Degrees of Lookahead in Context-free Infinite Games Joint work with - PowerPoint PPT Presentation

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 University Degrees of Lookahead in Context-free Infinite Games 1/15

Motivation Starting points: Walukiewicz: Solving games with deterministic context-free winning conditions in exponential time. Martin Zimmermann RWTH Aachen University Degrees of Lookahead in Context-free Infinite Games 2/15

Motivation Starting points: Walukiewicz: Solving games with deterministic context-free winning conditions in exponential time. Hosch & Landweber; Holtmann, Kaiser & Thomas: Delay games with regular winning conditions. Martin Zimmermann RWTH Aachen University Degrees of Lookahead in Context-free Infinite Games 2/15

Motivation Starting points: Walukiewicz: Solving games with deterministic context-free winning conditions in exponential time. Hosch & Landweber; Holtmann, Kaiser & Thomas: Delay games with regular winning conditions. Here: delay games with deterministic context-free winning conditions. Algorithmic properties. Bounds on delay. Martin Zimmermann RWTH Aachen University Degrees of Lookahead in Context-free Infinite Games 2/15

Outline 1. Definitions 2. Undecidability Results 3. Lower Bounds on Delay 4. Conclusion Martin Zimmermann RWTH Aachen University Degrees of Lookahead in Context-free Infinite Games 3/15

The Delay Game Γ f ( L ) Delay function: f : N → N + . ω -language L ⊆ (Σ I × Σ O ) ω . Two players: Input ( I ) vs. Output ( O ). Martin Zimmermann RWTH Aachen University Degrees of Lookahead in Context-free Infinite Games 4/15

The Delay Game Γ f ( L ) Delay function: f : N → N + . ω -language L ⊆ (Σ I × Σ O ) ω . Two players: Input ( I ) vs. Output ( O ). In round i : Player I picks word u i ∈ Σ f ( i ) (building α = u 0 u 1 · · · ). I Player O picks letter v i ∈ Σ O (building β = v 0 v 1 · · · ). Martin Zimmermann RWTH Aachen University Degrees of Lookahead in Context-free Infinite Games 4/15

The Delay Game Γ f ( L ) Delay function: f : N → N + . ω -language L ⊆ (Σ I × Σ O ) ω . Two players: Input ( I ) vs. Output ( O ). In round i : Player I picks word u i ∈ Σ f ( i ) (building α = u 0 u 1 · · · ). I Player O picks letter v i ∈ Σ O (building β = v 0 v 1 · · · ). � α (0) �� α (1) � Player O wins iff · · · ∈ L . β (0) β (1) Martin Zimmermann RWTH Aachen University Degrees of Lookahead in Context-free Infinite Games 4/15

The Delay Game Γ f ( L ) Delay function: f : N → N + . ω -language L ⊆ (Σ I × Σ O ) ω . Two players: Input ( I ) vs. Output ( O ). In round i : Player I picks word u i ∈ Σ f ( i ) (building α = u 0 u 1 · · · ). I Player O picks letter v i ∈ Σ O (building β = v 0 v 1 · · · ). � α (0) �� α (1) � Player O wins iff · · · ∈ L . β (0) β (1) Example � ω or � 0 0 Martin Zimmermann RWTH Aachen University Degrees of Lookahead in Context-free Infinite Games 4/15

The Delay Game Γ f ( L ) Delay function: f : N → N + . ω -language L ⊆ (Σ I × Σ O ) ω . Two players: Input ( I ) vs. Output ( O ). In round i : Player I picks word u i ∈ Σ f ( i ) (building α = u 0 u 1 · · · ). I Player O picks letter v i ∈ Σ O (building β = v 0 v 1 · · · ). � α (0) �� α (1) � Player O wins iff · · · ∈ L . β (0) β (1) Example � ω or � ω or � n � 0 � n � 1 � 0 � 0 �� ∗ 0 0 1 ∗ ∗ Martin Zimmermann RWTH Aachen University Degrees of Lookahead in Context-free Infinite Games 4/15

The Delay Game Γ f ( L ) Delay function: f : N → N + . ω -language L ⊆ (Σ I × Σ O ) ω . Two players: Input ( I ) vs. Output ( O ). In round i : Player I picks word u i ∈ Σ f ( i ) (building α = u 0 u 1 · · · ). I Player O picks letter v i ∈ Σ O (building β = v 0 v 1 · · · ). � α (0) �� α (1) � Player O wins iff · · · ∈ L . β (0) β (1) Example � ω or � ω or � ω and � n � 0 � n � 1 � n +1 � 0 � n � 1 � 0 � 0 � 0 �� ∗ �� ∗ 0 0 1 0 1 ∗ ∗ ∗ ∗ Martin Zimmermann RWTH Aachen University Degrees of Lookahead in Context-free Infinite Games 4/15

The Delay Game Γ f ( L ) Delay function: f : N → N + . ω -language L ⊆ (Σ I × Σ O ) ω . Two players: Input ( I ) vs. Output ( O ). In round i : Player I picks word u i ∈ Σ f ( i ) (building α = u 0 u 1 · · · ). I Player O picks letter v i ∈ Σ O (building β = v 0 v 1 · · · ). � α (0) �� α (1) � Player O wins iff · · · ∈ L . β (0) β (1) Example � ω or � ω or � ω and f ( i ) = 2 for all i � n � 0 � n � 1 � n +1 � 0 � n � 1 � 0 � 0 � 0 �� ∗ �� ∗ 0 0 1 0 1 ∗ ∗ ∗ ∗ Martin Zimmermann RWTH Aachen University Degrees of Lookahead in Context-free Infinite Games 4/15

