Examples � α (0) �� α (1) � · · · ∈ L 1 ⊆ ( { a , b } × { a , b } ) ω , if β ( i ) = α ( i + 2). β (0) β (1) I : b a b I : b a b b a b a O : a a O : b b a b No delay: I wins f (0) = 3, f ( i + 1) = 1 Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 4/23
Examples � α (0) �� α (1) � · · · ∈ L 1 ⊆ ( { a , b } × { a , b } ) ω , if β ( i ) = α ( i + 2). β (0) β (1) I : b a b I : b a b b a b a · · · O : a a O : b b a b a · · · No delay: I wins f (0) = 3, f ( i + 1) = 1: O wins Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 4/23
Examples � α (0) �� α (1) � · · · ∈ L 1 ⊆ ( { a , b } × { a , b } ) ω , if β ( i ) = α ( i + 2). β (0) β (1) I : b a b I : b a b b a b a · · · O : a a O : b b a b a · · · No delay: I wins f (0) = 3, f ( i + 1) = 1: O wins � α (0) �� α (1) � · · · ∈ L 2 ⊆ ( { a , b , c } × { a , b , c } ) ω , if β (0) β (1) α ( i ) = a for every i , or β (0) = α ( i ), where i is minimal with α ( i ) � = a . Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 4/23
Examples � α (0) �� α (1) � · · · ∈ L 1 ⊆ ( { a , b } × { a , b } ) ω , if β ( i ) = α ( i + 2). β (0) β (1) I : b a b I : b a b b a b a · · · O : a a O : b b a b a · · · No delay: I wins f (0) = 3, f ( i + 1) = 1: O wins � α (0) �� α (1) � · · · ∈ L 2 ⊆ ( { a , b , c } × { a , b , c } ) ω , if β (0) β (1) α ( i ) = a for every i , or β (0) = α ( i ), where i is minimal with α ( i ) � = a . f (0) � �� � I : a · · · a Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 4/23
Examples � α (0) �� α (1) � · · · ∈ L 1 ⊆ ( { a , b } × { a , b } ) ω , if β ( i ) = α ( i + 2). β (0) β (1) I : b a b I : b a b b a b a · · · O : a a O : b b a b a · · · No delay: I wins f (0) = 3, f ( i + 1) = 1: O wins � α (0) �� α (1) � · · · ∈ L 2 ⊆ ( { a , b , c } × { a , b , c } ) ω , if β (0) β (1) α ( i ) = a for every i , or β (0) = α ( i ), where i is minimal with α ( i ) � = a . f (0) � �� � I : a · · · a O : b Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 4/23
Examples � α (0) �� α (1) � · · · ∈ L 1 ⊆ ( { a , b } × { a , b } ) ω , if β ( i ) = α ( i + 2). β (0) β (1) I : b a b I : b a b b a b a · · · O : a a O : b b a b a · · · No delay: I wins f (0) = 3, f ( i + 1) = 1: O wins � α (0) �� α (1) � · · · ∈ L 2 ⊆ ( { a , b , c } × { a , b , c } ) ω , if β (0) β (1) α ( i ) = a for every i , or β (0) = α ( i ), where i is minimal with α ( i ) � = a . f (0) � �� � I : a · · · a c O : b I wins for every f Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 4/23
Previous Results Theorem (Hosch & Landweber ’72) The following problem is decidable: Given ω -regular L, does O win Γ f ( L ) for some constant f ? Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 5/23
Previous Results Theorem (Hosch & Landweber ’72) The following problem is decidable: Given ω -regular L, does O win Γ f ( L ) for some constant f ? Theorem (Holtmann, Kaiser & Thomas ’10) 1. TFAE for L given by deterministic parity automaton A : O wins Γ f ( L ) for some f . f ( L ) for some constant f with f (0) ≤ 2 2 |A| . O wins Γ 2. Deciding whether this is the case is in 2ExpTime . Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 5/23
Previous Results Theorem (Hosch & Landweber ’72) The following problem is decidable: Given ω -regular L, does O win Γ f ( L ) for some constant f ? Theorem (Holtmann, Kaiser & Thomas ’10) 1. TFAE for L given by deterministic parity automaton A : O wins Γ f ( L ) for some f . f ( L ) for some constant f with f (0) ≤ 2 2 |A| . O wins Γ 2. Deciding whether this is the case is in 2ExpTime . Theorem (Fridman, L¨ oding & Z. ’11) The following problem is undecidable: Given (one-counter, weak, and deterministic) context-free L, does O win Γ f ( L ) for some f ? Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 5/23
