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Easy to Win, Hard to Master: Optimal Strategies in Parity Games with - - PowerPoint PPT Presentation

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Easy to Win, Hard to Master: Optimal Strategies in Parity Games with Costs Joint work with Martin Zimmermann Alexander Weinert Saarland University December 13th, 2016 MFV Seminar, ULB, Brussels, Belgium Alexander Weinert Saarland University

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SLIDE 1

Easy to Win, Hard to Master:

Optimal Strategies in Parity Games with Costs

Joint work with Martin Zimmermann

Alexander Weinert

Saarland University

December 13th, 2016

MFV Seminar, ULB, Brussels, Belgium

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 1/23

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SLIDE 2

Parity Games

1 3 2 4 1 4 3 4 · · · 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 2/23

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SLIDE 3

Parity Games

1 3 2 4 1 4 3 4 · · · 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 2/23

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SLIDE 4

Parity Games

1 3 2 4 1 4 3 4 · · · 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 2/23

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SLIDE 5

Parity Games

1 3 2 4 1 4 3 4 · · · 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 2/23

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SLIDE 6

Parity Games

1 3 2 4 1 4 3 4 · · · 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 2/23

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SLIDE 7

Parity Games

1 3 2 4 1 4 3 4 · · · 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 2/23

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SLIDE 8

Parity Games

1 3 2 4 1 4 3 4 · · · 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 2/23

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SLIDE 9

Parity Games

1 3 2 4 1 4 3 4 · · · 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 2/23

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SLIDE 10

Parity Games

1 3 2 4 1 4 3 4 · · · 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 2/23

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SLIDE 11

Parity Games

1 3 2 4 1 4 3 4 · · · 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 2/23

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SLIDE 12

Parity Games

1 3 2 4 1 4 3 4 · · · 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 2/23

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SLIDE 13

Parity Games

1 3 2 4 1 4 3 4 · · · 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 2/23

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SLIDE 14

Parity Games

1 3 2 4 1 4 3 4 · · · 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 2/23

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SLIDE 15

Parity Games

1 3 2 4 1 4 3 4 · · · 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 2/23

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SLIDE 16

Parity Games

1 3 2 4 1 4 3 4 · · · 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 2/23

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SLIDE 17

Parity Games

1 3 2 4 1 4 3 4 · · · 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 2/23

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SLIDE 18

Parity Games

1 3 2 4 1 4 3 4 · · · 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 2/23

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SLIDE 19

Parity Games

1 3 2 4 1 4 3 4 · · · 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 2/23

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SLIDE 20

Finitary Parity / Parity Response Games

1 3 2 4 1 4 3 4 · · · 4 steps 3 steps 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 3/23

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SLIDE 21

Finitary Parity / Parity Response Games

1 3 2 4 1 4 3 4 · · · 4 steps 3 steps 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 3/23

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SLIDE 22

Finitary Parity / Parity Response Games

1 3 2 4 1 4 3 4 · · · 4 steps 3 steps 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 3/23

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SLIDE 23

Finitary Parity / Parity Response Games

1 3 2 4 1 4 3 4 · · · 4 steps 3 steps 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 3/23

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SLIDE 24

Finitary Parity / Parity Response Games

1 3 2 4 1 4 3 4 · · · 4 steps 3 steps 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 3/23

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SLIDE 25

Another Example

1 2

1 2 1 2 1 2 · · ·

Parity ✓ Finitary Parity ✓

1 2 1 2 1 2 · · ·

Parity ✓ Finitary Parity ✗ Player 1 wins from every vertex, but needs to stay longer and longer in vertex of color 0 ⇒ requires infinite memory

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 4/23

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SLIDE 26

Another Example

1 2

1 2 1 2 1 2 · · ·

Parity ✓ Finitary Parity ✓

1 2 1 2 1 2 · · ·

Parity ✓ Finitary Parity ✗ Player 1 wins from every vertex, but needs to stay longer and longer in vertex of color 0 ⇒ requires infinite memory

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 4/23

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SLIDE 27

Another Example

1 2

1 2 1 2 1 2 · · ·

Parity ✓ Finitary Parity ✓

1 2 1 2 1 2 · · ·

Parity ✓ Finitary Parity ✗ Player 1 wins from every vertex, but needs to stay longer and longer in vertex of color 0 ⇒ requires infinite memory

