Reactive Synthesis Without Regret

Transcription

1 Reactive Synthesis Without Regret (Non, rien de rien... ) Guillermo A. Pérez Prague September, 2015

2 That feeling when... Example: doing your laundry sc s G.A. Pérez (ULB) Reactive Synthesis Without Regret September, / 10

3 That feeling when... Example: doing your laundry sc s 4 e 2 ok 0 G.A. Pérez (ULB) Reactive Synthesis Without Regret September, / 10

4 That feeling when... Example: doing your laundry sc s 4 e 2 4 e ok 0 G.A. Pérez (ULB) Reactive Synthesis Without Regret September, / 10

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6 That feeling when... Example: doing your laundry sc s 4 e 2 4 e 4 e ok e e e bl 0 G.A. Pérez (ULB) Reactive Synthesis Without Regret September, / 10

7 That feeling when... Example: doing your laundry I should have... sc s 4 e 2 4 e 4 e ok e e e bl 0 G.A. Pérez (ULB) Reactive Synthesis Without Regret September, / 10

8 What do you mean by regret minimisation? In words... We want to find the strategy of ve that minimises the difference between her actual payoff and the payoff she could have achieved if she had known the strategy of dam in advance. G.A. Pérez (ULB) Reactive Synthesis Without Regret September, / 10

9 What do you mean by regret minimisation? In words... We want to find the strategy of ve that minimises the difference between her actual payoff and the payoff she could have achieved if she had known the strategy of dam in advance. Halpern and Pass: A better solution concept than NE Zwick and Paterson: Competitive analysis of online metrical task systems finite window online string matching selection with limited storage G.A. Pérez (ULB) Reactive Synthesis Without Regret September, / 10

10 What do you mean by regret minimisation? In words... We want to find the strategy of ve that minimises the difference between her actual payoff and the payoff she could have achieved if she had known the strategy of dam in advance. Halpern and Pass: A better solution concept than NE Zwick and Paterson: Competitive analysis of online metrical task systems finite window online string matching selection with limited storage What is a good strategy? On-the-fly determinisation of weighted automata G.A. Pérez (ULB) Reactive Synthesis Without Regret September, / 10

11 Some definitions Key Words Weighted arenas, ve, dam, Infinite plays, Strategies, and Payoff functions G.A. Pérez (ULB) Reactive Synthesis Without Regret September, / 10

12 Some definitions Key Words Weighted arenas, ve, dam, Infinite plays, Strategies, and Payoff functions a, 5 Σ, 10 b, 5 Σ, 5 Σ, 10 q G.A. Pérez (ULB) Reactive Synthesis Without Regret September, / 10

13 Some definitions Key Words Weighted arenas, ve, dam, Infinite plays, Strategies, and Payoff functions a, 5 Σ, 10 b, 5 Σ, 5 Σ, 10 q A payoff function is a mapping Val : Q ω R. Classical ones: sup, inf, lim sup, lim inf, mean-payoff, discounted-sum... G.A. Pérez (ULB) Reactive Synthesis Without Regret September, / 10

14 Choosing amongst optimal strategies Σ, 5 a, 5 Σ, 10 Σ, 0 Σ, 0 b, 0 what is the max value ve can ensure? G.A. Pérez (ULB) Reactive Synthesis Without Regret September, / 10

15 Choosing amongst optimal strategies Σ, 5 a, 5 Σ, 10 Σ, 0 Σ, 0 b, 0 what is the max value ve can ensure? can we do better than going left? G.A. Pérez (ULB) Reactive Synthesis Without Regret September, / 10

16 On-the-fly determinisation Restrict dam to play words, i.e. a strategy τ : N Σ. Σ, 0 Σ, 10 a, 0 Σ, 0 Σ, 0 a, 0 Σ \ a, 0 Σ \ a, 0 Σ, 5 Σ, 4 G.A. Pérez (ULB) Reactive Synthesis Without Regret September, / 10

17 On-the-fly determinisation Restrict dam to play words, i.e. a strategy τ : N Σ. can ve resolve non-determinism so that for every word x, our run ρ x has the same value as A(x)? Σ, 0 Σ, 10 a, 0 Σ, 0 Σ, 0 a, 0 Σ \ a, 0 Σ \ a, 0 Σ, 5 Σ, 4 G.A. Pérez (ULB) Reactive Synthesis Without Regret September, / 10

18 On-the-fly determinisation Restrict dam to play words, i.e. a strategy τ : N Σ. can ve resolve non-determinism so that for every word x, our run ρ x has the same value as A(x)? can she do it so that x : A(x) Val(ρ x ) 1? Σ, 0 Σ, 10 a, 0 Σ, 0 Σ, 0 a, 0 Σ \ a, 0 Σ \ a, 0 Σ, 5 Σ, 4 G.A. Pérez (ULB) Reactive Synthesis Without Regret September, / 10

19 More formally... Let G be a weighted arena, Val be a payoff function, and Σ and Σ be sets of strategies for ve and dam respectively. The regret of σ Σ reg σ Σ,Σ (G) := sup τ Σ ( sup σ Σ Val(σ, τ) Val(σ, τ)) G.A. Pérez (ULB) Reactive Synthesis Without Regret September, / 10

20 More formally... Let G be a weighted arena, Val be a payoff function, and Σ and Σ be sets of strategies for ve and dam respectively. The regret of σ Σ reg σ Σ,Σ (G) := sup τ Σ ( sup σ Σ Val(σ, τ) Val(σ, τ)) The regret of ve in G Reg Σ,Σ (G) := inf σ Σ reg σ Σ,Σ (G) G.A. Pérez (ULB) Reactive Synthesis Without Regret September, / 10

21 More formally... Let G be a weighted arena, Val be a payoff function, and Σ and Σ be sets of strategies for ve and dam respectively. The regret of σ Σ reg σ Σ,Σ (G) := sup τ Σ ( sup σ Σ Val(σ, τ) Val(σ, τ)) The regret of ve in G Reg Σ,Σ (G) := inf σ Σ reg σ Σ,Σ (G) We make assumptions about Σ and let Σ be the set of all strategies of ve. G.A. Pérez (ULB) Reactive Synthesis Without Regret September, / 10

22 Any questions? Summary sup inf lim sup lim inf MP DS Any poly-time equiv to regular game EXPSPACE, DS games hard Positional PSPACE, PSPACE-c EXPSPACE, conp-h PSPACE-h Word EXP-c undec.?? Thank you for your attention, I hope you don t regret having attended this talk. G.A. Pérez (ULB) Reactive Synthesis Without Regret September, / 10