Uniformization of Relations A strategy σ for O in Γ f ( L ) induces a mapping f σ : Σ ω I → Σ ω O � � α | α ∈ Σ ω σ is winning ⇔ { I } ⊆ L ( f σ uniformizes L ) f σ ( α ) Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 6/23
Uniformization of Relations A strategy σ for O in Γ f ( L ) induces a mapping f σ : Σ ω I → Σ ω O � � α | α ∈ Σ ω σ is winning ⇔ { I } ⊆ L ( f σ uniformizes L ) f σ ( α ) Continuity in terms of strategies: Strategy without lookahead: i -th letter of f σ ( α ) only depends on first i letters of α (very strong notion of continuity). Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 6/23
Uniformization of Relations A strategy σ for O in Γ f ( L ) induces a mapping f σ : Σ ω I → Σ ω O � � α | α ∈ Σ ω σ is winning ⇔ { I } ⊆ L ( f σ uniformizes L ) f σ ( α ) Continuity in terms of strategies: Strategy without lookahead: i -th letter of f σ ( α ) only depends on first i letters of α (very strong notion of continuity). Strategy with constant delay: f σ Lipschitz-continuous. Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 6/23
Uniformization of Relations A strategy σ for O in Γ f ( L ) induces a mapping f σ : Σ ω I → Σ ω O � � α | α ∈ Σ ω σ is winning ⇔ { I } ⊆ L ( f σ uniformizes L ) f σ ( α ) Continuity in terms of strategies: Strategy without lookahead: i -th letter of f σ ( α ) only depends on first i letters of α (very strong notion of continuity). Strategy with constant delay: f σ Lipschitz-continuous. Strategy with arbitrary (finite) delay: f σ (uniformly) continuous. Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 6/23
Uniformization of Relations A strategy σ for O in Γ f ( L ) induces a mapping f σ : Σ ω I → Σ ω O � � α | α ∈ Σ ω σ is winning ⇔ { I } ⊆ L ( f σ uniformizes L ) f σ ( α ) Continuity in terms of strategies: Strategy without lookahead: i -th letter of f σ ( α ) only depends on first i letters of α (very strong notion of continuity). Strategy with constant delay: f σ Lipschitz-continuous. Strategy with arbitrary (finite) delay: f σ (uniformly) continuous. Holtmann, Kaiser, Thomas : for ω -regular L L uniformizable by continuous function ⇔ L uniformizable by Lipschitz-continuous function Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 6/23
Open Questions No known (non-trivial) lower bounds on computational complexity and necessary lookahead. No results for subclasses of ω -regular conditions. We consider two subclasses: Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 7/23
Open Questions No known (non-trivial) lower bounds on computational complexity and necessary lookahead. No results for subclasses of ω -regular conditions. We consider two subclasses: Fix A = ( Q , Σ , q 0 , ∆ , F ) Reachability acceptance: L ∃ ( A ) = { w ∈ Σ ω | A has run on w that visits F } Safety acceptance: L ∀ ( A ) = { w ∈ Σ ω | A has run on w that never visits V \ F } Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 7/23
Outline 1. Lower Bounds on Lookahead 2. Complexity: Reachability Conditions 3. Complexity: Safety Conditions 4. Complexity: ω -regular Conditions 5. Beyond ω -regularity: WMSO+U conditions 6. Conclusion Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 8/23
Lower Bounds for Reachability Conditions Theorem For every n > 1 there is a language L n such that L n = L ∃ ( A n ) for some deterministic reachability automaton A n with |A n | ∈ O ( n ) , O wins Γ f ( L n ) for some constant delay function f , but f ( L n ) for every delay function f with f (0) ≤ 2 n . I wins Γ Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 9/23
Lower Bounds for Reachability Conditions Theorem For every n > 1 there is a language L n such that L n = L ∃ ( A n ) for some deterministic reachability automaton A n with |A n | ∈ O ( n ) , O wins Γ f ( L n ) for some constant delay function f , but f ( L n ) for every delay function f with f (0) ≤ 2 n . I wins Γ Proof: Σ I = Σ O = { 1 , . . . , n } . w ∈ Σ ∗ I contains bad j-pair (j ∈ Σ I ) if there are two occurrences of j in w such that no j ′ > j occurs in between. Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 9/23
