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Tradeoffs in Infinite Games Martin Zimmermann Saarland University May 15th, 2018 Scientific Talk in the Habilitation Process Martin Zimmermann Saarland University Tradeoffs in Infinite Games 1/36 Churchs Synthesis Problem o 1 i 1 . ?

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SLIDE 1

Tradeoffs in Infinite Games

Martin Zimmermann

Saarland University

May 15th, 2018

Scientific Talk in the Habilitation Process

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 1/36

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SLIDE 2

Church’s Synthesis Problem

?

i1 . . . in

  • 1

. . .

  • n

Church 1957: Given a specification on the input/output behavior

  • f a circuit (in some suitable logical language), decide whether

such a circuit exists, and, if yes, compute one.

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 2/36

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SLIDE 3

Church’s Synthesis Problem

?

i1 . . . in

  • 1

. . .

  • n

Example

Interpret input ij = 1 as client j requesting a shared resource and

  • utput oj = 1 as the corresponding grant to client j.

Typical properties:

  • 1. If there are infinitely many requests of client j, then also

infinitely many grants for client j.

  • 2. At most one grant at a time (mutual exclusion).
  • 3. No spurious grants.

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 2/36

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SLIDE 4

Church’s Synthesis Problem

?

i1 . . . in

  • 1

. . .

  • n

Solved by B¨ uchi & Landweber in 1969. Insight: Problem can be expressed as two-player game of infinite duration between the environment (producing inputs) and the circuit (producing outputs).

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 2/36

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SLIDE 5

Back to the Example

Consider the one-client case!

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 3/36

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SLIDE 6

Back to the Example

Consider the one-client case! s i i

  • Martin Zimmermann

Saarland University Tradeoffs in Infinite Games 3/36

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SLIDE 7

Back to the Example

Consider the one-client case! s i i

  • Input:

Output:

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 3/36

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SLIDE 8

Back to the Example

Consider the one-client case! s i i

  • Input:

Output:

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 3/36

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SLIDE 9

Back to the Example

Consider the one-client case! s i i

  • Input:

Output:

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 3/36

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SLIDE 10

Back to the Example

Consider the one-client case! s i i

  • Input:

Output:

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 3/36

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SLIDE 11

Back to the Example

Consider the one-client case! s i i

  • Input:

Output:

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 3/36

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SLIDE 12

Back to the Example

Consider the one-client case! s i i

  • Input:

Output:

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 3/36

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SLIDE 13

Back to the Example

Consider the one-client case! s i i

  • Input:

Output: 1

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 3/36

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SLIDE 14

Back to the Example

Consider the one-client case! s i i

  • Input:

1 Output: 1

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 3/36

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SLIDE 15

Back to the Example

Consider the one-client case! s i i

  • Input:

1 Output: 1 1

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 3/36

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SLIDE 16

Back to the Example

Consider the one-client case! s i i

  • Input:

1 1 Output: 1 1

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 3/36

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SLIDE 17

Back to the Example

Consider the one-client case! s i i

  • Input:

1 1 Output: 1 1 1

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 3/36

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SLIDE 18

Back to the Example

Consider the one-client case! s i i

  • Input:

1 1 · · · Output: 1 1 1 · · ·

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 3/36

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SLIDE 19

Back to the Example

Consider the one-client case! s i i

  • Winning plays for circuit player have to satisfy
  • 1. if i is visited infinitely often, then o as well, and
  • 2. if o is visited, then it has not been visited since the last visit
  • f i.

This requires an expressive specification language, e.g., Linear Temporal Logic (LTL).

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 3/36

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SLIDE 20

Back to the Example

Consider the one-client case! ℓ i s i

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 3/36

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SLIDE 21

Back to the Example

Consider the one-client case! ℓ i s i

* 1 1 1 1 1

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 3/36

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SLIDE 22

Back to the Example

Consider the one-client case! ℓ i s i

* 1 1 1 1 1

Now, the winning plays for the circuit player have to satisfy

  • 1. if i is visited infinitely often, then s as well, and
  • 2. ℓ is never visited.

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 3/36

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SLIDE 23

Back to the Example

Consider the one-client case! 3 2 1 Now, the winning plays for the circuit player have to satisfy

  • 1. if i is visited infinitely often, then s as well, and
  • 2. ℓ is never visited.

Equivalently: color the vertices by natural numbers as above and require that almost all odd colors are followed by a larger even one. This is the classical parity condition for ω-automata.

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 3/36

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SLIDE 24

B¨ uchi-Landweber in a Nutshell

2 3 1

* 1 1 1 1 1

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 4/36

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SLIDE 25

B¨ uchi-Landweber in a Nutshell

2 3 1

* 1 1 1 1 1

Player 0 has a memoryless winning strategy,

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 4/36

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SLIDE 26

B¨ uchi-Landweber in a Nutshell

0/0 1/1 Player 0 has a memoryless winning strategy, which can be turned into an automaton with output,

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 4/36

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SLIDE 27

B¨ uchi-Landweber in a Nutshell

0/0 1/1 i1

  • 1

Player 0 has a memoryless winning strategy, which can be turned into an automaton with output, which can be turned into a circuit satisfying the specification.

