Easy to Win, Hard to Master: Optimal Strategies in Parity Games with Costs Joint work with Martin Zimmermann Alexander Weinert Saarland University August 31st, 2016 CSL 2016 - Marseille, France Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 1/12 Parity Games 1 2 0 0 0 0 0 3 0 1 0 3 4 0 0 0 4 0 0 0 4 0 Deciding winner in UP ∩ co-UP ··· Positional Strategies Example due to (Fijalkow and Chatterjee, Infinite-state games, 2013) Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 2/12 Parity Games 1 2 0 0 0 0 0 3 0 1 0 3 4 0 0 0 4 0 0 0 4 0 Deciding winner in UP ∩ co-UP ··· Positional Strategies Example due to (Fijalkow and Chatterjee, Infinite-state games, 2013) Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 2/12 Parity Games 1 2 0 0 0 0 0 3 0 1 0 3 4 0 0 0 4 0 0 0 4 0 Deciding winner in UP ∩ co-UP ··· Positional Strategies Example due to (Fijalkow and Chatterjee, Infinite-state games, 2013) Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 2/12 Parity Games 1 2 0 0 0 0 0 3 0 1 0 3 4 0 0 0 4 0 0 0 4 0 Deciding winner in UP ∩ co-UP ··· Positional Strategies Example due to (Fijalkow and Chatterjee, Infinite-state games, 2013) Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 2/12 Parity Games 1 2 0 0 0 0 0 3 0 1 0 3 4 0 0 0 4 0 0 0 4 0 Deciding winner in UP ∩ co-UP ··· Positional Strategies Example due to (Fijalkow and Chatterjee, Infinite-state games, 2013) Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 2/12 Parity Games 1 2 0 0 0 0 0 3 0 1 0 3 4 0 0 0 4 0 0 0 4 0 Deciding winner in UP ∩ co-UP ··· Positional Strategies Example due to (Fijalkow and Chatterjee, Infinite-state games, 2013) Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 2/12 Parity Games 1 2 0 0 0 0 0 3 0 1 0 3 4 0 0 0 4 0 0 0 4 0 Deciding winner in UP ∩ co-UP ··· Positional Strategies Example due to (Fijalkow and Chatterjee, Infinite-state games, 2013) Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 2/12 Parity Games 1 2 0 0 0 0 0 3 0 1 0 3 4 0 0 0 4 0 0 0 4 0 Deciding winner in UP ∩ co-UP ··· Positional Strategies Example due to (Fijalkow and Chatterjee, Infinite-state games, 2013) Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 2/12 Parity Games 1 2 0 0 0 0 0 3 0 1 0 3 4 0 0 0 4 0 0 0 4 0 Deciding winner in UP ∩ co-UP ··· Positional Strategies Example due to (Fijalkow and Chatterjee, Infinite-state games, 2013) Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 2/12 Parity Games 1 2 0 0 0 0 0 3 0 1 0 3 4 0 0 0 4 0 0 0 4 0 Deciding winner in UP ∩ co-UP ··· Positional Strategies Example due to (Fijalkow and Chatterjee, Infinite-state games, 2013) Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 2/12 Parity Games 1 2 0 0 0 0 0 3 0 1 0 3 4 0 0 0 4 0 0 0 4 0 Deciding winner in UP ∩ co-UP ··· Positional Strategies Example due to (Fijalkow and Chatterjee, Infinite-state games, 2013) Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 2/12 Parity Games 1 2 0 0 0 0 0 3 0 1 0 3 4 0 0 0 4 0 0 0 4 0 Deciding winner in UP ∩ co-UP ··· Positional Strategies Example due to (Fijalkow and Chatterjee, Infinite-state games, 2013) Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 2/12 Parity Games 1 2 0 0 0 0 0 3 0 1 0 3 4 0 0 0 4 0 0 0 4 0 Deciding winner in UP ∩ co-UP ··· Positional Strategies Example due to (Fijalkow and Chatterjee, Infinite-state games, 2013) Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 2/12 Parity Games 1 2 0 0 0 0 0 3 0 1 0 3 4 0 0 0 4 0 0 0 4 0 Deciding winner in UP ∩ co-UP ··· Positional Strategies Example due to (Fijalkow and