry Parity games in finite time: play until first loop is closed, minimal color in loop determines winner. Positional determinacy ⇒ winning regions preserved No longer works for Muller games. Need scoring functions: 0 0 1 1 0 0 1 2 w 1 2 0 0 1 2 0 0 Sc { 0 } ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ Acc { 0 } Sc { 0 , 1 , 2 } 0 0 0 0 0 0 0 1 Acc { 0 , 1 , 2 } { 0 } { 0 } { 0 , 1 } { 0 , 1 } { 0 , 1 } { 0 , 1 } { 0 , 1 } ∅ Theorem Player i has strategy to bound the opponent’s scores by two when starting in W i ( G ) . Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 30/35

Playing Infinite Games in a Hurry Parity games in finite time: play until first loop is closed, minimal color in loop determines winner. Positional determinacy ⇒ winning regions preserved No longer works for Muller games. Need scoring functions: 0 0 1 1 0 0 1 2 w 1 2 0 0 1 2 0 0 Sc { 0 } ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ Acc { 0 } Sc { 0 , 1 , 2 } 0 0 0 0 0 0 0 1 Acc { 0 , 1 , 2 } { 0 } { 0 } { 0 , 1 } { 0 , 1 } { 0 , 1 } { 0 , 1 } { 0 , 1 } ∅ Theorem Player i has strategy to bound the opponent’s scores by two when starting in W i ( G ) . Corollary: Stopping play after first score reaches value three preserves winning regions (at most exponential play length) Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 30/35

Games with Costs Parity game: Player 0 wins from everywhere, but it takes arbitrarily long to “answer” 1 with 0. 1 2 0 Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 31/35

Games with Costs Parity game: Player 0 wins from everywhere, but it takes arbitrarily long to “answer” 1 with 0. 0 1 2 0 0 0 1 Add edge-costs: Player 0 wins if there is a bound b and a position n such that every odd color after n is followed by a smaller even color with cost ≤ b in between ⇒ Player 1 wins example from everywhere (stay longer and longer in 2). Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 31/35

Games with Costs Parity game: Player 0 wins from everywhere, but it takes arbitrarily long to “answer” 1 with 0. 0 1 2 0 0 0 1 Add edge-costs: Player 0 wins if there is a bound b and a position n such that every odd color after n is followed by a smaller even color with cost ≤ b in between ⇒ Player 1 wins example from everywhere (stay longer and longer in 2). Theorem Parity games with costs are determined, Player 0 has positional winning strategies, and they can be solved in NP ∩ co-NP . Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 31/35

Tradeoffs Every edge has cost 1 3 1 3 1 2 0 2 0 Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 32/35

Tradeoffs Every edge has cost 1 3 1 3 1 2 0 2 0 Player 0 has: Positional winning strategy with bound 9. Finite-state strategy of size 2 with bound 8. Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 32/35

Tradeoffs Every edge has cost 1 3 1 3 1 2 0 2 0 Player 0 has: Positional winning strategy with bound 9. Finite-state strategy of size 2 with bound 8. With d odd colors and d gadgets for each player: Player 0 has: Positional winning strategy with bound d 2 + 3 d − 1. Finite-state strategy of size 2 d − 2 with bound d 2 + 2 d . Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 32/35

Many other variants More winning conditions: various quantitative conditions Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 33/35

Many other variants More winning conditions: various quantitative conditions Games on timed automata ⇒ uncountable arenas Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 33/35

Many other variants More winning conditions: various quantitative conditions Games on timed automata ⇒ uncountable arenas Play even longer: games of ordinal length Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 33/35

Many other variants More winning conditions: various quantitative conditions Games on timed automata ⇒ uncountable arenas Play even longer: games of ordinal length Games with delay: Player 0 is allowed to skip some moves to obtain lookahead on Player 1’s moves. Basic question: what kind of lookahead is necessary to win. Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 33/35

Many other variants More winning conditions: various quantitative conditions Games on timed automata ⇒ uncountable arenas Play even longer: games of ordinal length Games with delay: Player 0 is allowed to skip some moves to obtain lookahead on Player 1’s moves. Basic question: what kind of lookahead is necessary to win. More than two players ⇒ no longer zero-sum games. Requires whole new theory (equilibria). Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 33/35