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 4/36

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SLIDE 28

Not Everyone is Equal

Quality of winning strategies is measured in multiple dimensions:

  • 1. Memory requirements
  • 2. Degree of satisfaction of a (quantitative) winning condition
  • 3. Computational complexity of computing such a strategy
  • 4. Use of lookahead
  • 5. Use of randomization
  • 6. Informedness
  • 7. Robustness
  • 8. Non-determinism

In previous work, each dimension was studied in isolation.

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 5/36

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SLIDE 29

Not Everyone is Equal

Quality of winning strategies is measured in multiple dimensions:

  • 1. Memory requirements
  • 2. Degree of satisfaction of a (quantitative) winning condition
  • 3. Computational complexity of computing such a strategy
  • 4. Use of lookahead
  • 5. Use of randomization
  • 6. Informedness
  • 7. Robustness
  • 8. Non-determinism

In previous work, each dimension was studied in isolation. Goal of this thesis: Understand the tradeoffs between (some of) these dimensions.

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 5/36

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SLIDE 30

Overview

Typical questions we answered in this thesis: What is the price of optimality? Is it harder to compute an optimal winning strategy in a quantitative game than an arbitrary winning strategy? Does optimality increase the memory requirements of winning strategies?

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 6/36

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SLIDE 31

Overview

Typical questions we answered in this thesis: What is the price of optimality? Is it harder to compute an optimal winning strategy in a quantitative game than an arbitrary winning strategy? Does optimality increase the memory requirements of winning strategies? Can the expressiveness of LTL be increased without increasing the complexity of solving games?

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 6/36

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SLIDE 32

Overview

Typical questions we answered in this thesis: What is the price of optimality? Is it harder to compute an optimal winning strategy in a quantitative game than an arbitrary winning strategy? Does optimality increase the memory requirements of winning strategies? Can the expressiveness of LTL be increased without increasing the complexity of solving games? How does the addition of lookahead change the characteristics

  • f games?

Does lookahead increase the complexity of solving games? Does lookahead allow to improve the quality of strategies in quantitative games?

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 6/36

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SLIDE 33

Outline

  • 1. Playing Optimally in Variations of Parity Games
  • 2. Playing (Approximatively) Optimally in LTL Games
  • 3. More Tradeoffs

Lookahead vs. Quality Lookahead vs. Memory Expressiveness vs. Complexity Expressiveness vs. Memory

  • 4. Conclusion

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 7/36

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SLIDE 34

Outline

  • 1. Playing Optimally in Variations of Parity Games
  • 2. Playing (Approximatively) Optimally in LTL Games
  • 3. More Tradeoffs
  • 4. Conclusion

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 8/36

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SLIDE 35

A Parity Game

1 2

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 9/36

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SLIDE 36

A Parity Game

1 2 1 2 1 2 1 2 · · ·

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 9/36

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SLIDE 37

A Parity Game

1 2 1 2 1 2 1 2 · · · 2 steps 3 steps 4 steps

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 9/36

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SLIDE 38

A Parity Game

1 2 1 2 1 2 1 2 · · · 2 steps 3 steps 4 steps Player 0 wins from every vertex, but Player 1 can delay between color 1 and color 2 longer and longer. ⇒ undesired behavior.

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 9/36

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SLIDE 39

Parity Games go Quantitative

During the last two decades, various quantitative variants of parity games have been introduced: Mean-payoff parity games [CHJ05] Finitary parity games [CH06] Energy parity games [CD11]

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 10/36

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SLIDE 40

Parity Games go Quantitative

During the last two decades, various quantitative variants of parity games have been introduced: Mean-payoff parity games [CHJ05] Finitary parity games [CH06] Energy parity games [CD11] Finitary parity games are distinguished, as here the quantitative aspect measures the satisfaction of the qualitative one.

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 10/36

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SLIDE 41

Parity Games go Quantitative

During the last two decades, various quantitative variants of parity games have been introduced: Mean-payoff parity games [CHJ05] Finitary parity games [CH06] Energy parity games [CD11] Window parity games [BHR16] Finitary parity games are distinguished, as here the quantitative aspect measures the satisfaction of the qualitative one.

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 10/36

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SLIDE 42

Finitary Parity Games

Parity: Almost all odd colors are followed by larger even one. Finitary Parity: There is a bound b such that almost all odd colors are followed by larger even one within b steps.

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 11/36

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SLIDE 43

Finitary Parity Games

Parity: Almost all odd colors are followed by larger even one. Finitary Parity: There is a bound b such that almost all odd colors are followed by larger even one within b steps. Condition Complexity Memory Pl. 0 Memory Pl. 1 Parity quasi-poly Memoryless Memoryless Finitary Parity PTime Memoryless Infinite

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 11/36

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SLIDE 44

Boundedness vs. Optimization

The bound b in the definition of finitary parity games is existentially quantified (and may depend on the play).

Corollary

If Player 0 wins a finitary parity game G, then a uniform bound b ≤ |G| suffices. A trivial example shows that the upper bound |G| is tight.

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 12/36

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SLIDE 45

Boundedness vs. Optimization

The bound b in the definition of finitary parity games is existentially quantified (and may depend on the play).