Chatterjee, Infinite-state games, 2013) Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 2/12 Parity Games 1 2 0 0 0 0 0 3 0 1 0 3 4 0 0 0 4 0 0 0 4 0 Deciding winner in UP ∩ co-UP ··· Positional Strategies Example due to (Fijalkow and Chatterjee, Infinite-state games, 2013) Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 2/12 Parity Games 1 2 0 0 0 0 0 3 0 1 0 3 4 0 0 0 4 0 0 0 4 0 Deciding winner in UP ∩ co-UP ··· Positional Strategies Example due to (Fijalkow and Chatterjee, Infinite-state games, 2013) Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 2/12 Parity Games 1 2 0 0 0 0 0 3 0 1 0 3 4 0 0 0 4 0 0 0 4 0 Deciding winner in UP ∩ co-UP ··· Positional Strategies Example due to (Fijalkow and Chatterjee, Infinite-state games, 2013) Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 2/12 Parity Games 1 2 0 0 0 0 0 3 0 1 0 3 4 0 0 0 4 0 0 0 4 0 Deciding winner in UP ∩ co-UP ··· Positional Strategies Example due to (Fijalkow and Chatterjee, Infinite-state games, 2013) Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 2/12 Finitary Parity Games 1 2 0 0 0 0 0 3 4 4 0 1 0 3 0 0 0 4 0 0 0 4 0 ··· 3 Goal for Player 0: Bound response times Example due to (Fijalkow and Chatterjee, Infinite-state games, 2013) Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 3/12 Finitary Parity Games 1 2 0 0 0 0 0 3 4 4 0 1 0 3 0 0 0 4 0 0 0 4 0 ··· 3 Goal for Player 0: Bound response times Example due to (Fijalkow and Chatterjee, Infinite-state games, 2013) Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 3/12 Finitary Parity Games 1 2 0 0 0 0 0 3 4 4 0 1 0 3 0 0 0 4 0 0 0 4 0 ··· 3 Goal for Player 0: Bound response times Example due to (Fijalkow and Chatterjee, Infinite-state games, 2013) Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 3/12 Finitary Parity Games 1 2 0 0 0 0 0 3 4 4 0 1 0 3 0 0 0 4 0 0 0 4 0 ··· 3 Goal for Player 0: Bound response times Example due to (Fijalkow and Chatterjee, Infinite-state games, 2013) Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 3/12 Finitary Parity Games 1 2 0 0 0 0 0 3 4 4 0 1 0 3 0 0 0 4 0 0 0 4 0 ··· 3 Goal for Player 0: Bound response times Example due to (Fijalkow and Chatterjee, Infinite-state games, 2013) Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 3/12 Decision Problem Theorem (Chatterjee et al., Finitary Winning, 2009) The following decision problem is in PTime: Input: Finitary parity game G = (A, FinParity(Ω)) Question: Does there exist a strategy σ with Cst(σ) < ∞? Theorem The following decision problem is PSpace-complete: Input: Finitary parity game G = (A, FinParity(Ω)), bound b ∈ N Question: Does there exist a strategy σ with Cst(σ) ≤ b? Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 4/12 Decision Problem Theorem (Chatterjee et al., Finitary Winning, 2009) The following decision problem is in PTime: Input: Finitary parity game G = (A, FinParity(Ω)) Question: Does there exist a strategy σ with Cst(σ) < ∞? Theorem The following decision problem is PSpace-complete: Input: Finitary parity game G = (A, FinParity(Ω)), bound b ∈ N Question: Does there exist a strategy σ with Cst(σ) ≤ b? Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 4/12 Introduction 3 Complexity in PSpace Exponential Memory Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 5/12 Introduction 3 Complexity in PSpace Exponential Memory Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 5/12 Introduction 3 Complexity in PSpace Exponential Memory Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 5/12 Introduction 3 Complexity in PSpace PSpace-hard Exponential Memory Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 5/12 Introduction 3 Complexity in PSpace PSpace-hard Exponential Memory Sufficient Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 5/12 Introduction 3 Complexity in PSpace PSpace-hard Exponential Memory Sufficient Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 5/12 Introduction 3 Complexity in PSpace PSpace-hard Exponential Memory Sufficient Alexander Weinert Saarland University Necessary Optimal Strategies in Parity Games with Costs 5/12 From Finitary Parity to Parity Given: Finitary parity game G = (A, FinParity(Ω)), bound b ∈ N. Lemma Deciding if Player 0 has strategy σ with Cst(σ) ≤ b is in PSpace. Idea: Simulate game, keeping track of open requests. Lemma Player 0 has such a strategy iff she “survives” p(|G|) steps in extended game G 0 . Algorithm: Simulate all plays in G 0 on-the-fly for p(|G|) steps using an alternating Turing machine. ⇒ Problem is in APTime (Chandra et al., Alternation, 1981) ⇒ Problem is in PSpace Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 6/12 From Finitary Parity to Parity Given: Finitary parity game G = (A, FinParity(Ω)), bound b ∈ N. Lemma Deciding if Player 0 has strategy σ with Cst(σ) ≤ b is in PSpace. Idea: Simulate game, keeping track of open requests. Lemma Player 0 has such a strategy iff she “survives” p(|G|) steps in extended game G 0 . Algorithm: Simulate all plays in G 0 on-the-fly for p(|G|) steps using an alternating Turing machine. ⇒ Problem is in APTime (Chandra et al., Alternation, 1981) ⇒ Problem is in PSpace Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 6/12 From Finitary Parity to Parity Given: Finitary parity game G = (A, FinParity(Ω)), bound b ∈ N. Lemma Deciding if Player 0 has strategy σ with Cst(σ) ≤ b is in PSpace. Idea: Simulate game, keeping track of open requests. Lemma Player 0 has such a strategy iff she “survives” p(|G|) steps in extended game G 0 . Algorithm: Simulate all plays in G 0 on-the-fly for p(|G|) steps using an alternating Turing machine. ⇒ Problem is in APTime (Chandra et al., Alternation, 1981) ⇒ Problem is in PSpace Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 6/12 From Finitary Parity to Parity Given: Finitary parity game G = (A, FinParity(Ω)), bound b ∈ N. Lemma Deciding if Player 0 has strategy σ with Cst(σ) ≤ b is in PSpace. Idea: Simulate game, keeping track of open requests. Lemma Player 0 has such a strategy iff she “survives” p(|G|) steps in extended game G 0 . Algorithm: Simulate all plays in G 0 on-the-fly for p(|G|) steps using an alternating Turing machine. ⇒ Problem is in APTime (Chandra et al., Alternation, 1981) ⇒ Problem is in PSpace Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 6/12 From Finitary Parity to Parity Given: Finitary parity game G = (A, FinParity(Ω)), bound b ∈ N. Lemma Deciding if Player 0 has strategy σ with Cst(σ) ≤ b is in PSpace. Idea: Simulate game, keeping track of open requests. Lemma Player 0 has such a strategy iff she “survives” p(|G|) steps in extended game G 0 . Algorithm: Simulate all plays in G 0 on-the-fly for p(|G|) steps using an alternating Turing machine. ⇒ Problem is in APTime (Chandra et al., Alternation, 1981) ⇒ Problem is in PSpace Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 6/12 From Finitary Parity to Parity Given: Finitary parity game G = (A, FinParity(Ω)), bound b ∈ N. Lemma Deciding if Player 0 has strategy σ with Cst(σ) ≤ b is in PSpace. Idea: Simulate game, keeping track of open