Many other variants More winning conditions: various quantitative conditions Games on timed automata ⇒ uncountable arenas Play even longer: games of ordinal length Games with delay: Player 0 is allowed to skip some moves to obtain lookahead on Player 1’s moves. Basic question: what kind of lookahead is necessary to win. More than two players ⇒ no longer zero-sum games. Requires whole new theory (equilibria). And: any combination of extensions discussed above. Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 33/35

Thesis Topics DFG project TriCS : Tradeoffs in Controller Synthesis. How to compute optimal strategies? Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 34/35

Thesis Topics DFG project TriCS : Tradeoffs in Controller Synthesis. How to compute optimal strategies? Games with delay: how much lookahead is necessary for different winning conditions? Tradeoffs between lookahead and memory? Temporal logics for the specification of reactive systems. ... Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 34/35

Thesis Topics DFG project TriCS : Tradeoffs in Controller Synthesis. How to compute optimal strategies? Games with delay: how much lookahead is necessary for different winning conditions? Tradeoffs between lookahead and memory? Temporal logics for the specification of reactive systems. ... Your own idea? Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 34/35

Thesis Topics DFG project TriCS : Tradeoffs in Controller Synthesis. How to compute optimal strategies? Games with delay: how much lookahead is necessary for different winning conditions? Tradeoffs between lookahead and memory? Temporal logics for the specification of reactive systems. ... Your own idea? Or what about some open problems: Generalized reachability games with sets of size two: P , NP , or PSPACE ? Exact complexity of parity games. Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 34/35

Thesis Topics DFG project TriCS : Tradeoffs in Controller Synthesis. How to compute optimal strategies? Games with delay: how much lookahead is necessary for different winning conditions? Tradeoffs between lookahead and memory? Temporal logics for the specification of reactive systems. ... Your own idea? Or what about some open problems: Generalized reachability games with sets of size two: P , NP , or PSPACE ? Exact complexity of parity games. If you are interested in working on current research topics, contact us! Martin Zimmermann Saarland University Infinite Games - Recap and Outlook 34/35

Infinite Games Recap and Outlook Martin Zimmermann Saarland - PowerPoint PPT Presentation

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

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

Infinite Games Martin Zimmermann Saarland University October 21st, 2014 Research Training Group

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Safe Harbor 401(k) Plans and Automatic Enrollment Lisa Barton Sarah Fry April 12, 2011

Matching projects at the BLS Michael Horrigan Associate Commissioner Office of Employment and

Make Me a Match! Friendship Force International planning@friendshipforce.org WHAT IS JOURNEY

Early Childhood Matching Project, Phase II Presentation by Charlie Rosemond In 2016, what

This Changes (Almost) Everything: Mentor, Youth, Parent & Staff Perspectives on Youth

LOTTERY Scott Hankins Mark Hoekstra Paige Marta Skiba University of Kentucky University of

NCFC YOUTH FOOTBALL CLUB Recreation Coaches Meeting August 5 th 2017 1 SCOTT BECKER NCFC YOUTH

Aspire Johnson County January 25, 2017 Jeff Owen welcomed everyone then asked for round robin

Infinite Games Recap and Outlook Martin Zimmermann Saarland - PowerPoint PPT Presentation

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

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

Infinite Games Martin Zimmermann Saarland University October 21st, 2014 Research Training Group

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Safe Harbor 401(k) Plans and Automatic Enrollment Lisa Barton Sarah Fry April 12, 2011

Matching projects at the BLS Michael Horrigan Associate Commissioner Office of Employment and

Make Me a Match! Friendship Force International planning@friendshipforce.org WHAT IS JOURNEY

Early Childhood Matching Project, Phase II Presentation by Charlie Rosemond In 2016, what

This Changes (Almost) Everything: Mentor, Youth, Parent &amp; Staff Perspectives on Youth

LOTTERY Scott Hankins Mark Hoekstra Paige Marta Skiba University of Kentucky University of

NCFC YOUTH FOOTBALL CLUB Recreation Coaches Meeting August 5 th 2017 1 SCOTT BECKER NCFC YOUTH

Aspire Johnson County January 25, 2017 Jeff Owen welcomed everyone then asked for round robin

This Changes (Almost) Everything: Mentor, Youth, Parent & Staff Perspectives on Youth