Corollary

If Player 0 wins a finitary parity game G, then a uniform bound b ≤ |G| suffices. A trivial example shows that the upper bound |G| is tight. Questions

  • 1. Does Player 0 need memory to achieve the optimal bound?
  • 2. Is it harder to compute the optimal bound than checking

whether a bound exists?

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 12/36

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SLIDE 46

Chatterjee & Fijalkow

1 3 2 4

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 13/36

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SLIDE 47

Chatterjee & Fijalkow

1 3 2 4 Player 0 has a unique memoryless winning strategy, which achieves the bound five: from the 1 it takes five steps to the 4.

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 13/36

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SLIDE 48

Chatterjee & Fijalkow

1 3 2 4 Player 0 has a unique memoryless winning strategy, which achieves the bound five: from the 1 it takes five steps to the 4. With two memory states, she can achieve the bound four: “answer” a 1 by a 2 and a 3 by a 4.

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 13/36

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SLIDE 49

Chatterjee & Fijalkow

1 3 2 4 Player 0 has a unique memoryless winning strategy, which achieves the bound five: from the 1 it takes five steps to the 4. With two memory states, she can achieve the bound four: “answer” a 1 by a 2 and a 3 by a 4. It is trivial to extend this example to d odd colors and d even colors requiring d memory states to play optimally. ⇒ In general, playing optimally requires memory; but how much?

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 13/36

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SLIDE 50

Memory Requirements

2 4 2d . . . . . . . . . 1 3 2d − 1 . . . . . . . . .

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 14/36

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SLIDE 51

Memory Requirements

. . . . . .

d request gadgets with d colors

  • d response gadgets with d colors
  • Martin Zimmermann

Saarland University Tradeoffs in Infinite Games 15/36

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SLIDE 52

Memory Requirements

. . . . . .

d request gadgets with d colors

  • d response gadgets with d colors
  • Player 0 has winning strategy with cost d2 + 2d: answer j-th

unique request in j-th response-gadget. ⇒ requires exponential memory (in d).

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 15/36

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SLIDE 53

Memory Requirements

. . . . . .

d request gadgets with d colors

  • d response gadgets with d colors
  • Player 0 has winning strategy with cost d2 + 2d: answer j-th

unique request in j-th response-gadget. ⇒ requires exponential memory (in d). Against a smaller strategy Player 1 can enforce a larger cost, as Player 0 cannot store every sequence of requests.

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 15/36

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SLIDE 54

Memory Requirements

. . . . . .

d request gadgets with d colors

  • d response gadgets with d colors
  • Player 0 has winning strategy with cost d2 + 2d: answer j-th

unique request in j-th response-gadget. ⇒ requires exponential memory (in d). Against a smaller strategy Player 1 can enforce a larger cost, as Player 0 cannot store every sequence of requests.

Theorem (WZ16)

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−1.

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 15/36

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SLIDE 55

PSPACE-Hardness

Lemma (WZ16)

The following problem is PSpace-hard: “Given a finitary parity game G and a bound b ∈ N, does Player 0 have a strategy for G whose cost is at most b?”

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 16/36

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SLIDE 56

PSPACE-Hardness

Lemma (WZ16)

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

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 16/36

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SLIDE 57

PSPACE-Hardness

Lemma (WZ16)

The following problem is PSpace-hard: “Given a finitary parity game G and a bound b ∈ N, does Player 0 have a strategy for G whose cost is at most b?” Proof By a reduction from QBF (w.l.o.g. in CNF). 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.

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 16/36

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SLIDE 58

The Reduction

ϕ = ∀x ∃y .

ψ

  • ( x ∨ ¬y ) ∧ ( ¬x ∨ y )

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 17/36

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SLIDE 59

The Reduction

ϕ = ∀x ∃y .

ψ

  • ( x ∨ ¬y ) ∧ ( ¬x ∨ y )

1 x 3 ¬x

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 17/36

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SLIDE 60

The Reduction

ϕ = ∀x ∃y .

ψ

  • ( x ∨ ¬y ) ∧ ( ¬x ∨ y )

1 x 3 ¬x 5 y 7 ¬y

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 17/36

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SLIDE 61

The Reduction

ϕ = ∀x ∃y .

ψ

  • ( x ∨ ¬y ) ∧ ( ¬x ∨ y )

1 x 3 ¬x 5 y 7 ¬y ψ

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 17/36

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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)

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 17/36

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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

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 17/36

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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

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 17/36

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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

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 17/36

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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

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 17/36

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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 . . . . . . . . . . . .

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 17/36

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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

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 17/36

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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

For a well-chosen bound b, a strategy for Player 0 with cost at most b witnesses the truth of ϕ and vice versa.

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 17/36

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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

For a well-chosen bound b, a strategy for Player 0 with cost at most b witnesses the truth of ϕ and vice versa.

1 x 5 y · · · 2 x 10

  • b steps

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 17/36

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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

For a well-chosen bound b, a strategy for Player 0 with cost at most b witnesses the truth of ϕ and vice versa.

1 x 5 y · · · 4 ¬x 10

  • b steps

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 17/36

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SLIDE 72

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

For a well-chosen bound b, a strategy for Player 0 with cost at most b witnesses the truth of ϕ and vice versa.

