Easy to Win, Hard to Master:
Playing Infinite Games Optimally
Alexander Weinert
Saarland University
April 26th, 2017
Thesis Proposal Talk
Alexander Weinert
Saarland University
Playing Infinite Games Optimally
1/13
Programming
Program
Alexander Weinert
Saarland University
Playing Infinite Games Optimally
2/13
Program Verification
Specification ϕ
Program
Verifier
3
7
Alexander Weinert
Saarland University
Playing Infinite Games Optimally
3/13
Program Synthesis
Specification ϕ
Synthesis
3
Program
7
Alexander Weinert
Saarland University
Playing Infinite Games Optimally
4/13
Alexander Weinert
Saarland University
Playing Infinite Games Optimally
SNALUBMA
SNALUBMA
SNALUBMA
Example
5/13
Program Synthesis
Specification ϕ
Synthesis
3
Program
7
Alexander Weinert
Saarland University
Playing Infinite Games Optimally
6/13
SNALUBMA
Parity Games
1
2
0
0
0
0
0
3
0
1
0
3
4
0
0
0
0
2
0
4
0
···
Example due to (Fijalkow and Chatterjee, Infinite-state games, 2013)
Deciding winner in NP ∩ co-NP
Alexander Weinert
Saarland University
Positional Strategies
Playing Infinite Games Optimally
7/13
SNALUBMA
Finitary Parity Games
1
2
0
0
0
0
0
3
4
3
0
1
0
3
0
0
0
0
2
0
4
0
···
3
Goal for Player 0: Bound response times
Alexander Weinert
Saarland University
Playing Infinite Games Optimally
8/13
Decision Problem
Theorem (Chatterjee, Henzinger, Horn, 2009)
The following decision problem is in PTime:
Input: Finitary parity game G
Question: Does there exist a strategy σ with Cst(σ) < ∞?
Theorem (W., Zimmermann, 2016)
The following decision problem is PSpace-complete:
Input: Finitary parity game G, bound b ∈ N
Question: Does there exist a strategy σ with Cst(σ) ≤ b?
Alexander Weinert
Saarland University
Playing Infinite Games Optimally
9/13
Memory Requirements (for Player 0)
Theorem (W., Zimmermann, 2016)
Optimal strategies for finitary parity games need exponential
memory
Sufficiency: Corollary of proof of PSpace-membership
Necessity:
G1
···
d times
G1
G0
···
G0
d times
Player 0 needs to recall d positions with d possible values
⇒ Player 0 requires ≈ 2d many memory states
Alexander Weinert
Saarland University
Playing Infinite Games Optimally
10/13
Results so far
Parity
Complexity
Strategy Size
Alexander Weinert
NP ∩ co-NP
1
Saarland University
Finitary Parity
Winning
Optimal
PTime
1
PSpace-comp.
Exp.
Playing Infinite Games Optimally
11/13
Outlook
So far
Imperfect Information
SNALUBMA
Multi-Dimensional Games
Alexander Weinert
Saarland University
Playing Infinite Games Optimally
12/13
Conclusion
Results so far: 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
Guiding Question: What costs does playing games optimally
incur
in terms of computing a strategy?
in terms of the complexity of strategies?
Alexander Weinert
Saarland University
Playing Infinite Games Optimally
13/13