Lower Bounds for Reachability Conditions Theorem For every n > 1 there is a language L n such that L n = L ∃ ( A n ) for some deterministic reachability automaton A n with |A n | ∈ O ( n ) , O wins Γ f ( L n ) for some constant delay function f , but f ( L n ) for every delay function f with f (0) ≤ 2 n . I wins Γ Proof: Σ I = Σ O = { 1 , . . . , n } . w ∈ Σ ∗ I contains bad j-pair (j ∈ Σ I ) if there are two occurrences of j in w such that no j ′ > j occurs in between. O has no bad j-pair for any j ⇒ | w | ≤ 2 n − 1 . w ∈ Σ ∗ O with | w n | = 2 n − 1 and without bad j-pair. Exists w n ∈ Σ ∗ Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 9/23
Lower Bounds for Reachability Conditions � α (0) �� α (1) � · · · ∈ L n iff α (1) α (2) · · · contains a bad β (0)-pair. β (0) β (1) Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 10/23
Lower Bounds for Reachability Conditions � α (0) �� α (1) � · · · ∈ L n iff α (1) α (2) · · · contains a bad β (0)-pair. β (0) β (1) Σ I \ { j } < j Σ I � ∗ � � a � 1 B 1 [ a \ ] ∗ j j . . . > j � ∗ � � a � B n [ a \ ] n ∗ Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 10/23
Lower Bounds for Reachability Conditions � α (0) �� α (1) � · · · ∈ L n iff α (1) α (2) · · · contains a bad β (0)-pair. β (0) β (1) Σ I \ { j } < j Σ I � ∗ � � a � 1 B 1 [ a \ ] ∗ j j . . . > j � ∗ � � a � B n [ a \ ] n ∗ f ( L n ), if f (0) > 2 n : In first round, I picks u 0 s.t. u 0 O wins Γ without its first letter has bad j -pair. O picks j in first round. Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 10/23
Lower Bounds for Reachability Conditions � α (0) �� α (1) � · · · ∈ L n iff α (1) α (2) · · · contains a bad β (0)-pair. β (0) β (1) Σ I \ { j } < j Σ I � ∗ � � a � 1 B 1 [ a \ ] ∗ j j . . . > j � ∗ � � a � B n [ a \ ] n ∗ f ( L n ), if f (0) > 2 n : In first round, I picks u 0 s.t. u 0 O wins Γ without its first letter has bad j -pair. O picks j in first round. f ( L n ), if f (0) ≤ 2 n : I wins Γ I picks prefix of 1 w n of length f (0) in first round, O answers by some j . I finishes w n and then picks some j ′ � = j ad infinitum. Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 10/23
Remarks The automata are deterministic. Similar construction works for safety, too. Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 11/23
Remarks The automata are deterministic. Similar construction works for safety, too. Alphabet size grows in n . Constant-size alphabets possible using binary encoding. Requires automata of size ( n log n ). Open question: constant-size alphabet and automata of size O ( n ) simultaneously achievable. Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 11/23
Outline 1. Lower Bounds on Lookahead 2. Complexity: Reachability Conditions 3. Complexity: Safety Conditions 4. Complexity: ω -regular Conditions 5. Beyond ω -regularity: WMSO+U conditions 6. Conclusion Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 12/23
A Sufficient Condition O wins Γ f ( L ) for some f ⇒ projection pr 0 ( L ) to Σ I universal. Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 13/23
A Sufficient Condition O wins Γ f ( L ) for some f ⇒ projection pr 0 ( L ) to Σ I universal. Theorem Let L = L ∃ ( A ) , where A is a non-deterministic reachability automaton. The following are equivalent: 1. O wins Γ f ( L ) for some delay function f . 2. O wins Γ f ( L ) for some constant delay function f with f (0) ≤ 2 |A| . 3. pr 0 ( L ) is universal. Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 13/23
