Easy to Win, Hard to Master:
Optimal Strategies in Parity Games with Costs
Joint work with Martin Zimmermann
Alexander Weinert
Saarland University
September 9th, 2016
Highlights 2016 - Brussels
Alexander Weinert
Saarland University
Optimal Strategies in Parity Games with Costs
1/8
Parity Games
1
2
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1
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3
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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
2/8
Parity Games
1
2
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1
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3
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0
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0
4
0
0
0
4
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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
2/8
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
2/8
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
2/8
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
2/8
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
2/8
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
2/8
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
2/8
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
2/8
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
2/8
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
2/8
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
2/8
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
2/8
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
2/8
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
2/8
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
2/8
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
2/8
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
2/8
Finitary Parity Games
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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
3/8
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
3/8
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
3/8
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
3/8
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
3/8
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
4/8
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
4/8
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.
Lemma
Player 0 has such a strategy iff she “survives” p(|G|) steps.
Algorithm:
Simulate all plays in G 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
5/8
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.
Lemma
Player 0 has such a strategy iff she “survives” p(|G|) steps.
Algorithm:
Simulate all plays in G 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
5/8
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.
Lemma
Player 0 has such a strategy iff she “survives” p(|G|) steps.
Algorithm:
Simulate all plays in G 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
5/8
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.
Lemma
Player 0 has such a strategy iff she “survives” p(|G|) steps.
Algorithm:
Simulate all plays in G 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
5/8
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.
Lemma
Player 0 has such a strategy iff she “survives” p(|G|) steps.
Algorithm:
Simulate all plays in G 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
5/8
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.
Lemma
Player 0 has such a strategy iff she “survives” p(|G|) steps.
Algorithm:
Simulate all plays in G 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
5/8
PSpace-completeness
Lemma
The given decision problem is in PSpace.
Lemma
The given decision problem is PSpace-hard.
Proof: By reduction from True Quantified Boolean Formulas
⇒ The given decision problem is PSpace-complete
Alexander Weinert
Saarland University
Optimal Strategies in Parity Games with Costs
6/8
PSpace-completeness
Lemma
The given decision problem is in PSpace.
Lemma
The given decision problem is PSpace-hard.
Proof: By reduction from True Quantified Boolean Formulas
⇒ The given decision problem is PSpace-complete
Alexander Weinert
Saarland University
Optimal Strategies in Parity Games with Costs
6/8
PSpace-completeness
Lemma
The given decision problem is in PSpace.
Lemma
The given decision problem is PSpace-hard.
Proof: By reduction from True Quantified Boolean Formulas
⇒ The given decision problem is PSpace-complete
Alexander Weinert
Saarland University
Optimal Strategies in Parity Games with Costs
6/8
PSpace-completeness
Lemma
The given decision problem is in PSpace.
Lemma
The given decision problem is PSpace-hard.
Proof: By reduction from True Quantified Boolean Formulas
⇒ The given decision problem is PSpace-complete
Alexander Weinert
Saarland University
Optimal Strategies in Parity Games with Costs
6/8
Memory Requirements (for Player 0)
Theorem
Optimal strategies for Parity Games require exponential memory
Sufficiency: Corollary of proof of PSpace-membership
Necessity:
1
2
0
0
0
0
0
3
For given parameter d:
Generalize to d colors
Repeat d times
Alexander Weinert
Saarland University
4
Player 1 has d choices of d actions
⇒
Player 0 needs ≈ 2d memory states
Optimal Strategies in Parity Games with Costs
7/8
Memory Requirements (for Player 0)
Theorem
Optimal strategies for Parity Games require exponential memory
Sufficiency: Corollary of proof of PSpace-membership
Necessity:
1
2
0
0
0
0
0
3
For given parameter d:
Generalize to d colors
Repeat d times
Alexander Weinert
Saarland University
4
Player 1 has d choices of d actions
⇒
Player 0 needs ≈ 2d memory states
Optimal Strategies in Parity Games with Costs
7/8
Memory Requirements (for Player 0)
Theorem
Optimal strategies for Parity Games require exponential memory
Sufficiency: Corollary of proof of PSpace-membership
Necessity:
1
2
0
0
0
0
0
3
For given parameter d:
Generalize to d colors
Repeat d times
Alexander Weinert
Saarland University
4
Player 1 has d choices of d actions
⇒
Player 0 needs ≈ 2d memory states
Optimal Strategies in Parity Games with Costs
7/8
Memory Requirements (for Player 0)
Theorem
Optimal strategies for Parity Games require exponential memory
Sufficiency: Corollary of proof of PSpace-membership
Necessity:
1
2
0
0
0
0
0
3
For given parameter d:
Generalize to d colors
Repeat d times
Alexander Weinert
Saarland University
4
Player 1 has d choices of d actions
⇒
Player 0 needs ≈ 2d memory states
Optimal Strategies in Parity Games with Costs
7/8
Memory Requirements (for Player 0)
Theorem
Optimal strategies for Parity Games require exponential memory
Sufficiency: Corollary of proof of PSpace-membership
Necessity:
1
2
0
0
0
0
0
3
For given parameter d:
Generalize to d colors
Repeat d times
Alexander Weinert
Saarland University
4
Player 1 has d choices of d actions
⇒
Player 0 needs ≈ 2d memory states
Optimal Strategies in Parity Games with Costs
7/8
Memory Requirements (for Player 0)
Theorem
Optimal strategies for Parity Games require exponential memory
Sufficiency: Corollary of proof of PSpace-membership
Necessity:
1
2
0
0
0
0
0
3
For given parameter d:
Generalize to d colors
Repeat d times
Alexander Weinert
Saarland University
4
Player 1 has d choices of d actions
⇒
Player 0 needs ≈ 2d memory states
Optimal Strategies in Parity Games with Costs
7/8
Conclusion
Parity
Complexity
Strategies
Finitary Parity
UP ∩ co-UP
1
Winning
Optimal
UP ∩ co-UP
1
PSpace-comp.
Exp.
Questions?
Alexander Weinert
Saarland University
Optimal Strategies in Parity Games with Costs
8/8
Conclusion
Finitary Parity
Parity
Complexity
Strategies
UP ∩ co-UP
1
Winning
Optimal
UP ∩ co-UP
1
PSpace-comp.
Exp.
Questions?
Alexander Weinert
Saarland University
Optimal Strategies in Parity Games with Costs
8/8
Conclusion
Finitary Parity
Parity
Complexity
Strategies
UP ∩ co-UP
1
Winning
Optimal
UP ∩ co-UP
1
PSpace-comp.
Exp.
Questions?
Alexander Weinert
Saarland University
Optimal Strategies in Parity Games with Costs
8/8
Conclusion
Finitary Parity
Parity
Complexity
Strategies
UP ∩ co-UP
1
Winning
Optimal
UP ∩ co-UP
1
PSpace-comp.
Exp.
Questions?
Alexander Weinert
Saarland University
Optimal Strategies in Parity Games with Costs
8/8