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 4/23

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SLIDE 28

Another Example

1 2

1 2 1 2 1 2 · · ·

Parity ✓ Finitary Parity ✓

1 2 1 2 1 2 · · ·

Parity ✓ Finitary Parity ✗ Player 1 wins from every vertex, but needs to stay longer and longer in vertex of color 0 ⇒ requires infinite memory

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 4/23

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SLIDE 29

Another Example

1 2

1 2 1 2 1 2 · · ·

Parity ✓ Finitary Parity ✓

1 2 1 2 1 2 · · ·

Parity ✓ Finitary Parity ✗ Player 1 wins from every vertex, but needs to stay longer and longer in vertex of color 0 ⇒ requires infinite memory

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 4/23

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SLIDE 30

Another Example

1 2

1 2 1 2 1 2 · · ·

Parity ✓ Finitary Parity ✓

1 2 1 2 1 2 · · ·

Parity ✓ Finitary Parity ✗ Player 1 wins from every vertex, but needs to stay longer and longer in vertex of color 0 ⇒ requires infinite memory

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 4/23

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SLIDE 31

Another Example

1 2

1 2 1 2 1 2 · · ·

Parity ✓ Finitary Parity ✓

1 2 1 2 1 2 · · ·

Parity ✓ Finitary Parity ✗ Player 1 wins from every vertex, but needs to stay longer and longer in vertex of color 0 ⇒ requires infinite memory

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 4/23

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SLIDE 32

Another Example

1 2

1 2 1 2 1 2 · · ·

Parity ✓ Finitary Parity ✓

1 2 1 2 1 2 · · ·

Parity ✓ Finitary Parity ✗ Player 1 wins from every vertex, but needs to stay longer and longer in vertex of color 0 ⇒ requires infinite memory

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 4/23

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SLIDE 33

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 5/23

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SLIDE 34

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 5/23

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SLIDE 35

Introduction ✓ Complexity in PSpace Exponential Memory Extensions

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 6/23

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SLIDE 36

Introduction ✓ Complexity in PSpace Exponential Memory Extensions

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 6/23

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SLIDE 37

Introduction ✓ Complexity in PSpace Exponential Memory Extensions

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 6/23

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SLIDE 38

Introduction ✓ Complexity in PSpace PSpace-hard Exponential Memory Extensions

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 6/23

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SLIDE 39

Introduction ✓ Complexity in PSpace PSpace-hard Sufficient Exponential Memory Extensions

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 6/23

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SLIDE 40

Introduction ✓ Complexity in PSpace PSpace-hard Sufficient Exponential Memory Extensions

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 6/23

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SLIDE 41

Introduction ✓ Complexity in PSpace PSpace-hard Sufficient Exponential Memory Necessary Extensions

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 6/23

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SLIDE 42

Introduction ✓ Complexity in PSpace PSpace-hard Sufficient Exponential Memory Necessary Tradeoffs Extensions

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 6/23

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SLIDE 43

Introduction ✓ Complexity in PSpace PSpace-hard Sufficient Exponential Memory Necessary Tradeoffs Extensions

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 6/23

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SLIDE 44

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 G, keeping track of open requests explicitly. Result: Parity game G′ of exponential size.

Lemma

The winner of a play in G′ can be decided after 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 7/23

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SLIDE 45

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 G, keeping track of open requests explicitly. Result: Parity game G′ of exponential size.

Lemma

The winner of a play in G′ can be decided after 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 7/23

slide-46
SLIDE 46

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 G, keeping track of open requests explicitly. Result: Parity game G′ of exponential size.

Lemma

The winner of a play in G′ can be decided after 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 7/23

slide-47
SLIDE 47

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 G, keeping track of open requests explicitly. Result: Parity game G′ of exponential size.

Lemma

The winner of a play in G′ can be decided after 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 7/23

slide-48
SLIDE 48

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 G, keeping track of open requests explicitly. Result: Parity game G′ of exponential size.

Lemma

The winner of a play in G′ can be decided after 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 7/23

slide-49
SLIDE 49

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 G, keeping track of open requests explicitly. Result: Parity game G′ of exponential size.

Lemma

The winner of a play in G′ can be decided after 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 7/23

slide-50
SLIDE 50

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 G, keeping track of open requests explicitly. Result: Parity game G′ of exponential size.