requests. Lemma Player 0 has such a strategy iff she “survives” p(|G|) steps in extended game G 0 . Algorithm: Simulate all plays in G 0 on-the-fly for p(|G|) steps using an alternating Turing machine. ⇒ Problem is in APTime (Chandra et al., Alternation, 1981) ⇒ Problem is in PSpace Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 6/12 From Finitary Parity to Parity Given: Finitary parity game G = (A, FinParity(Ω)), bound b ∈ N. Lemma Deciding if Player 0 has strategy σ with Cst(σ) ≤ b is in PSpace. Idea: Simulate game, keeping track of open requests. Lemma Player 0 has such a strategy iff she “survives” p(|G|) steps in extended game G 0 . Algorithm: Simulate all plays in G 0 on-the-fly for p(|G|) steps using an alternating Turing machine. ⇒ Problem is in APTime (Chandra et al., Alternation, 1981) ⇒ Problem is in PSpace Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 6/12 Introduction 3 Complexity in PSpace PSpace-hard Exponential Memory Sufficient Alexander Weinert Saarland University Necessary Optimal Strategies in Parity Games with Costs 7/12 Introduction 3 Complexity in PSpace 3 PSpace-hard Exponential Memory Sufficient Alexander Weinert Saarland University Necessary Optimal Strategies in Parity Games with Costs 7/12 PSPACE-completeness Lemma The given decision problem is PSpace-hard. Idea: Reduction from Quantified Boolean Formulas, e.g.: ∀x ∃y . ( x ∨ ¬y ) ∧ ( ¬x ∨ y ) y x 0 1 0 0 5 0 0 3 0 0 7 0 x̄ Alexander Weinert Saarland University ȳ Optimal Strategies in Parity Games with Costs 8/12 PSPACE-completeness Lemma The given decision problem is PSpace-hard. Idea: Reduction from Quantified Boolean Formulas, e.g.: ∀x ∃y . ( x ∨ ¬y ) ∧ ( ¬x ∨ y ) y x 0 1 0 0 5 0 0 3 0 0 7 0 x̄ Alexander Weinert Saarland University ȳ Optimal Strategies in Parity Games with Costs 8/12 PSPACE-completeness Lemma The given decision problem is PSpace-hard. Idea: Reduction from Quantified Boolean Formulas, e.g.: ∀x ∃y . ( x ∨ ¬y ) ∧ ( ¬x ∨ y ) y x 0 1 0 0 5 0 0 3 0 0 7 0 x̄ Alexander Weinert Saarland University ȳ Optimal Strategies in Parity Games with Costs 8/12 PSPACE-completeness Lemma The given decision problem is PSpace-hard. Idea: Reduction from Quantified Boolean Formulas, e.g.: ∀x ∃y . ( x ∨ ¬y ) ∧ ( ¬x ∨ y ) y x 0 1 0 0 5 0 0 3 0 0 7 0 x̄ Alexander Weinert Saarland University ȳ Optimal Strategies in Parity Games with Costs 8/12 PSPACE-completeness Lemma The given decision problem is PSpace-hard. Idea: Reduction from Quantified Boolean Formulas, e.g.: ∀x ∃y . ( x ∨ ¬y ) ∧ ( ¬x ∨ y ) y x 0 1 0 0 5 0 0 3 0 0 7 0 x̄ Alexander Weinert Saarland University ȳ Optimal Strategies in Parity Games with Costs 8/12 PSPACE-completeness Lemma The given decision problem is PSpace-hard. Idea: Reduction from Quantified Boolean Formulas, e.g.: ∀x ∃y . ( x ∨ ¬y ) ∧ ( ¬x ∨ y ) x x 0 1 0 0 3 0 x̄ 5 0 0 7 0 0 4 6 0 0 8 ȳ ȳ Alexander Weinert 0 y x̄ y 0 2 Saarland University Optimal Strategies in Parity Games with Costs 8/12 PSPACE-completeness Lemma The given decision problem is PSpace-hard. Idea: Reduction from Quantified Boolean Formulas, e.g.: ∀x ∃y . ( x ∨ ¬y ) ∧ ( ¬x ∨ y ) x x 0 1 0 0 3 0 x̄ 5 0 0 7 0 0 4 6 0 0 8 ȳ ȳ Alexander Weinert 0 y x̄ y 0 2 Saarland University Optimal Strategies in Parity Games with Costs 8/12 PSPACE-completeness Lemma The given decision problem is PSpace-hard. Idea: Reduction from Quantified Boolean Formulas, e.g.: ∀x ∃y . ( x ∨ ¬y ) ∧ ( ¬x ∨ y ) x x 0 1 0 0 3 0 x̄ 5 0 0 7 0 0 4 6 0 0 8 ȳ ȳ Alexander Weinert 0 y x̄ y 0 2 Saarland University Optimal Strategies