The Delay Game Γ f ( L ) Delay function: f : N → N + . ω -language L ⊆ (Σ I × Σ O ) ω . Two players: Input ( I ) vs. Output ( O ). In round i : Player I picks word u i ∈ Σ f ( i ) (building α = u 0 u 1 · · · ). I Player O picks letter v i ∈ Σ O (building β = v 0 v 1 · · · ). � α (0) �� α (1) � Player O wins iff · · · ∈ L . β (0) β (1) Example � ω or � ω or � ω and f ( i ) = 2 for all i � n � 0 � n � 1 � n +1 � 0 � n � 1 � 0 � 0 � 0 �� ∗ �� ∗ 0 0 1 0 1 ∗ ∗ ∗ ∗ I : 0 0 Martin Zimmermann RWTH Aachen University Degrees of Lookahead in Context-free Infinite Games 4/15

The Delay Game Γ f ( L ) Delay function: f : N → N + . ω -language L ⊆ (Σ I × Σ O ) ω . Two players: Input ( I ) vs. Output ( O ). In round i : Player I picks word u i ∈ Σ f ( i ) (building α = u 0 u 1 · · · ). I Player O picks letter v i ∈ Σ O (building β = v 0 v 1 · · · ). � α (0) �� α (1) � Player O wins iff · · · ∈ L . β (0) β (1) Example � ω or � ω or � ω and f ( i ) = 2 for all i � n � 0 � n � 1 � n +1 � 0 � n � 1 � 0 � 0 � 0 �� ∗ �� ∗ 0 0 1 0 1 ∗ ∗ ∗ ∗ I : 0 0 O : 0 Martin Zimmermann RWTH Aachen University Degrees of Lookahead in Context-free Infinite Games 4/15

The Delay Game Γ f ( L ) Delay function: f : N → N + . ω -language L ⊆ (Σ I × Σ O ) ω . Two players: Input ( I ) vs. Output ( O ). In round i : Player I picks word u i ∈ Σ f ( i ) (building α = u 0 u 1 · · · ). I Player O picks letter v i ∈ Σ O (building β = v 0 v 1 · · · ). � α (0) �� α (1) � Player O wins iff · · · ∈ L . β (0) β (1) Example � ω or � ω or � ω and f ( i ) = 2 for all i � n � 0 � n � 1 � n +1 � 0 � n � 1 � 0 � 0 � 0 �� ∗ �� ∗ 0 0 1 0 1 ∗ ∗ ∗ ∗ I : 0 0 0 0 O : 0 Martin Zimmermann RWTH Aachen University Degrees of Lookahead in Context-free Infinite Games 4/15

The Delay Game Γ f ( L ) Delay function: f : N → N + . ω -language L ⊆ (Σ I × Σ O ) ω . Two players: Input ( I ) vs. Output ( O ). In round i : Player I picks word u i ∈ Σ f ( i ) (building α = u 0 u 1 · · · ). I Player O picks letter v i ∈ Σ O (building β = v 0 v 1 · · · ). � α (0) �� α (1) � Player O wins iff · · · ∈ L . β (0) β (1) Example � ω or � ω or � ω and f ( i ) = 2 for all i � n � 0 � n � 1 � n +1 � 0 � n � 1 � 0 � 0 � 0 �� ∗ �� ∗ 0 0 1 0 1 ∗ ∗ ∗ ∗ I : 0 0 0 0 O : 0 0 Martin Zimmermann RWTH Aachen University Degrees of Lookahead in Context-free Infinite Games 4/15

The Delay Game Γ f ( L ) Delay function: f : N → N + . ω -language L ⊆ (Σ I × Σ O ) ω . Two players: Input ( I ) vs. Output ( O ). In round i : Player I picks word u i ∈ Σ f ( i ) (building α = u 0 u 1 · · · ). I Player O picks letter v i ∈ Σ O (building β = v 0 v 1 · · · ). � α (0) �� α (1) � Player O wins iff · · · ∈ L . β (0) β (1) Example � ω or � ω or � ω and f ( i ) = 2 for all i � n � 0 � n � 1 � n +1 � 0 � n � 1 � 0 � 0 � 0 �� ∗ �� ∗ 0 0 1 0 1 ∗ ∗ ∗ ∗ I : 0 0 0 0 0 0 O : 0 0 Martin Zimmermann RWTH Aachen University Degrees of Lookahead in Context-free Infinite Games 4/15

The Delay Game Γ f ( L ) Delay function: f : N → N + . ω -language L ⊆ (Σ I × Σ O ) ω . Two players: Input ( I ) vs. Output ( O ). In round i : Player I picks word u i ∈ Σ f ( i ) (building α = u 0 u 1 · · · ). I Player O picks letter v i ∈ Σ O (building β = v 0 v 1 · · · ). � α (0) �� α (1) � Player O wins iff · · · ∈ L . β (0) β (1) Example � ω or � ω or � ω and f ( i ) = 2 for all i � n � 0 � n � 1 � n +1 � 0 � n � 1 � 0 � 0 � 0 �� ∗ �� ∗ 0 0 1 0 1 ∗ ∗ ∗ ∗ I : 0 0 0 0 0 0 O : 0 0 0 Martin Zimmermann RWTH Aachen University Degrees of Lookahead in Context-free Infinite Games 4/15