3 ¬x 5 y · · · 4 ¬x 10

  • b steps

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 17/36

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SLIDE 73

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

For a well-chosen bound b, a strategy for Player 0 with cost at most b witnesses the truth of ϕ and vice versa.

3 ¬x 5 y · · · 2 x 10

  • b steps

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 17/36

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SLIDE 74

PSPACE-Membership

Lemma (WZ16)

The following problem is in PSpace: “Given a finitary parity game G and a bound b ∈ N, does Player 0 have a strategy for G whose cost is at most b?”

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 18/36

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SLIDE 75

PSPACE-Membership

Lemma (WZ16)

The following problem is in PSpace: “Given a finitary parity game G and a bound b ∈ N, does Player 0 have a strategy for G whose cost is at most b?” Proof Sketch Fix G and b (w.l.o.g. b ≤ |G|).

  • 1. Construct equivalent parity game G′ storing the costs of open

requests (up to bound b) and the number of “overflows” (up to bound |G|) ⇒ |G′| ∈ |G|O(d).

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 18/36

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SLIDE 76

PSPACE-Membership

Lemma (WZ16)

The following problem is in PSpace: “Given a finitary parity game G and a bound b ∈ N, does Player 0 have a strategy for G whose cost is at most b?” Proof Sketch Fix G and b (w.l.o.g. b ≤ |G|).

  • 1. Construct equivalent parity game G′ storing the costs of open

requests (up to bound b) and the number of “overflows” (up to bound |G|) ⇒ |G′| ∈ |G|O(d).

  • 2. Define equivalent finite-duration variant G′

f of G′ with

polynomial play-length.

  • 3. G′

f can be solved on alternating polynomial-time Turing

machine.

  • 4. APTime = PSpace concludes the proof.

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 18/36

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SLIDE 77

Upper Bounds on Memory

Equivalence between finitary parity game G w.r.t. bound b and parity game G′ yields upper bounds on memory requirements.

Corollary

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

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 19/36

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SLIDE 78

Upper Bounds on Memory

Equivalence between finitary parity game G w.r.t. bound b and parity game G′ yields upper bounds on memory requirements.

Corollary

Let G be a finitary parity game with costs with d odd colors. If Player 0 has a strategy for G with cost b, then she also has a strategy with cost b and size (b + 2)d = 2d log(b+2). Recall: lower bound 2d−1. The same bounds hold for Player 1.

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 19/36

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SLIDE 79

Tradeoffs

Theorem (WZ16)

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, every strategy σ′ for Player 0 in Gd whose cost is at most the cost of σi has at least the size of σi.

129 128 127 126 125 124 123 122 121 120 119 1 1022

cost size

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 20/36

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SLIDE 80

Generalizations

We generalized finitary parity games to finitary parity parity

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 21/36

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SLIDE 81

Generalizations

We generalized finitary parity games to parity games with costs (by allowing non-negative weights on the edges) [FZ14], and finitary parity parity parity with costs

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SLIDE 82

Generalizations

We generalized finitary parity games to parity games with costs (by allowing non-negative weights on the edges) [FZ14], and parity games with weights (by allowing arbitrary weights on the edges) [SWZ18]. finitary parity parity parity with costs parity with weights

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SLIDE 83

Generalizations

We generalized finitary parity games to parity games with costs (by allowing non-negative weights on the edges) [FZ14], and parity games with weights (by allowing arbitrary weights on the edges) [SWZ18]. Boundedness Condition Complexity Memory Pl. 0 Memory Pl. 1 Parity quasi-poly Memoryless Memoryless Finitary Parity PTime Memoryless Infinite Parity w. Costs quasi-poly Memoryless Infinite Parity w. Weights NP ∩ co-NP Exponential Infinite

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SLIDE 84

Generalizations

We generalized finitary parity games to parity games with costs (by allowing non-negative weights on the edges) [FZ14], and parity games with weights (by allowing arbitrary weights on the edges) [SWZ18]. Optimization Condition Complexity Memory Pl. 0 & 1 Finitary Parity PSpace-complete Exponential Parity w. Costs PSpace-complete Exponential Parity w. Weights PSpace-hard ≥ Exponential The results for parity games with costs hold for unary and binary encodings of the weights.

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SLIDE 85

Outline

  • 1. Playing Optimally in Variations of Parity Games
  • 2. Playing (Approximatively) Optimally in LTL Games
  • 3. More Tradeoffs
  • 4. Conclusion

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SLIDE 86

Introducing LTL by Examples

Atomic propositions ri for requests and gi for grants.

  • 1. Answer every request:

i G (ri → F gi)

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SLIDE 87

Introducing LTL by Examples

Atomic propositions ri for requests and gi for grants.

  • 1. Answer every request:

i G (ri → F gi)

  • 2. At most one grant at a time: G

i=j ¬(gi ∧ gj)

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SLIDE 88

Introducing LTL by Examples

Atomic propositions ri for requests and gi for grants.

  • 1. Answer every request:

i G (ri → F gi)

  • 2. At most one grant at a time: G

i=j ¬(gi ∧ gj)

  • 3. No spurious grants:
  • i

¬[ (¬ri U (¬ri ∧ gi)) ] ∧ ¬[ F (gi ∧ X (¬ri U (¬ri ∧ gi))) ]

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 23/36

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SLIDE 89

Introducing LTL by Examples

Atomic propositions ri for requests and gi for grants.