in Parity Games with Costs 8/12 PSPACE-completeness Lemma The given decision problem is PSpace-hard. Idea: Reduction from Quantified Boolean Formulas, e.g.: ∀x ∃y . ( x ∨ ¬y ) ∧ ( ¬x ∨ y ) x x 0 1 0 0 3 0 x̄ 5 0 0 7 0 0 4 6 0 0 8 ȳ ȳ Alexander Weinert 0 y x̄ y 0 2 Saarland University Optimal Strategies in Parity Games with Costs 8/12 PSPACE-completeness Lemma The given decision problem is PSpace-hard. Idea: Reduction from Quantified Boolean Formulas, e.g.: ∀x ∃y . ( x ∨ ¬y ) ∧ ( ¬x ∨ y ) x x 0 1 0 0 3 0 x̄ 5 0 0 7 0 0 4 6 0 0 8 ȳ ȳ Alexander Weinert 0 y x̄ y 0 2 Saarland University Optimal Strategies in Parity Games with Costs 8/12 PSPACE-completeness Lemma The given decision problem is PSpace-hard. Idea: Reduction from Quantified Boolean Formulas, e.g.: ∀x ∃y . ( x ∨ ¬y ) ∧ ( ¬x ∨ y ) (x ∨ ¬y ) 0 x 0 1 0 0 3 0 0 x x̄ x̄ y 0 0 5 7 0 0 0 (¬x ∨ y ) ȳ Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 2 0 0 4 6 0 0 8 y ȳ 8/12 PSPACE-completeness Lemma The given decision problem is PSpace-hard. Idea: Reduction from Quantified Boolean Formulas, e.g.: ∀x ∃y . ( x ∨ ¬y ) ∧ ( ¬x ∨ y ) (x ∨ ¬y ) 0 x 0 1 0 0 3 0 0 x x̄ x̄ y 0 0 5 7 0 0 0 (¬x ∨ y ) ȳ Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 2 0 0 4 6 0 0 8 y ȳ 8/12 PSPACE-completeness Lemma The given decision problem is PSpace-hard. Idea: Reduction from Quantified Boolean Formulas, e.g.: ∀x ∃y . ( x ∨ ¬y ) ∧ ( ¬x ∨ y ) x 0 0 1 3 0 (x ∨ ¬y ) 0 ... 0 ... x̄ 0 y ... 0 5 0 0 7 0 0 (¬x ∨ y ) ... x x̄ 2 0 0 4 ··· y 6 0 0 8 ȳ ȳ Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 8/12 PSPACE-completeness Lemma The given decision problem is PSpace-hard. Idea: Reduction from Quantified Boolean Formulas, e.g.: ∀x ∃y . ( x ∨ ¬y ) ∧ ( ¬x ∨ y ) x 0 0 1 3 0 (x ∨ ¬y ) 0 ... 0 ... x̄ 0 y ... 0 5 0 0 7 0 0 (¬x ∨ y ) ... x x̄ 2 0 0 4 ··· y 6 0 0 8 ȳ ȳ Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 8/12 PSPACE-completeness Lemma The given decision problem is PSpace-hard. Idea: Reduction from Quantified Boolean Formulas, e.g.: ∀x ∃y . ( x ∨ ¬y ) ∧ ( ¬x ∨ y ) x 0 0 1 3 0 (x ∨ ¬y ) 0 ... 0 ... x̄ 0 y ... 0 5 0 0 7 0 0 (¬x ∨ y ) ... x x̄ 2 0 0 4 ··· y 6 0 0 8 ȳ ȳ Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 8/12 PSPACE-completeness Lemma The given decision problem is PSpace-hard. Idea: Reduction from Quantified Boolean Formulas, e.g.: ∀x ∃y . ( x ∨ ¬y ) ∧ ( ¬x ∨ y ) x 0 0 1 3 0 (x ∨ ¬y ) 0 ... 0 ... x̄ 0 y ... 0 5 0 0 7 0 0 (¬x ∨ y ) ... x x̄ 2 0 0 4 ··· y 6 0 0 8 ȳ ȳ Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 8/12 Introduction 3 Complexity in PSpace 3 PSpace-hard Exponential Memory Sufficient Alexander Weinert Saarland University Necessary Optimal Strategies in Parity Games with Costs 9/12 Introduction 3 Complexity in PSpace 3 PSpace-hard 3 Exponential Memory Sufficient Alexander Weinert Saarland University Necessary Optimal Strategies in Parity Games with Costs 9/12 Memory Requirements (for Player 0) Theorem Optimal strategies for parity games require exponential memory. Sufficiency: Corollary of proof of PSpace-membership Necessity: Construct family Gd : 0 0 0 (Fijalkow and Chatterjee, Infinite-state games, 2013) Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 10/12 Memory Requirements (for Player 0) Theorem Optimal strategies for parity games require exponential memory. Sufficiency: Corollary of proof of PSpace-membership Necessity: Construct family Gd : 0 0 0 (Fijalkow and Chatterjee, Infinite-state games, 2013) Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 10/12 Memory Requirements (for