A Sufficient Condition O wins Γ f ( L ) for some f ⇒ projection pr 0 ( L ) to Σ I universal. Theorem Let L = L ∃ ( A ) , where A is a non-deterministic reachability automaton. The following are equivalent: 1. O wins Γ f ( L ) for some delay function f . 2. O wins Γ f ( L ) for some constant delay function f with f (0) ≤ 2 |A| . 3. pr 0 ( L ) is universal. Corollary The following problem is PSpace -complete: Given a non-deterministic reachability automaton A , does O win Γ f ( L ∃ ( A )) for some f ? Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 13/23
Outline 1. Lower Bounds on Lookahead 2. Complexity: Reachability Conditions 3. Complexity: Safety Conditions 4. Complexity: ω -regular Conditions 5. Beyond ω -regularity: WMSO+U conditions 6. Conclusion Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 14/23
Hardness of Safety Conditions Theorem The following problem is ExpTime -hard: Given a deterministic safety automaton A , does O win Γ f ( L ∀ ( A )) for some f ? Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 15/23
Hardness of Safety Conditions Theorem The following problem is ExpTime -hard: Given a deterministic safety automaton A , does O win Γ f ( L ∀ ( A )) for some f ? Proof: By a reduction from alternating polynomial space Turing machines. I produces configurations, picks existential transitions: has to start with initial configuration, and either copies the current configuration or gives a new one. O checks copies for correctness, picks universal transitions. Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 15/23
Hardness of Safety Conditions I : O : Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 16/23
Hardness of Safety Conditions c 0 I : N ∃ O : Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 16/23
Hardness of Safety Conditions c 0 c 1 I : N ∃ N ∀ O : Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 16/23
Hardness of Safety Conditions c 0 c 1 c 1 I : N ∃ N ∀ C ∀ O : Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 16/23
Hardness of Safety Conditions c 0 c 1 c 1 I : N ∃ N ∀ C ∀ O : Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 16/23
Hardness of Safety Conditions c 0 c 1 c 1 I : N ∃ N ∀ C ∀ τ O : Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 16/23
Hardness of Safety Conditions c 0 c 1 c 1 c 2 I : N ∃ N ∀ C ∀ N ∀ τ O : Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 16/23
Hardness of Safety Conditions c 0 c 1 c 1 c 2 I : N ∃ N ∀ C ∀ N ∀ τ O : To prevent I from cheating, O can claim errors: an incorrect copy by marking the position in the original. an incorrect update by marking the position in the original. Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 16/23
Hardness of Safety Conditions c 0 c 1 c 1 c 2 I : N ∃ N ∀ C ∀ N ∀ τ O : To prevent I from cheating, O can claim errors: an incorrect copy by marking the position in the original. an incorrect update by marking the position in the original. Winning condition checks: I always picks configurations of length p ( n ). c 0 is initial configuration on w . The first error claimed by O is not an actual error. Some c i is accepting. Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 16/23
Hardness of Safety Conditions c 0 c 1 c 1 c 2 I : N ∃ N ∀ C ∀ N ∀ τ O : To prevent I from cheating, O can claim errors: an incorrect copy by marking the position in the original. an incorrect update by marking the position in the original. Winning condition checks: I always picks configurations of length p ( n ). c 0 is initial configuration on w . The first error claimed by O is not an actual error. Some c i is accepting. If this is the case, play is not accepted, i.e., I wins. Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 16/23