  • 1. Answer every request:

i G (ri → F gi)

  • 2. At most one grant at a time: G

i=j ¬(gi ∧ gj)

  • 3. No spurious grants:
  • i

¬[ (¬ri U (¬ri ∧ gi)) ] ∧ ¬[ F (gi ∧ X (¬ri U (¬ri ∧ gi))) ] ≡

  • i

[ (ri R (ri ∨ ¬gi)) ] ∧ [ G (¬gi ∨ X (ri R (ri ∨ ¬gi))) ]

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SLIDE 90

A Problem with LTL

Answer every request:

i G (ri → F gi)

r0 g0 r0 g0 r0 g0 · · ·

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SLIDE 91

A Problem with LTL

Answer every request:

i G (ri → F gi)

r0 g0 r0 g0 r0 g0 · · · Problem: LTL is too weak to express timing-constraints: no guarantee when request is granted, only that it is granted eventually

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SLIDE 92

LTL goes Quantitative

During the last two decades, various quantitative variants of LTL have been introduced: Parametric LTL [AETP99] PROMPT–LTL [KPV07] Parametric MTL [GTN10]

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SLIDE 93

LTL goes Quantitative

During the last two decades, various quantitative variants of LTL have been introduced: Parametric LTL [AETP99] PROMPT–LTL [KPV07] Parametric MTL [GTN10] PROMPT–LTL is distinguished, as all problems for the more general Parametric LTL are reducible to those for PROMPT–LTL.

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SLIDE 94

Prompt-LTL

Syntax: Add prompt-eventually operator FP . ϕ ::= p | ¬p | ϕ ∧ ϕ | ϕ ∨ ϕ | X ϕ | ϕ U ϕ | ϕ R ϕ | FP ϕ

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SLIDE 95

Prompt-LTL

Syntax: Add prompt-eventually operator FP . ϕ ::= p | ¬p | ϕ ∧ ϕ | ϕ ∨ ϕ | X ϕ | ϕ U ϕ | ϕ R ϕ | FP ϕ Semantics: Defined with respect to a fixed bound k ∈ N. (ρ, n, k) | = FP ϕ: ρ n n + k ϕ

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SLIDE 96

Prompt-LTL

Syntax: Add prompt-eventually operator FP . ϕ ::= p | ¬p | ϕ ∧ ϕ | ϕ ∨ ϕ | X ϕ | ϕ U ϕ | ϕ R ϕ | FP ϕ Semantics: Defined with respect to a fixed bound k ∈ N. (ρ, n, k) | = FP ϕ: ρ n n + k ϕ Now:

i G (ri → FP gi)

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SLIDE 97

Prompt-LTL Games

Label the arena by atomic propositions. Winning condition: PROMPT–LTL formula ϕ. Player 0 wins if there is a uniform bound k and a strategy σ such that every play that is consistent with σ satisfies the winning condition ϕ w.r.t. k.

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SLIDE 98

Prompt-LTL Games

Label the arena by atomic propositions. Winning condition: PROMPT–LTL formula ϕ. Player 0 wins if there is a uniform bound k and a strategy σ such that every play that is consistent with σ satisfies the winning condition ϕ w.r.t. k. PROMPT–LTL games are not harder than LTL games...

Theorem (KPV07)

  • 1. Determining the winner of PROMPT–LTL games is

2ExpTime-complete.

  • 2. If Player 0 wins, then also with a finite-state strategy of

size 22|ϕ| and w.r.t. the bound kϕ = 22|ϕ|.

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SLIDE 99

Prompt-LTL Games

Label the arena by atomic propositions. Winning condition: PROMPT–LTL formula ϕ. Player 0 wins if there is a uniform bound k and a strategy σ such that every play that is consistent with σ satisfies the winning condition ϕ w.r.t. k. ...unless you optimize the bound.

Theorem (Z11)

  • 1. The PROMPT–LTL game optimization problem can be

solved in triply-exponential time.

  • 2. The bound kϕ is tight in general.

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SLIDE 100

Prompt-LTL Games

Label the arena by atomic propositions. Winning condition: PROMPT–LTL formula ϕ. Player 0 wins if there is a uniform bound k and a strategy σ such that every play that is consistent with σ satisfies the winning condition ϕ w.r.t. k. Questions

  • 1. Is the optimization problem harder than the boundedness

problem?

  • 2. Can the optimum be approximated?

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SLIDE 101

An Approximation Algorithm

Lemma (TWZ16)

Fix a PROMPT–LTL game G with winning condition ϕ and k ≤ kϕ. There is an LTL game Gk such that

  • 1. if Player 0 wins G w.r.t. k, then she wins Gk,
  • 2. if Player 0 wins Gk, then she wins G w.r.t. 2k, and
  • 3. Gk can be solved in doubly-exponential time in |G|.

