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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