Player 0) Theorem Optimal strategies for parity games require exponential memory. Sufficiency: Corollary of proof of PSpace-membership Necessity: Construct family Gd : 0 0 0 (Fijalkow and Chatterjee, Infinite-state games, 2013) Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 10/12 Memory Requirements (for Player 0) Theorem Optimal strategies for parity games require exponential memory. Sufficiency: Corollary of proof of PSpace-membership Necessity: Construct family Gd : 1 2 0 0 0 0 0 3 4 (Fijalkow and Chatterjee, Infinite-state games, 2013) Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 10/12 Memory Requirements (for Player 0) Theorem Optimal strategies for parity games require exponential memory. Sufficiency: Corollary of proof of PSpace-membership Necessity: Construct family Gd : 1 2 0 0 0 3 4 0 0 0 0 5 6 (Fijalkow and Chatterjee, Infinite-state games, 2013) Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 10/12 Memory Requirements (for Player 0) Theorem Optimal strategies for parity games require exponential memory. Sufficiency: Corollary of proof of PSpace-membership Necessity: Construct family Gd : 1 2 0 0 .. . ··· .. . 0 0 d 0 ··· d +1 (Fijalkow and Chatterjee, Infinite-state games, 2013) Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 10/12 Memory Requirements (for Player 0) Theorem Optimal strategies for parity games require exponential memory. Sufficiency: Corollary of proof of PSpace-membership Necessity: Construct family Gd : 1 2 0 G1 0 .. . ··· .. . 0 0 d Alexander Weinert Saarland University 0 G0 ··· d +1 Optimal Strategies in Parity Games with Costs 10/12 Memory Requirements (for Player 0) Theorem Optimal strategies for parity games require exponential memory. Sufficiency: Corollary of proof of PSpace-membership Necessity: Construct family Gd : G1 ··· d times G1 G0 ··· G0 d times Player 0 needs to store d choices of d possible values each ⇒ Player 0 requires ≈ 2d many memory states Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 10/12 Memory Requirements (for Player 0) Theorem Optimal strategies for parity games require exponential memory. Sufficiency: Corollary of proof of PSpace-membership Necessity: Construct family Gd : G1 ··· d times G1 G0 ··· G0 d times Player 0 needs to store d choices of d possible values each ⇒ Player 0 requires ≈ 2d many memory states Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 10/12 Memory Requirements (for Player 0) Theorem Optimal strategies for parity games require exponential memory. Sufficiency: Corollary of proof of PSpace-membership Necessity: Construct family Gd : G1 ··· d times G1 G0 ··· G0 d times Player 0 needs to store d choices of d possible values each ⇒ Player 0 requires ≈ 2d many memory states Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 10/12 Memory Requirements (for Player 0) Theorem Optimal strategies for parity games require exponential memory. Sufficiency: Corollary of proof of PSpace-membership Necessity: Construct family Gd : G1 ··· d times G1 G0 ··· G0 d times Player 0 needs to store d choices of d possible values each ⇒ Player 0 requires ≈ 2d many memory states Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 10/12 Introduction 3 Complexity in PSpace 3 PSpace-hard 3 Exponential Memory Sufficient Alexander Weinert Saarland University Necessary Optimal Strategies in Parity Games with Costs 11/12 Introduction 3 Complexity in PSpace 3 PSpace-hard 3 Exponential Memory Sufficient 3 Alexander Weinert Saarland University Necessary 3 Optimal Strategies in Parity