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SLIDE 102

An Approximation Algorithm

Lemma (TWZ16)

Fix a PROMPT–LTL game G with winning condition ϕ and k ≤ kϕ. There is an LTL game Gk such that

  • 1. if Player 0 wins G w.r.t. k, then she wins Gk,
  • 2. if Player 0 wins Gk, then she wins G w.r.t. 2k, and
  • 3. Gk can be solved in doubly-exponential time in |G|.

The algorithm:

1: for k = 0; k ≤ kϕ; k ← k + 1 do 2:

if Player 0 wins Gk then

3:

return 2k

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SLIDE 103

An Approximation Algorithm

Lemma (TWZ16)

Fix a PROMPT–LTL game G with winning condition ϕ and k ≤ kϕ. There is an LTL game Gk such that

  • 1. if Player 0 wins G w.r.t. k, then she wins Gk,
  • 2. if Player 0 wins Gk, then she wins G w.r.t. 2k, and
  • 3. Gk can be solved in doubly-exponential time in |G|.

The algorithm:

1: for k = 0; k ≤ kϕ; k ← k + 1 do 2:

if Player 0 wins Gk then

3:

return 2k Running time: doubly-exponential Approximation ratio: 2 Yields winning strategy

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SLIDE 104

Tradeoffs

memory size bound

Implementation via bounded synthesis: Search for finite-state strategy of size n achieving bound k. Complete due to upper bounds on n and k.

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SLIDE 105

Tradeoffs

memory size bound

Implementation via bounded synthesis: Search for finite-state strategy of size n achieving bound k. Complete due to upper bounds on n and k.

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SLIDE 106

Tradeoffs

memory size bound

Implementation via bounded synthesis: Search for finite-state strategy of size n achieving bound k. Complete due to upper bounds on n and k.

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SLIDE 107

Tradeoffs

memory size bound

Implementation via bounded synthesis: Search for finite-state strategy of size n achieving bound k. Complete due to upper bounds on n and k.

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SLIDE 108

Tradeoffs

memory size bound

Implementation via bounded synthesis: Search for finite-state strategy of size n achieving bound k. Complete due to upper bounds on n and k.

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SLIDE 109

Tradeoffs

memory size bound

Implementation via bounded synthesis: Search for finite-state strategy of size n achieving bound k. Complete due to upper bounds on n and k.

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SLIDE 110

Tradeoffs

memory size bound

Implementation via bounded synthesis: Search for finite-state strategy of size n achieving bound k. Complete due to upper bounds on n and k.

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SLIDE 111

Tradeoffs

memory size bound

Implementation via bounded synthesis: Search for finite-state strategy of size n achieving bound k. Complete due to upper bounds on n and k.

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SLIDE 112

Tradeoffs

memory size bound

Implementation via bounded synthesis: Search for finite-state strategy of size n achieving bound k. Complete due to upper bounds on n and k.

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SLIDE 113

Tradeoffs

memory size bound

Implementation via bounded synthesis: Search for finite-state strategy of size n achieving bound k. Complete due to upper bounds on n and k.

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SLIDE 114

Tradeoffs

memory size bound

Implementation via bounded synthesis: Search for finite-state strategy of size n achieving bound k. Complete due to upper bounds on n and k.

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SLIDE 115

Tradeoffs

memory size bound

Implementation via bounded synthesis: Search for finite-state strategy of size n achieving bound k. Complete due to upper bounds on n and k.

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SLIDE 116

Tradeoffs

memory size bound

Implementation via bounded synthesis: Search for finite-state strategy of size n achieving bound k. Complete due to upper bounds on n and k.

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SLIDE 117

Tradeoffs

memory size bound

Implementation via bounded synthesis: Search for finite-state strategy of size n achieving bound k. Complete due to upper bounds on n and k.

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SLIDE 118

Tradeoffs

memory size bound

Implementation via bounded synthesis: Search for finite-state strategy of size n achieving bound k. Complete due to upper bounds on n and k.

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SLIDE 119

Tradeoffs

memory size bound

Implementation via bounded synthesis: Search for finite-state strategy of size n achieving bound k. Complete due to upper bounds on n and k.

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SLIDE 120

Tradeoffs

memory size bound

Implementation via bounded synthesis: Search for finite-state strategy of size n achieving bound k. Complete due to upper bounds on n and k.

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SLIDE 121

Tradeoffs

memory size bound

Implementation via bounded synthesis: Search for finite-state strategy of size n achieving bound k. Complete due to upper bounds on n and k.

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SLIDE 122

Tradeoffs

memory size bound

Implementation via bounded synthesis: Search for finite-state strategy of size n achieving bound k. Complete due to upper bounds on n and k.

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SLIDE 123

Tradeoffs

memory size bound

Implementation via bounded synthesis: Search for finite-state strategy of size n achieving bound k. Complete due to upper bounds on n and k.

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SLIDE 124

Tradeoffs

memory size bound

Implementation via bounded synthesis: Search for finite-state strategy of size n achieving bound k. Complete due to upper bounds on n and k.

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SLIDE 125

Tradeoffs

RG

memory size bound

Implementation via bounded synthesis: Search for finite-state strategy of size n achieving bound k. Complete due to upper bounds on n and k. RG: realizable combinations for game G.

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SLIDE 126

Tradeoffs

RG

memory size bound

Implementation via bounded synthesis: Search for finite-state strategy of size n achieving bound k. Complete due to upper bounds on n and k. RG: realizable combinations for game G.