Outline 1. Lower Bounds on Lookahead 2. Complexity: Reachability Conditions 3. Complexity: Safety Conditions 4. Complexity: ω -regular Conditions 5. Beyond ω -regularity: WMSO+U conditions 6. Conclusion Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 17/23
Upper Bounds for ω -regular Conditions Theorem The following problem is in ExpTime : Given a deterministic automaton A , does O win Γ f ( L ( A )) for some f ? Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 18/23
Upper Bounds for ω -regular Conditions Theorem The following problem is in ExpTime : Given a deterministic automaton A , does O win Γ f ( L ( A )) for some f ? Proof Idea: Define abstract game G ( A ): Define equivalence relation on Σ ∗ I : x ≡ x ′ , if x and x ′ induce the same behavior on projection of A to Σ I . In G ( A ), Player I picks ≡ -equivalence classes, Player O constructs a run of A on representatives of the picked classes (one move delay). Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 18/23
Upper Bounds for ω -regular Conditions Theorem The following problem is in ExpTime : Given a deterministic automaton A , does O win Γ f ( L ( A )) for some f ? Proof Idea: Define abstract game G ( A ): Define equivalence relation on Σ ∗ I : x ≡ x ′ , if x and x ′ induce the same behavior on projection of A to Σ I . In G ( A ), Player I picks ≡ -equivalence classes, Player O constructs a run of A on representatives of the picked classes (one move delay). G ( A ) can be encoded as parity game of exponential size with the same colors as A . Such a game can be solved in exponential time in |A| . Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 18/23
Upper Bounds for ω -regular Conditions Equivalence classes have “short” representatives, as they are recognized by “small” automata. Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 19/23
Upper Bounds for ω -regular Conditions Equivalence classes have “short” representatives, as they are recognized by “small” automata. Corollary Let L = L ( A ) where A is a deterministic parity automaton with k colors. The following are equivalent: 1. O wins Γ f ( L ) for some delay function f . 2. O wins Γ f ( L ) for some constant delay function f with f (0) ≤ 2 ( |A| k ) 2 +1 . Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 19/23
Upper Bounds for ω -regular Conditions Equivalence classes have “short” representatives, as they are recognized by “small” automata. Corollary Let L = L ( A ) where A is a deterministic parity automaton with k colors. The following are equivalent: 1. O wins Γ f ( L ) for some delay function f . 2. O wins Γ f ( L ) for some constant delay function f with f (0) ≤ 2 ( |A| k ) 2 +1 . Note: f (0) ≤ 2 2 | A | k +2 + 2 achievable by direct pumping argument. Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 19/23
Outline 1. Lower Bounds on Lookahead 2. Complexity: Reachability Conditions 3. Complexity: Safety Conditions 4. Complexity: ω -regular Conditions 5. Beyond ω -regularity: WMSO+U conditions 6. Conclusion Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 20/23
Delay Games with WMSO+U conditions Two equivalent definitions: 1. WMSO+U : weak monadic second-order logic with the unbounding quantifier U . UX ϕ ( X ): there are arbitrarily large finite sets X s.t. ϕ ( X ) holds. Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 21/23
Delay Games with WMSO+U conditions Two equivalent definitions: 1. WMSO+U : weak monadic second-order logic with the unbounding quantifier U . UX ϕ ( X ): there are arbitrarily large finite sets X s.t. ϕ ( X ) holds. 2. Max-automata Deterministic finite automata with counters; actions: incr, reset, max. Acceptance: boolean combination of “counter γ is bounded”. Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 21/23
