Easy to Win, Hard to Master:

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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, 216 MFV Seminar, ULB, Brussels, Belgium Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 1/23

2 Parity Games Deciding winner in UP co-up Positional Strategies Example due to (Fijalkow and Chatterjee, Infinite-state games, 213) Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 2/23

3 Finitary Parity / Parity Response Games steps steps Goal for Player : Bound response times Example due to (Fijalkow and Chatterjee, Infinite-state games, 213) Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 3/23

4 Another Example Parity Finitary Parity Parity Finitary Parity Player 1 wins from every vertex, but needs to stay longer and longer in vertex of color requires infinite memory Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 4/23

5 Decision Problem Theorem (Chatterjee et al., Finitary Winning, 29) 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

6 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

7 From Finitary Parity to Parity Given: Finitary parity game G = (A, FinParity(Ω)), bound b N. Lemma Deciding if Player 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

8 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

9 PSPACE-Hardness Lemma The following problem is PSpace-hard: Given a finitary parity game G and a bound b N, does Player 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 wants to prove truth of ϕ): 1. Player 1 picks truth value for x 2. Player picks truth value for y 3. Player 1 picks clause C 4. Player picks literal l from C 5. Player 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

10 ϕ = x y. The Reduction ψ {}}{ ( x y ) ( x y ) x 1 x 3 y 5 y 7 (x y) ψ ( x y) x... x... y... y Choose bound b such that it enforces the following: x 1 y 3 2 x 1 b steps Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 1/23

11 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

12 Sufficient Memory (for Player ) Corollary Let G be a parity game with costs with d odd colors. If Player 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 = 2 d 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

13 Memory Requirements (for Player ) Theorem Optimal strategies for parity games require exponential memory. Necessity: Construct family G d : d times d times (Fijalkow and Chatterjee, Infinite-state games, 213) For optimal play: Player needs to store d choices of d possible values each Player requires 2 d many memory states Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 13/23

14 Memory Requirements (cont.) Theorem For every d > 1, there exists a finitary parity game G d such that G d O(d 2 ) and G d has d odd colors, and every optimal strategy for Player has at least size 2 d 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 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 = 2 d log(b+2). Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 14/23

15 Introduction Complexity in PSpace PSpace-hard Exponential Memory Sufficient Necessary Tradeoffs Extensions Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 15/23

16 Results so far Parity Winning Finitary Parity Optimal Complexity UP co-up PTime PSpace-comp. Strategies Pos. Pos. Exp. Take-away: Forcing Player 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

17 Introduction Complexity in PSpace PSpace-hard Exponential Memory Sufficient Necessary Tradeoffs Extensions Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 17/23

18 Tradeoffs d request gadgets with d colors d response gadgets with d colors Recall: Player has winning strategy with cost d 2 + 2d and size 2 d 2: store all d requests made by Player 1. Smaller strategy: Only store first i unique requests, then default to largest answer. achieves cost d 2 + 3d i and size i 1 ( n ) j=1 j These are the smallest strategies achieving this cost. Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 18/23

19 Tradeoffs Theorem Fix some finitary parity game G d as before. For every i with 1 i d there exists a strategy σ i for Player in G d such that σ i has cost d 2 + 3d i and size i 1 ( d ) j=1 j. Also, all these strategies are minimal for their respective cost. size cost Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 19/23

20 Introduction Complexity in PSpace PSpace-hard Exponential Memory Sufficient Necessary Tradeoffs Extensions Alexander Weinert Saarland University Optimal Strategies in Parity Games with Costs 2/23

21 Extension 1: Parity Games with Costs 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

22 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 {, 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

23 Conclusion Parity Winning Parity with Costs Optimal Complexity UP co-up UP co-up PSpace-comp. Strategies Pos. Pos. Exp. Streett Winning Streett with Costs 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

The Complexity of Request-Response Games

The Complexity of Request-Response Games Krishnendu Chatterjee 1, Thomas A. Henzinger 1, and Florian Horn 1,2 1 IST (Institute of Science and Technology), Austria {krish.chat,tah}@ist.ac.at 2 LIAFA, CNRS