Lemma

The winner of a play in G′ can be decided after 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 7/23

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SLIDE 51

Introduction ✓ Complexity in PSpace PSpace-hard Sufficient Exponential Memory Necessary Tradeoffs Extensions

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 8/23

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SLIDE 52

Introduction ✓ Complexity in PSpace PSpace-hard Sufficient Exponential Memory Necessary Tradeoffs Extensions ✓

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 8/23

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SLIDE 53

PSPACE-Hardness

Lemma

The following problem is PSpace-hard: “Given a finitary parity game G and a bound b ∈ N, does Player 0 have a strategy σ for G with Cst(σ) ≤ b?” Proof By reduction from QBF Checking the truth of ϕ = ∀x∃y. (x ∨ ¬y) ∧ (¬x ∨ y) as a two-player game (Player 0 wants to prove truth of ϕ):

  • 1. Player 1 picks truth value for x
  • 2. Player 0 picks truth value for y
  • 3. Player 1 picks clause C
  • 4. Player 0 picks literal ℓ from C
  • 5. Player 0 wins ⇔ ℓ is picked to be satisfied in step 1 or 2

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 9/23

slide-54
SLIDE 54

PSPACE-Hardness

Lemma

The following problem is PSpace-hard: “Given a finitary parity game G and a bound b ∈ N, does Player 0 have a strategy σ for G with Cst(σ) ≤ b?” Proof By reduction from QBF Checking the truth of ϕ = ∀x∃y. (x ∨ ¬y) ∧ (¬x ∨ y) as a two-player game (Player 0 wants to prove truth of ϕ):

  • 1. Player 1 picks truth value for x
  • 2. Player 0 picks truth value for y
  • 3. Player 1 picks clause C
  • 4. Player 0 picks literal ℓ from C
  • 5. Player 0 wins ⇔ ℓ is picked to be satisfied in step 1 or 2

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 9/23

slide-55
SLIDE 55

PSPACE-Hardness

Lemma

The following problem is PSpace-hard: “Given a finitary parity game G and a bound b ∈ N, does Player 0 have a strategy σ for G with Cst(σ) ≤ b?” Proof By reduction from QBF Checking the truth of ϕ = ∀x∃y. (x ∨ ¬y) ∧ (¬x ∨ y) as a two-player game (Player 0 wants to prove truth of ϕ):

  • 1. Player 1 picks truth value for x
  • 2. Player 0 picks truth value for y
  • 3. Player 1 picks clause C
  • 4. Player 0 picks literal ℓ from C
  • 5. Player 0 wins ⇔ ℓ is picked to be satisfied in step 1 or 2

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 9/23

slide-56
SLIDE 56

The Reduction

ϕ = ∀x ∃y .

ψ

  • ( x ∨ ¬y ) ∧ ( ¬x ∨ y )

1 x 3 ¬x 5 y 7 ¬y ψ (x ∨ ¬y) (¬x ∨ y) 2 x ¬y 8 ¬x 4 6 y . . . . . . . . . . . . 10

Choose bound b such that it enforces the following:

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 10/23

slide-57
SLIDE 57

The Reduction

ϕ = ∀x ∃y .

ψ

  • ( x ∨ ¬y ) ∧ ( ¬x ∨ y )

1 x 3 ¬x 5 y 7 ¬y ψ (x ∨ ¬y) (¬x ∨ y) 2 x ¬y 8 ¬x 4 6 y . . . . . . . . . . . . 10

Choose bound b such that it enforces the following:

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 10/23

slide-58
SLIDE 58

The Reduction

ϕ = ∀x ∃y .

ψ

  • ( x ∨ ¬y ) ∧ ( ¬x ∨ y )

1 x 3 ¬x 5 y 7 ¬y ψ (x ∨ ¬y) (¬x ∨ y) 2 x ¬y 8 ¬x 4 6 y . . . . . . . . . . . . 10

Choose bound b such that it enforces the following:

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 10/23

slide-59
SLIDE 59

The Reduction

ϕ = ∀x ∃y .

ψ

  • ( x ∨ ¬y ) ∧ ( ¬x ∨ y )

1 x 3 ¬x 5 y 7 ¬y ψ (x ∨ ¬y) (¬x ∨ y) 2 x ¬y 8 ¬x 4 6 y . . . . . . . . . . . . 10

Choose bound b such that it enforces the following:

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 10/23

slide-60
SLIDE 60

The Reduction

ϕ = ∀x ∃y .

ψ

  • ( x ∨ ¬y ) ∧ ( ¬x ∨ y )

1 x 3 ¬x 5 y 7 ¬y ψ (x ∨ ¬y) (¬x ∨ y) 2 x ¬y 8 ¬x 4 6 y . . . . . . . . . . . . 10

Choose bound b such that it enforces the following:

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 10/23

slide-61
SLIDE 61

The Reduction

ϕ = ∀x ∃y .