Games with Costs 11/12 Conclusion Parity Complexity Strategies UP ∩ co-UP 1 Finitary Parity Winning Optimal PTime 1 PSpace-comp. Exp. Take-away: Forcing Player 0 to answer quickly in parity games makes it harder to decide whether she can satisfy the bound for her to play the game Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 12/12 Conclusion Parity Complexity Strategies UP ∩ co-UP 1 Finitary Parity Winning Optimal PTime 1 PSpace-comp. Exp. Take-away: Forcing Player 0 to answer quickly in parity games makes it harder to decide whether she can satisfy the bound for her to play the game Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 12/12 Conclusion Parity Complexity Strategies UP ∩ co-UP 1 Finitary Parity Winning Optimal PTime 1 PSpace-comp. Exp. Take-away: Forcing Player 0 to answer quickly in parity games makes it harder to decide whether she can satisfy the bound for her to play the game Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 12/12 Conclusion Parity Complexity Strategies UP ∩ co-UP 1 Finitary Parity Winning Optimal PTime 1 PSpace-comp. Exp. Take-away: Forcing Player 0 to answer quickly in parity games makes it harder to decide whether she can satisfy the bound for her to play the game Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 12/12 Introduction 3 Complexity in PSpace 3 PSpace-hard 3 Exponential Memory Necessary 3 Sufficient 3 Tradeoffs Parity Games with Costs Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 1/5 Introduction 3 Complexity in PSpace 3 PSpace-hard 3 Exponential Memory Necessary 3 Sufficient 3 Tradeoffs Parity Games with Costs Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 1/5 Introduction 3 Complexity in PSpace 3 PSpace-hard 3 Exponential Memory Necessary 3 Sufficient 3 Tradeoffs Parity Games with Costs Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 1/5 Tradeoffs G1 ··· G1 G0 ··· Winning Size Cost Alexander Weinert 1 3d Saarland University G0 Optimal d 3d − 1 ··· ··· 2d−1 2d + 1 Optimal Strategies in Parity Games with Costs 2d 2d 2/5 Tradeoffs G1 ··· G1 G0 ··· Winning Size Cost Alexander Weinert 1 3d Saarland University G0 Optimal d 3d − 1 ··· ··· 2d−1 2d + 1 Optimal Strategies in Parity Games with Costs 2d 2d 2/5 Tradeoffs G1 ··· G1 G0 ··· Winning Size Cost Alexander Weinert 1 3d Saarland University G0 Optimal d 3d − 1 ··· ··· 2d−1 2d + 1 Optimal Strategies in Parity Games with Costs 2d 2d 2/5 Tradeoffs G1 ··· G1 G0 ··· Winning Size Cost Alexander Weinert 1 3d Saarland University G0 Optimal d 3d − 1 ··· ··· 2d−1 2d + 1 Optimal Strategies in Parity Games with Costs 2d 2d 2/5 Tradeoffs G1 ··· G1 G0 ··· Winning Size Cost Alexander Weinert 1 3d Saarland University G0 Optimal d 3d − 1 ··· ··· 2d−1 2d + 1 Optimal Strategies in Parity Games with Costs 2d 2d 2/5 Tradeoffs G1 ··· G1 G0 ··· Winning Size Cost Alexander Weinert 1 3d Saarland University G0 Optimal d 3d − 1 ··· ··· 2d−1 2d + 1 Optimal Strategies in Parity Games with Costs 2d 2d 2/5 Tradeoffs G1 ··· G1 G0 ··· Winning Size Cost Alexander Weinert 1 3d Saarland University G0 Optimal d 3d − 1 ··· ··· 2d−1 2d + 1 Optimal Strategies in Parity Games with Costs 2d 2d 2/5 Introduction 3 Complexity in PSpace 3 PSpace-hard 3 Exponential Memory Necessary 3 Sufficient 3 Tradeoffs 3 Parity Games with Costs Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 3/5 Introduction 3 Complexity in PSpace 3 PSpace-hard 3 Exponential Memory Necessary 3 Sufficient 3 Tradeoffs 3 Parity Games with Costs Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 3/5 Parity