Delay Games with WMSO+U conditions Two equivalent definitions: 1. WMSO+U : weak monadic second-order logic with the unbounding quantifier U . UX ϕ ( X ): there are arbitrarily large finite sets X s.t. ϕ ( X ) holds. 2. Max-automata Deterministic finite automata with counters; actions: incr, reset, max. Acceptance: boolean combination of “counter γ is bounded”. Example: L = { α ∈ { a , b , c } ω | a n b infix of α for every n } Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 21/23
Delay Games with WMSO+U conditions Two equivalent definitions: 1. WMSO+U : weak monadic second-order logic with the unbounding quantifier U . UX ϕ ( X ): there are arbitrarily large finite sets X s.t. ϕ ( X ) holds. 2. Max-automata Deterministic finite automata with counters; actions: incr, reset, max. Acceptance: boolean combination of “counter γ is bounded”. Example: L = { α ∈ { a , b , c } ω | a n b infix of α for every n } Theorem The following problem is decidable: Given a max-automaton A , does Player O win Γ f ( L ( A )) for some constant f ? Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 21/23
Delay Games with WMSO+U conditions Two equivalent definitions: 1. WMSO+U : weak monadic second-order logic with the unbounding quantifier U . UX ϕ ( X ): there are arbitrarily large finite sets X s.t. ϕ ( X ) holds. 2. Max-automata Deterministic finite automata with counters; actions: incr, reset, max. Acceptance: boolean combination of “counter γ is bounded”. Example: L = { α ∈ { a , b , c } ω | a n b infix of α for every n } Theorem The following problem is decidable: Given a max-automaton A , does Player O win Γ f ( L ( A )) for some constant f ? But constant delay is not always sufficient. Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 21/23
Outline 1. Lower Bounds on Lookahead 2. Complexity: Reachability Conditions 3. Complexity: Safety Conditions 4. Complexity: ω -regular Conditions 5. Beyond ω -regularity: WMSO+U conditions 6. Conclusion Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 22/23
Conclusion Results: automaton lookahead complexity exponential ∗ (non)det. reachability PSpace -complete Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 23/23
Conclusion Results: automaton lookahead complexity exponential ∗ (non)det. reachability PSpace -complete exponential ∗ det. safety ExpTime -complete exponential ∗ det. parity ExpTime -complete Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 23/23
Conclusion Results: automaton lookahead complexity exponential ∗ (non)det. reachability PSpace -complete exponential ∗ det. safety ExpTime -complete exponential ∗ det. parity ExpTime -complete Π P safety ∩ det. reach. polynomial 2 ∗ : tight bound. Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 23/23
Conclusion Results: automaton lookahead complexity exponential ∗ (non)det. reachability PSpace -complete exponential ∗ det. safety ExpTime -complete exponential ∗ det. parity ExpTime -complete Π P safety ∩ det. reach. polynomial 2 ∗ : tight bound. Open questions: Consider non-deterministic automata and Rabin, Streett, Muller automata. Can we determine minimal lookahead that is sufficient to win? Weak MSO+U w.r.t. arbitrary delay functions. Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 23/23
Outline 7. Backup Slides Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 24/23
Upper Bounds for ω -regular Conditions Theorem The following problem is in ExpTime : Given a deterministic automaton A , does O win Γ f ( L ( A )) for some f ? Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 25/23
Upper Bounds for ω -regular Conditions Theorem The following problem is in ExpTime : Given a deterministic automaton A , does O win Γ f ( L ( A )) for some f ? Proof: Extend A to C to keep track of maximal color seen during run using states of the form ( q , c ). Note: L ( C ) � = L ( A ). Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 25/23