ψ

  • ( x ∨ ¬y ) ∧ ( ¬x ∨ y )

1 x 3 ¬x 5 y 7 ¬y ψ (x ∨ ¬y) (¬x ∨ y) 2 x ¬y 8 ¬x 4 6 y . . . . . . . . . . . . 10

Choose bound b such that it enforces the following:

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 10/23

slide-62
SLIDE 62

The Reduction

ϕ = ∀x ∃y .

ψ

  • ( x ∨ ¬y ) ∧ ( ¬x ∨ y )

1 x 3 ¬x 5 y 7 ¬y ψ (x ∨ ¬y) (¬x ∨ y) 2 x ¬y 8 ¬x 4 6 y . . . . . . . . . . . . 10

Choose bound b such that it enforces the following:

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 10/23

slide-63
SLIDE 63

The Reduction

ϕ = ∀x ∃y .

ψ

  • ( x ∨ ¬y ) ∧ ( ¬x ∨ y )

1 x 3 ¬x 5 y 7 ¬y ψ (x ∨ ¬y) (¬x ∨ y) 2 x ¬y 8 ¬x 4 6 y . . . . . . . . . . . . 10

Choose bound b such that it enforces the following:

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 10/23

slide-64
SLIDE 64

The Reduction

ϕ = ∀x ∃y .

ψ

  • ( x ∨ ¬y ) ∧ ( ¬x ∨ y )

1 x 3 ¬x 5 y 7 ¬y ψ (x ∨ ¬y) (¬x ∨ y) 2 x ¬y 8 ¬x 4 6 y . . . . . . . . . . . . 10

Choose bound b such that it enforces the following:

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 10/23

slide-65
SLIDE 65

The Reduction

ϕ = ∀x ∃y .

ψ

  • ( x ∨ ¬y ) ∧ ( ¬x ∨ y )

1 x 3 ¬x 5 y 7 ¬y ψ (x ∨ ¬y) (¬x ∨ y) 2 x ¬y 8 ¬x 4 6 y . . . . . . . . . . . . 10

Choose bound b such that it enforces the following:

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 10/23

slide-66
SLIDE 66

The Reduction

ϕ = ∀x ∃y .

ψ

  • ( x ∨ ¬y ) ∧ ( ¬x ∨ y )

1 x 3 ¬x 5 y 7 ¬y ψ (x ∨ ¬y) (¬x ∨ y) 2 x ¬y 8 ¬x 4 6 y . . . . . . . . . . . . 10

Choose bound b such that it enforces the following:

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 10/23

slide-67
SLIDE 67

The Reduction

ϕ = ∀x ∃y .

ψ

  • ( x ∨ ¬y ) ∧ ( ¬x ∨ y )

1 x 3 ¬x 5 y 7 ¬y ψ (x ∨ ¬y) (¬x ∨ y) 2 x ¬y 8 ¬x 4 6 y . . . . . . . . . . . . 10

Choose bound b such that it enforces the following:

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 10/23

slide-68
SLIDE 68

The Reduction

ϕ = ∀x ∃y .

ψ

  • ( x ∨ ¬y ) ∧ ( ¬x ∨ y )

1 x 3 ¬x 5 y 7 ¬y ψ (x ∨ ¬y) (¬x ∨ y) 2 x ¬y 8 ¬x 4 6 y . . . . . . . . . . . . 10

Choose bound b such that it enforces the following:

1 x 3 y · · · 2 x 10 b steps

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 10/23

slide-69
SLIDE 69

The Reduction

ϕ = ∀x ∃y .

ψ

  • ( x ∨ ¬y ) ∧ ( ¬x ∨ y )

1 x 3 ¬x 5 y 7 ¬y ψ (x ∨ ¬y) (¬x ∨ y) 2 x ¬y 8 ¬x 4 6 y . . . . . . . . . . . . 10

Choose bound b such that it enforces the following:

1 x 3 y · · · 4 ¬x 10 b steps

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 10/23

slide-70
SLIDE 70

The Reduction

ϕ = ∀x ∃y .

ψ

  • ( x ∨ ¬y ) ∧ ( ¬x ∨ y )

1 x 3 ¬x 5 y 7 ¬y ψ (x ∨ ¬y) (¬x ∨ y) 2 x ¬y 8 ¬x 4 6 y . . . . . . . . . . . . 10

Choose bound b such that it enforces the following:

3 ¬x 3 y · · · 4 ¬x 10 b steps

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 10/23

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SLIDE 71

The Reduction

ϕ = ∀x ∃y .