Games with Cost 0 1 1 0 0 2 1 1 0 0 0 3 0 1 1 0 1 4 0 0 Finitary parity games are special case ⇒ PSpace-hard ⇒ Exp. memory necessary Algorithm for solving finitary games works as well ⇒ In PSpace Alexander Weinert Saarland University ⇒ Exp. memory sufficient Optimal Strategies in Parity Games with Costs 4/5 Parity Games with Cost 0 1 1 0 0 2 1 1 0 0 0 3 0 1 1 0 1 4 0 0 Finitary parity games are special case ⇒ PSpace-hard ⇒ Exp. memory necessary Algorithm for solving finitary games works as well ⇒ In PSpace Alexander Weinert Saarland University ⇒ Exp. memory sufficient Optimal Strategies in Parity Games with Costs 4/5 Parity Games with Cost 0 1 1 0 0 2 1 1 0 0 0 3 0 1 1 0 1 4 0 0 Finitary parity games are special case ⇒ PSpace-hard ⇒ Exp. memory necessary Algorithm for solving finitary games works as well ⇒ In PSpace Alexander Weinert Saarland University ⇒ Exp. memory sufficient Optimal Strategies in Parity Games with Costs 4/5 Parity Games with Cost 0 1 1 0 0 2 1 1 0 0 0 3 0 1 1 0 1 4 0 0 Finitary parity games are special case ⇒ PSpace-hard ⇒ Exp. memory necessary Algorithm for solving finitary games works as well ⇒ In PSpace Alexander Weinert Saarland University ⇒ Exp. memory sufficient Optimal Strategies in Parity Games with Costs 4/5 Parity Games with Cost 0 1 1 0 0 2 1 1 0 0 0 3 0 1 1 0 1 4 0 0 Finitary parity games are special case ⇒ PSpace-hard ⇒ Exp. memory necessary Algorithm for solving finitary games works as well ⇒ In PSpace Alexander Weinert Saarland University ⇒ Exp. memory sufficient Optimal Strategies in Parity Games with Costs 4/5 Parity Games with Cost 0 1 1 0 0 2 1 1 0 0 0 3 0 1 1 0 1 4 0 0 Finitary parity games are special case ⇒ PSpace-hard ⇒ Exp. memory necessary Algorithm for solving finitary games works as well ⇒ In PSpace Alexander Weinert Saarland University ⇒ Exp. memory sufficient Optimal Strategies in Parity Games with Costs 4/5 Parity Games with Cost 0 1 1 0 0 2 1 1 0 0 0 3 0 1 1 0 1 4 0 0 Finitary parity games are special case ⇒ PSpace-hard ⇒ Exp. memory necessary Algorithm for solving finitary games works as well ⇒ In PSpace Alexander Weinert Saarland University ⇒ Exp. memory sufficient Optimal Strategies in Parity Games with Costs 4/5 Conclusion Parity Complexity Strategies Cost-Parity UP ∩ co-UP 1 Winning Optimal UP ∩ co-UP 1 PSpace-comp. Exp. Take-away: Forcing Player 0 to answer quickly in parity games makes it harder to decide whether she can satisfy the bound for her to play the game Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 5/5 Conclusion Cost-Parity Parity Complexity Strategies UP ∩ co-UP 1 Winning Optimal UP ∩ co-UP 1 PSpace-comp. Exp. Take-away: Forcing Player 0 to answer quickly in parity games makes it harder to decide whether she can satisfy the bound for her to play the game Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 5/5 Conclusion Cost-Parity Parity Complexity Strategies UP ∩ co-UP 1 Winning Optimal UP ∩ co-UP 1 PSpace-comp. Exp. Take-away: Forcing Player 0 to answer quickly in parity games makes it harder to decide whether she can satisfy the bound for her to play the game Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 5/5 Conclusion Cost-Parity Parity Complexity Strategies UP ∩ co-UP 1 Winning Optimal UP ∩ co-UP 1 PSpace-comp. Exp. Take-away: Forcing Player 0 to answer quickly in parity games makes it harder to decide whether she can satisfy the bound for her to play the game Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 5/5

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