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SLIDE 127

Tradeoffs

RG

memory size bound

Implementation via bounded synthesis: Search for finite-state strategy of size n achieving bound k. Complete due to upper bounds on n and k. RG: realizable combinations for game G.

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SLIDE 128

Tradeoffs

RG

memory size bound

Implementation via bounded synthesis: Search for finite-state strategy of size n achieving bound k. Complete due to upper bounds on n and k. RG: realizable combinations for game G.

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SLIDE 129

Tradeoffs

RG

memory size bound

Implementation via bounded synthesis: Search for finite-state strategy of size n achieving bound k. Complete due to upper bounds on n and k. RG: realizable combinations for game G.

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SLIDE 130

Tradeoffs

RG

memory size bound

Implementation via bounded synthesis: Search for finite-state strategy of size n achieving bound k. Complete due to upper bounds on n and k. RG: realizable combinations for game G.

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SLIDE 131

Tradeoffs

RG

memory size bound

Implementation via bounded synthesis: Search for finite-state strategy of size n achieving bound k. Complete due to upper bounds on n and k. RG: realizable combinations for game G.

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SLIDE 132

Tradeoffs

RG

memory size bound

Implementation via bounded synthesis: Search for finite-state strategy of size n achieving bound k. Complete due to upper bounds on n and k. RG: realizable combinations for game G.

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SLIDE 133

Tradeoffs

RG

memory size bound

Implementation via bounded synthesis: Search for finite-state strategy of size n achieving bound k. Complete due to upper bounds on n and k. RG: realizable combinations for game G.

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SLIDE 134

Tradeoffs

RG

memory size bound

Implementation via bounded synthesis: Search for finite-state strategy of size n achieving bound k. Complete due to upper bounds on n and k. RG: realizable combinations for game G.

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SLIDE 135

Tradeoffs

RG

memory size bound

Implementation via bounded synthesis: Search for finite-state strategy of size n achieving bound k. Complete due to upper bounds on n and k. RG: realizable combinations for game G.

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SLIDE 136

Tradeoffs

RG

memory size bound

Implementation via bounded synthesis: Search for finite-state strategy of size n achieving bound k. Complete due to upper bounds on n and k. RG: realizable combinations for game G.

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SLIDE 137

Tradeoffs

RG

memory size bound

Implementation via bounded synthesis: Search for finite-state strategy of size n achieving bound k. Complete due to upper bounds on n and k. RG: realizable combinations for game G.

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SLIDE 138

Tradeoffs

RG

memory size bound

Implementation via bounded synthesis: Search for finite-state strategy of size n achieving bound k. Complete due to upper bounds on n and k. RG: realizable combinations for game G.

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SLIDE 139

Tradeoffs

RG

memory size bound

Implementation via bounded synthesis: Search for finite-state strategy of size n achieving bound k. Complete due to upper bounds on n and k. RG: realizable combinations for game G.

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SLIDE 140

Tradeoffs

RG

memory size bound

Implementation via bounded synthesis: Search for finite-state strategy of size n achieving bound k. Complete due to upper bounds on n and k. RG: realizable combinations for game G.

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SLIDE 141

Tradeoffs

RG

memory size bound

Implementation via bounded synthesis: Search for finite-state strategy of size n achieving bound k. Complete due to upper bounds on n and k. RG: realizable combinations for game G.

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SLIDE 142

Tradeoffs

RG

memory size bound

Implementation via bounded synthesis: Search for finite-state strategy of size n achieving bound k. Complete due to upper bounds on n and k. RG: realizable combinations for game G.

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SLIDE 143

Tradeoffs

RG

memory size bound

Lemma

  • 1. If (n, k) ∈ RG, then (n, n2|ϕ|) ∈ RG.
  • 2. If (n, k) ∈ RG, then (22k|ϕ|, k) ∈ RG.

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SLIDE 144

Tradeoffs

b + 1 Pareto points 2b 2b RGb

memory size bound

Theorem (TWZ16)

For every b, there is a game Gb of size O(b) such that (2j, 2b−j) is a Pareto point for every j ≤ b.

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SLIDE 145

Tradeoffs

c 22b 2b RGb

memory size bound

Theorem (TWZ16)

For every b, there is a game Gb of size O(b) such that (22b, 0) and (c, 2b) are Pareto points for some constant c.

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SLIDE 146

An Example

Five clients:

  • 1. Answer every request of client 1 promptly: G (r1 → FP g1)
  • 2. Answer every other request eventually:

i>1 G (ri → F gi)

  • 3. At most one grant at a time: G

i=j ¬(gi ∧ gj)

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 30/36

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SLIDE 147

An Example

Five clients:

  • 1. Answer every request of client 1 promptly: G (r1 → FP g1)
  • 2. Answer every other request eventually:

i>1 G (ri → F gi)

  • 3. At most one grant at a time: G

i=j ¬(gi ∧ gj)

4 5 6 7 8 9 10 1 2 3 4 5 0.4 0.6 0.8 1 n k Running time [sec] 1

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SLIDE 148

An Example

Five clients:

  • 1. Answer every request of client 1 promptly: G (r1 → FP g1)
  • 2. Answer every other request eventually:

i>1 G (ri → F gi)

  • 3. At most one grant at a time: G

i=j ¬(gi ∧ gj)

s0: g4 s1: g2 True s2: g5,p True s4: g1,p True s3: g3 True True

s0: p,g2 s2: g1 True s3: p,g3 True s1: g4 s5: g1,p True s4: g5 True True True

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SLIDE 149

Generalizations

LTL LDL PLTL

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SLIDE 150

Generalizations

Parametric LDL [FZ14]: full expressive power of the ω-regular languages and bounded operators.