ψ

  • ( x ∨ ¬y ) ∧ ( ¬x ∨ y )

1 x 3 ¬x 5 y 7 ¬y ψ (x ∨ ¬y) (¬x ∨ y) 2 x ¬y 8 ¬x 4 6 y . . . . . . . . . . . . 10

Choose bound b such that it enforces the following:

3 ¬x 3 y · · · 2 x 10 b steps

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 10/23

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SLIDE 72

Introduction ✓ Complexity in PSpace PSpace-hard Sufficient Exponential Memory Necessary Tradeoffs Extensions ✓

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 11/23

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SLIDE 73

Introduction ✓ Complexity in PSpace PSpace-hard Sufficient Exponential Memory Necessary Tradeoffs Extensions ✓ ✓

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 11/23

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SLIDE 74

Sufficient Memory (for Player 0)

Corollary

Let G be a parity game with costs with d odd colors. If Player 0 has a strategy σ for G with Cst(σ) = b, then she also has a strategy σ′ with Cst(σ′) = b and size (b + 2)d = 2d log(b+2). Follows from proof of PSpace-membership and positional strategies for parity games.

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 12/23

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SLIDE 75

Sufficient Memory (for Player 0)

Corollary

Let G be a parity game with costs with d odd colors. If Player 0 has a strategy σ for G with Cst(σ) = b, then she also has a strategy σ′ with Cst(σ′) = b and size (b + 2)d = 2d log(b+2). Follows from proof of PSpace-membership and positional strategies for parity games.

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 12/23

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SLIDE 76

Memory Requirements (for Player 0)

Theorem

Optimal strategies for parity games require exponential memory. Necessity: Construct family Gd:

(Fijalkow and Chatterjee, Infinite-state games, 2013)

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 13/23

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SLIDE 77

Memory Requirements (for Player 0)

Theorem

Optimal strategies for parity games require exponential memory. Necessity: Construct family Gd:

(Fijalkow and Chatterjee, Infinite-state games, 2013)

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 13/23

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SLIDE 78

Memory Requirements (for Player 0)

Theorem

Optimal strategies for parity games require exponential memory. Necessity: Construct family Gd: 1 3 2 4

(Fijalkow and Chatterjee, Infinite-state games, 2013)

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 13/23

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SLIDE 79

Memory Requirements (for Player 0)

Theorem

Optimal strategies for parity games require exponential memory. Necessity: Construct family Gd: 1 3 5 2 4 6

(Fijalkow and Chatterjee, Infinite-state games, 2013)

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 13/23

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SLIDE 80

Memory Requirements (for Player 0)

Theorem

Optimal strategies for parity games require exponential memory. Necessity: Construct family Gd: 1 · · · . . . d 2 d + 1 · · · . . .

(Fijalkow and Chatterjee, Infinite-state games, 2013)

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 13/23

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SLIDE 81

Memory Requirements (for Player 0)

Theorem

Optimal strategies for parity games require exponential memory. Necessity: Construct family Gd: 1 · · · . . . d 2 d + 1 · · · . . .

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 13/23

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SLIDE 82

Memory Requirements (for Player 0)

Theorem

Optimal strategies for parity games require exponential memory. Necessity: Construct family Gd: · · · · · · d times d times For optimal play: 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 13/23

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SLIDE 83

Memory Requirements (for Player 0)

Theorem

Optimal strategies for parity games require exponential memory. Necessity: Construct family Gd: · · · · · · d times d times For optimal play: 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 13/23

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SLIDE 84

Memory Requirements (for Player 0)

Theorem

Optimal strategies for parity games require exponential memory. Necessity: Construct family Gd: · · · · · · d times d times For optimal play: 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 13/23

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SLIDE 85

Memory Requirements (for Player 0)

Theorem

Optimal strategies for parity games require exponential memory. Necessity: Construct family Gd: · · · · · · d times d times For optimal play: 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 13/23

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SLIDE 86

Memory Requirements (cont.)

Theorem

For every d > 1, there exists a finitary parity game Gd such that |Gd| ∈ O(d2) and Gd has d odd colors, and every optimal strategy for Player 0 has at least size 2d − 2. Similar bounds (upper and lower) hold true for Player 1.

Corollary

Let G be a parity game with costs with d odd colors. If Player 0 has a strategy σ for G with Cst(σ) = b, then she also has a strategy σ′ with Cst(σ′) = b and size (b + 2)d = 2d log(b+2).