Upper Bounds for ω -regular Conditions Theorem The following problem is in ExpTime : Given a deterministic automaton A , does O win Γ f ( L ( A )) for some f ? Proof: Extend A to C to keep track of maximal color seen during run using states of the form ( q , c ). Note: L ( C ) � = L ( A ). α (0) α ( i ) α ( j ) I : β (0) β ( i ) O : Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 25/23
Upper Bounds for ω -regular Conditions Theorem The following problem is in ExpTime : Given a deterministic automaton A , does O win Γ f ( L ( A )) for some f ? Proof: Extend A to C to keep track of maximal color seen during run using states of the form ( q , c ). Note: L ( C ) � = L ( A ). α (0) α ( i ) α ( j ) I : q 0 q β (0) β ( i ) O : � α (0) � � α ( i ) � q : state reached by A after processing · · · . β (0) β ( i ) Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 25/23
Upper Bounds for ω -regular Conditions Theorem The following problem is in ExpTime : Given a deterministic automaton A , does O win Γ f ( L ( A )) for some f ? Proof: Extend A to C to keep track of maximal color seen during run using states of the form ( q , c ). Note: L ( C ) � = L ( A ). α (0) α ( i ) α ( j ) I : q 0 q P β (0) β ( i ) O : � α (0) � � α ( i ) � q : state reached by A after processing · · · . β (0) β ( i ) P : set of states reachable by pr 0 ( C ) from ( q , Ω( q )) after processing α ( i + 1) · · · α ( j ). Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 25/23
Proof Continued δ P : transition function of powerset automaton of pr 0 ( C ). Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 26/23
Proof Continued δ P : transition function of powerset automaton of pr 0 ( C ). w : D → 2 Q C via Let w ∈ Σ ∗ I : define r D w ( q , c ) = δ ∗ r D P ( { ( q , Ω( q )) } , w ) Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 26/23
Proof Continued δ P : transition function of powerset automaton of pr 0 ( C ). w : D → 2 Q C via Let w ∈ Σ ∗ I : define r D w ( q , c ) = δ ∗ r D P ( { ( q , Ω( q )) } , w ) w is witness for r D w ⇒ Language W r of witnesses. R = { r | W r infinite } . Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 26/23
Proof Continued δ P : transition function of powerset automaton of pr 0 ( C ). w : D → 2 Q C via Let w ∈ Σ ∗ I : define r D w ( q , c ) = δ ∗ r D P ( { ( q , Ω( q )) } , w ) w is witness for r D w ⇒ Language W r of witnesses. R = { r | W r infinite } . Lemma Fix domain D. If | w | ≥ 2 |C| 2 , then w is witness of a unique r ∈ R with domain D. Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 26/23
The Game G ( A ) Define new game G ( A ) between I and O : In round 0: I has to pick r 0 ∈ R with dom ( r 0 ) = { q C I } , O has to pick q 0 ∈ dom ( r 0 ) (i.e., q 0 = q C I ). Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 27/23
The Game G ( A ) Define new game G ( A ) between I and O : In round 0: I has to pick r 0 ∈ R with dom ( r 0 ) = { q C I } , O has to pick q 0 ∈ dom ( r 0 ) (i.e., q 0 = q C I ). Round i > 0 with play prefix r 0 q 0 · · · r i − 1 q i − 1 : I has to pick r i ∈ R with dom ( r i ) = r i − 1 ( q i − 1 ), O has to pick q i ∈ dom ( r i ). Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 27/23
The Game G ( A ) Define new game G ( A ) between I and O : In round 0: I has to pick r 0 ∈ R with dom ( r 0 ) = { q C I } , O has to pick q 0 ∈ dom ( r 0 ) (i.e., q 0 = q C I ). Round i > 0 with play prefix r 0 q 0 · · · r i − 1 q i − 1 : I has to pick r i ∈ R with dom ( r i ) = r i − 1 ( q i − 1 ), O has to pick q i ∈ dom ( r i ). Let q i = ( q ′ i , c i ). O wins play if c 0 c 1 c 2 · · · satisfies parity condition. Martin Zimmermann Saarland University How Much Lookahead is Needed to Win Infinite Games? 27/23
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