LTL LDL PLTL PLDL

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SLIDE 151

Generalizations

Parametric LDL [FZ14]: full expressive power of the ω-regular languages and bounded operators. Parametric LTL and LDL with costs [Z15]: replace unit cost by non-negative weights.

LTL LDL PLTL PLDL cPLDL cPLTL

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SLIDE 152

Generalizations

Parametric LDL [FZ14]: full expressive power of the ω-regular languages and bounded operators. Parametric LTL and LDL with costs [Z15]: replace unit cost by non-negative weights. Logic Complexity Memory Pl. 0 & 1 LTL 2ExpTime-complete Doubly-exponential LDL 2ExpTime-complete Doubly-exponential PLTL 2ExpTime-complete Doubly-exponential PLDL 2ExpTime-complete Doubly-exponential cPLTL 2ExpTime-complete Doubly-exponential cPLDL 2ExpTime-complete Doubly-exponential

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SLIDE 153

Generalizations

Parametric LDL [FZ14]: full expressive power of the ω-regular languages and bounded operators. Parametric LTL and LDL with costs [Z15]: replace unit cost by non-negative weights. Visibly LDL [WZ15]: full expressive power of the ω-visibly pushdown languages. Logic Complexity Memory Pl. 0 & 1 LTL 2ExpTime-complete Doubly-exponential LDL 2ExpTime-complete Doubly-exponential PLTL 2ExpTime-complete Doubly-exponential PLDL 2ExpTime-complete Doubly-exponential cPLTL 2ExpTime-complete Doubly-exponential cPLDL 2ExpTime-complete Doubly-exponential VLDL 3ExpTime-complete pushdown transducer

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SLIDE 154

Outline

  • 1. Playing Optimally in Variations of Parity Games
  • 2. Playing (Approximatively) Optimally in LTL Games
  • 3. More Tradeoffs
  • 4. Conclusion

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SLIDE 155

Delay Games

Landweber & Hosch: Allow one player to delay her moves to obtain a lookahead on the opponent’s moves. This thesis presents the first in-depth study of delay games.

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SLIDE 156

Delay Games

Landweber & Hosch: Allow one player to delay her moves to obtain a lookahead on the opponent’s moves. This thesis presents the first in-depth study of delay games.

Theorem (KZ15)

Solving ω-regular delay games is ExpTime-complete and exponential lookahead is always sufficient and in general necessary. Best previous result: in 2ExpTime and doubly-exponential upper bound, no non-trivial lower bounds [HKT10].

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SLIDE 157

Delay Games

Landweber & Hosch: Allow one player to delay her moves to obtain a lookahead on the opponent’s moves. This thesis presents the first in-depth study of delay games. More results (incomplete) Solving LTL delay games is 3ExpTime-complete, triply-exponential lookahead sufficient and necessary [KZ16]. Lookahead can be traded for quality in delay games with finitary parity conditions [Z17]. Finite-state strategies for delay games [WZ18]: lookahead can be traded for memory.

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SLIDE 158

More Results

In the thesis, but not covered in this talk: Solving an open problem on average-energy games [BHMRZ17] Optimal strategies for request-response games [HTWZ15] Distributed synthesis for PROMPT–LTL [JTZ16] A first-order logic for Hyperproperties [FZ17] Context-free delay games [FLZ11] Borel determinacy for delay games [KZ15] Delay games with WMSO+U winning conditions [Z15]

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SLIDE 159

Outline

  • 1. Playing Optimally in Variations of Parity Games
  • 2. Playing (Approximatively) Optimally in LTL Games
  • 3. More Tradeoffs
  • 4. Conclusion

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SLIDE 160

Conclusion

Tradeoffs in infinite games exist: Optimality can be prohibitively expensive, both in terms of computational complexity and in terms of memory requirements.

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SLIDE 161

Conclusion

Tradeoffs in infinite games exist: Optimality can be prohibitively expensive, both in terms of computational complexity and in terms of memory requirements. Positive results: Optimal bounds in PROMPT–LTL games can be approximated at no extra cost. The expressiveness of LTL can be increased considerably for free. Lookahead allows to improve strategies and decrease memory requirements.

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SLIDE 162

Conclusion

Tradeoffs in infinite games exist: Optimality can be prohibitively expensive, both in terms of computational complexity and in terms of memory requirements. Positive results: Optimal bounds in PROMPT–LTL games can be approximated at no extra cost. The expressiveness of LTL can be increased considerably for free. Lookahead allows to improve strategies and decrease memory requirements. ⇒ Need to take tradeoffs into account when solving games.

Martin Zimmermann Saarland University Tradeoffs in Infinite Games 36/36