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 14/23

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SLIDE 87

Memory Requirements (cont.)

Theorem

For every d > 1, there exists a finitary parity game Gd such that |Gd| ∈ O(d2) and Gd has d odd colors, and every optimal strategy for Player 0 has at least size 2d − 2. Similar bounds (upper and lower) hold true for Player 1.

Corollary

Let G be a parity game with costs with d odd colors. If Player 0 has a strategy σ for G with Cst(σ) = b, then she also has a strategy σ′ with Cst(σ′) = b and size (b + 2)d = 2d log(b+2).

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 14/23

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SLIDE 88

Memory Requirements (cont.)

Theorem

For every d > 1, there exists a finitary parity game Gd such that |Gd| ∈ O(d2) and Gd has d odd colors, and every optimal strategy for Player 0 has at least size 2d − 2. Similar bounds (upper and lower) hold true for Player 1.

Corollary

Let G be a parity game with costs with d odd colors. If Player 0 has a strategy σ for G with Cst(σ) = b, then she also has a strategy σ′ with Cst(σ′) = b and size (b + 2)d = 2d log(b+2).

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 14/23

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SLIDE 89

Introduction ✓ Complexity in PSpace PSpace-hard Sufficient Exponential Memory Necessary Tradeoffs Extensions ✓ ✓

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 15/23

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SLIDE 90

Introduction ✓ Complexity in PSpace PSpace-hard Sufficient Exponential Memory Necessary Tradeoffs Extensions ✓ ✓ ✓ ✓

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 15/23

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SLIDE 91

Results so far

Parity Finitary Parity Winning Optimal Complexity UP ∩ co-UP PTime PSpace-comp. Strategies Pos. Pos. Exp. Take-away: Forcing Player 0 to answer quickly in (finitary) 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 16/23

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SLIDE 92

Results so far

Parity Finitary Parity Winning Optimal Complexity UP ∩ co-UP PTime PSpace-comp. Strategies Pos. Pos. Exp. Take-away: Forcing Player 0 to answer quickly in (finitary) 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 16/23

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SLIDE 93

Results so far

Parity Finitary Parity Winning Optimal Complexity UP ∩ co-UP PTime PSpace-comp. Strategies Pos. Pos. Exp. Take-away: Forcing Player 0 to answer quickly in (finitary) 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 16/23

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SLIDE 94

Results so far

Parity Finitary Parity Winning Optimal Complexity UP ∩ co-UP PTime PSpace-comp. Strategies Pos. Pos. Exp. Take-away: Forcing Player 0 to answer quickly in (finitary) 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 16/23

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SLIDE 95

Introduction ✓ Complexity in PSpace PSpace-hard Sufficient Exponential Memory Necessary Tradeoffs Extensions ✓ ✓ ✓ ✓

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 17/23

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SLIDE 96

Tradeoffs

· · · · · ·

d request gadgets with d colors d response gadgets with d colors

Recall: Player 0 has winning strategy with cost d2 + 2d and size 2d − 2: store all d requests made by Player 1. Smaller strategy: Only store first i unique requests, then default to largest answer. ⇒ achieves cost d2 + 3d − i and size i−1

j=1

n

j

  • These are the smallest strategies achieving this cost.

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 18/23

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SLIDE 97

Tradeoffs

· · · · · ·

d request gadgets with d colors d response gadgets with d colors

Recall: Player 0 has winning strategy with cost d2 + 2d and size 2d − 2: store all d requests made by Player 1. Smaller strategy: Only store first i unique requests, then default to largest answer. ⇒ achieves cost d2 + 3d − i and size i−1

j=1

n

j

  • These are the smallest strategies achieving this cost.

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 18/23

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SLIDE 98

Tradeoffs

· · · · · ·

d request gadgets with d colors d response gadgets with d colors

Recall: Player 0 has winning strategy with cost d2 + 2d and size 2d − 2: store all d requests made by Player 1. Smaller strategy: Only store first i unique requests, then default to largest answer. ⇒ achieves cost d2 + 3d − i and size i−1

j=1

n

j

  • These are the smallest strategies achieving this cost.

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 18/23

slide-99
SLIDE 99

Tradeoffs

· · · · · ·

d request gadgets with d colors d response gadgets with d colors

Recall: Player 0 has winning strategy with cost d2 + 2d and size 2d − 2: store all d requests made by Player 1. Smaller strategy: Only store first i unique requests, then default to largest answer. ⇒ achieves cost d2 + 3d − i and size i−1

j=1

n

j

  • These are the smallest strategies achieving this cost.

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 18/23

slide-100
SLIDE 100

Tradeoffs

· · · · · ·

d request gadgets with d colors d response gadgets with d colors

Recall: Player 0 has winning strategy with cost d2 + 2d and size 2d − 2: store all d requests made by Player 1. Smaller strategy: Only store first i unique requests, then default to largest answer. ⇒ achieves cost d2 + 3d − i and size i−1

j=1

n

j

  • These are the smallest strategies achieving this cost.

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 18/23

slide-101
SLIDE 101

Tradeoffs

· · · · · ·

d request gadgets with d colors d response gadgets with d colors

Recall: Player 0 has winning strategy with cost d2 + 2d and size 2d − 2: store all d requests made by Player 1. Smaller strategy: Only store first i unique requests, then default to largest answer. ⇒ achieves cost d2 + 3d − i and size i−1

j=1

n

j

  • These are the smallest strategies achieving this cost.

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 18/23

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SLIDE 102

Tradeoffs

Theorem

Fix some finitary parity game Gd as before. For every i with 1 ≤ i ≤ d there exists a strategy σi for Player 0 in Gd such that σi has cost d2 + 3d − i and size i−1

j=1

d

j

  • .

Also, all these strategies are minimal for their respective cost.

129 128 127 126 125 124 123 122 121 120 1 1022

cost size

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 19/23

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SLIDE 103

Introduction ✓ Complexity in PSpace PSpace-hard Sufficient Exponential Memory Necessary Tradeoffs Extensions ✓ ✓ ✓ ✓ ✓

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 20/23

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SLIDE 104

Introduction ✓ Complexity in PSpace PSpace-hard Sufficient Exponential Memory Necessary Tradeoffs Extensions ✓ ✓ ✓ ✓ ✓

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 20/23

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SLIDE 105

Extension 1: Parity Games with Costs

1 3 2 4 1 1 1 1 1 1 Finitary parity games are special case ⇒ PSpace-hard ⇒ Exp. memory necessary Algorithm for finitary games works with some extensions ⇒ In PSpace ⇒ Exp. memory sufficient

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 21/23

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SLIDE 106

Extension 1: Parity Games with Costs

1 3 2 4 1 1 1 1 1 1 Finitary parity games are special case ⇒ PSpace-hard ⇒ Exp. memory necessary Algorithm for finitary games works with some extensions ⇒ In PSpace ⇒ Exp. memory sufficient

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 21/23

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SLIDE 107

Extension 1: Parity Games with Costs

1 3 2 4 1 1 1 1 1 1 Finitary parity games are special case ⇒ PSpace-hard ⇒ Exp. memory necessary Algorithm for finitary games works with some extensions ⇒ In PSpace ⇒ Exp. memory sufficient

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 21/23

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SLIDE 108

Extension 1: Parity Games with Costs

1 3 2 4 1 1 1 1 1 1 Finitary parity games are special case ⇒ PSpace-hard ⇒ Exp. memory necessary Algorithm for finitary games works with some extensions ⇒ In PSpace ⇒ Exp. memory sufficient

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 21/23

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SLIDE 109

Extension 1: Parity Games with Costs

1 3 2 4 1 1 1 1 1 1 Finitary parity games are special case ⇒ PSpace-hard ⇒ Exp. memory necessary Algorithm for finitary games works with some extensions ⇒ In PSpace ⇒ Exp. memory sufficient

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 21/23

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SLIDE 110

Extension 1: Parity Games with Costs

1 3 2 4 1 1 1 1 1 1 Finitary parity games are special case ⇒ PSpace-hard ⇒ Exp. memory necessary Algorithm for finitary games works with some extensions ⇒ In PSpace ⇒ Exp. memory sufficient

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 21/23

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SLIDE 111

Extension 1: Parity Games with Costs

1 3 2 4 1 1 1 1 1 1 Finitary parity games are special case ⇒ PSpace-hard ⇒ Exp. memory necessary Algorithm for finitary games works with some extensions ⇒ In PSpace ⇒ Exp. memory sufficient

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 21/23

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SLIDE 112

Extension 2: Streett

Finitary Streett Games in parity game, large responses answer all lower requests in Streett games, there are requests and responses, but not hierarchical Streett Games with Costs Streett condition and weights from {0, 1} No jump in complexity: Solving finitary Streett games is already ExpTime-complete and exponential memory is necessary ⇒ Appropriate G′ can be solved directly Streett Games with Costs Deciding winner ExpTime-complete Exponential memory necessary and sufficient

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 22/23

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SLIDE 113

Extension 2: Streett

Finitary Streett Games in parity game, large responses answer all lower requests in Streett games, there are requests and responses, but not hierarchical Streett Games with Costs Streett condition and weights from {0, 1} No jump in complexity: Solving finitary Streett games is already ExpTime-complete and exponential memory is necessary ⇒ Appropriate G′ can be solved directly Streett Games with Costs Deciding winner ExpTime-complete Exponential memory necessary and sufficient

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 22/23

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SLIDE 114

Extension 2: Streett

Finitary Streett Games in parity game, large responses answer all lower requests in Streett games, there are requests and responses, but not hierarchical Streett Games with Costs Streett condition and weights from {0, 1} No jump in complexity: Solving finitary Streett games is already ExpTime-complete and exponential memory is necessary ⇒ Appropriate G′ can be solved directly Streett Games with Costs Deciding winner ExpTime-complete Exponential memory necessary and sufficient

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 22/23

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SLIDE 115

Extension 2: Streett

Finitary Streett Games in parity game, large responses answer all lower requests in Streett games, there are requests and responses, but not hierarchical Streett Games with Costs Streett condition and weights from {0, 1} No jump in complexity: Solving finitary Streett games is already ExpTime-complete and exponential memory is necessary ⇒ Appropriate G′ can be solved directly Streett Games with Costs Deciding winner ExpTime-complete Exponential memory necessary and sufficient

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 22/23

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SLIDE 116

Conclusion

Parity Parity with Costs Winning Optimal Complexity UP ∩ co-UP UP ∩ co-UP PSpace-comp. Strategies Pos. Pos. Exp. Streett Streett with Costs Winning Optimal Complexity co-NP ExpTime ExpTime-comp. Strategies Exp. Exp. Exp.

Slides available at react.uni-saarland.de/people/weinert.html

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 23/23

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SLIDE 117

Conclusion

Parity Parity with Costs Winning Optimal Complexity UP ∩ co-UP UP ∩ co-UP PSpace-comp. Strategies Pos. Pos. Exp. Streett Streett with Costs Winning Optimal Complexity co-NP ExpTime ExpTime-comp. Strategies Exp. Exp. Exp.

Slides available at react.uni-saarland.de/people/weinert.html

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 23/23

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SLIDE 118

Conclusion

Parity Parity with Costs Winning Optimal Complexity UP ∩ co-UP UP ∩ co-UP PSpace-comp. Strategies Pos. Pos. Exp. Streett Streett with Costs Winning Optimal Complexity co-NP ExpTime ExpTime-comp. Strategies Exp. Exp. Exp.

Slides available at react.uni-saarland.de/people/weinert.html

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 23/23

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SLIDE 119

Conclusion

Parity Parity with Costs Winning Optimal Complexity UP ∩ co-UP UP ∩ co-UP PSpace-comp. Strategies Pos. Pos. Exp. Streett Streett with Costs Winning Optimal Complexity co-NP ExpTime ExpTime-comp. Strategies Exp. Exp. Exp.

Slides available at react.uni-saarland.de/people/weinert.html

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 23/23

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SLIDE 120

Conclusion

Parity Parity with Costs Winning Optimal Complexity UP ∩ co-UP UP ∩ co-UP PSpace-comp. Strategies Pos. Pos. Exp. Streett Streett with Costs Winning Optimal Complexity co-NP ExpTime ExpTime-comp. Strategies Exp. Exp. Exp.

Slides available at react.uni-saarland.de/people/weinert.html

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 23/23

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SLIDE 121

Conclusion

Parity Parity with Costs Winning Optimal Complexity UP ∩ co-UP UP ∩ co-UP PSpace-comp. Strategies Pos. Pos. Exp. Streett Streett with Costs Winning Optimal Complexity co-NP ExpTime ExpTime-comp. Strategies Exp. Exp. Exp.

Slides available at react.uni-saarland.de/people/weinert.html

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 23/23

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SLIDE 122

Conclusion

Parity Parity with Costs Winning Optimal Complexity UP ∩ co-UP UP ∩ co-UP PSpace-comp. Strategies Pos. Pos. Exp. Streett Streett with Costs Winning Optimal Complexity co-NP ExpTime ExpTime-comp. Strategies Exp. Exp. Exp.

Slides available at react.uni-saarland.de/people/weinert.html

Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 23/23