Bounding Average-Energy Games Patricia Bouyer 1 Piotr Hofman 1 , 2 Nicolas Markey 1 , 3 Mickael Randour 4 Martin Zimmermann 5 1 LSV - CNRS & ENS Cachan 2 University of Warsaw 3 IRISA - CNRS & INRIA & U. Rennes 4 ULB - Universit´ e libre de Bruxelles 5 Saarland University March 29, 2017 Formal Methods and Verification seminar — ULB
AE games AE L games Multi-dim. extensions Conclusion The talk in one slide Study of average-energy games: quantitative two-player games where the goal is to minimize the average energy level in the long-run . AE games studied in [BMR + 16], also in conjunction with energy constraints: EG L or EG LU (lower bound only, or lower + upper bounds). Goal of this work Solving a problem left open in [BMR + 16]: two-player games with conjunction of an AE constraint and an EG L one, i.e., AE L games. � To solve them, we make a detour by mean-payoff games on infinite arenas . � We also consider multi-dimensional extensions of AE games. Bounding Average-Energy Games Bouyer, Hofman, Markey, Randour, Zimmermann 1 / 26
AE games AE L games Multi-dim. extensions Conclusion Advertisement Featured in FoSSaCS’17 [BHM + 17]. Full paper available on arXiv [BHM + 16]: abs/1610.07858 Bounding Average-Energy Games Bouyer, Hofman, Markey, Randour, Zimmermann 2 / 26
AE games AE L games Multi-dim. extensions Conclusion 1 Average-energy games 2 Average-energy games with lower-bounded energy 3 Multi-dimensional extensions 4 Conclusion Bounding Average-Energy Games Bouyer, Hofman, Markey, Randour, Zimmermann 3 / 26
AE games AE L games Multi-dim. extensions Conclusion 1 Average-energy games 2 Average-energy games with lower-bounded energy 3 Multi-dimensional extensions 4 Conclusion Bounding Average-Energy Games Bouyer, Hofman, Markey, Randour, Zimmermann 4 / 26
AE games AE L games Multi-dim. extensions Conclusion General context: strategy synthesis in quantitative games system environment informal description description specification 1 How complex is it to decide if model as a model as a winning strategy exists? two-player a winning game objective 2 How complex such a strategy needs to be? Simpler is synthesis better . 3 Can we synthesize one efficiently? is there a winning strategy ? ⇒ Depends on the winning yes no objective . empower system capabilities strategy or weaken = specification controller requirements Bounding Average-Energy Games Bouyer, Hofman, Markey, Randour, Zimmermann 5 / 26
AE games AE L games Multi-dim. extensions Conclusion Motivating example for average-energy Hydac oil pump industrial case study [CJL + 09] (Quasimodo research project). Goals: 1 Keep the oil level in the safe zone. ֒ → Energy objective with lower and upper bounds: EG LU 2 Minimize the average oil level. ֒ → Average-energy objective: AE ⇒ Conjunction: AE LU Bounding Average-Energy Games Bouyer, Hofman, Markey, Randour, Zimmermann 6 / 26
AE games AE L games Multi-dim. extensions Conclusion Average-energy: illustration − 2 1 Two-player turn-based games with 0 integer weights. 2 0 Focus on two memoryless strategies. = ⇒ We look at the energy level ( EL ) along a play. Energy Energy 3 3 � ( EL ≥ 0) � ( EL ≥ 0) 2 2 1 1 Step Step 0 0 0 1 2 3 4 5 6 0 1 2 3 4 5 6 Energy objective ( EG L / EG LU ) : e.g., always maintain EL ≥ 0. Bounding Average-Energy Games Bouyer, Hofman, Markey, Randour, Zimmermann 7 / 26
AE games AE L games Multi-dim. extensions Conclusion Average-energy: illustration − 2 1 Two-player turn-based games with 0 integer weights. 2 0 Focus on two memoryless strategies. = ⇒ We look at the energy level ( EL ) along a play. Energy Energy 3 3 MP = 0 MP = 1 / 3 2 2 1 1 Step Step 0 0 0 1 2 3 4 5 6 0 1 2 3 4 5 6 Mean-payoff ( MP ) : long-run average payoff per transition. Bounding Average-Energy Games Bouyer, Hofman, Markey, Randour, Zimmermann 7 / 26
AE games AE L games Multi-dim. extensions Conclusion Average-energy: illustration − 2 1 Two-player turn-based games with 0 integer weights. 2 − 1 Focus on two memoryless strategies. = ⇒ We look at the energy level ( EL ) along a play. Energy Energy 3 3 MP = 0 MP = 0 2 2 1 1 Step Step 0 0 0 1 2 3 4 5 6 0 1 2 3 4 5 6 Mean-payoff ( MP ) : long-run average payoff per transition. = ⇒ Let’s change the weights of our game. Bounding Average-Energy Games Bouyer, Hofman, Markey, Randour, Zimmermann 7 / 26
AE games AE L games Multi-dim. extensions Conclusion Average-energy: illustration − 2 1 Two-player turn-based games with 0 integer weights. 2 − 1 Focus on two memoryless strategies. = ⇒ We look at the energy level ( EL ) along a play. Energy Energy 3 3 TP = 0 , TP = 2 TP = 0 , TP = 1 2 2 1 1 Step Step 0 0 0 1 2 3 4 5 6 0 1 2 3 4 5 6 Total-payoff ( TP ) refines MP in the case MP = 0 by looking at high/low points of the sequence. Bounding Average-Energy Games Bouyer, Hofman, Markey, Randour, Zimmermann 7 / 26
AE games AE L games Multi-dim. extensions Conclusion Average-energy: illustration − 2 2 Two-player turn-based games with 0 integer weights. 2 − 2 Focus on two memoryless strategies. = ⇒ We look at the energy level ( EL ) along a play. Energy Energy 3 3 TP = 0 , TP = 2 TP = 0 , TP = 2 2 2 1 1 Step Step 0 0 0 1 2 3 4 5 6 0 1 2 3 4 5 6 Total-payoff ( TP ) refines MP in the case MP = 0 by looking at high/low points of the sequence. = ⇒ Let’s change the weights again. Bounding Average-Energy Games Bouyer, Hofman, Markey, Randour, Zimmermann 7 / 26
AE games AE L games Multi-dim. extensions Conclusion Average-energy: illustration − 2 2 Two-player turn-based games with 0 integer weights. 2 − 2 Focus on two memoryless strategies. = ⇒ We look at the energy level ( EL ) along a play. Energy Energy 3 3 2 2 AE = 4 / 3 1 AE = 1 1 Step Step 0 0 0 1 2 3 4 5 6 0 1 2 3 4 5 6 Average-energy ( AE ) further refines TP: average EL along a play. Bounding Average-Energy Games Bouyer, Hofman, Markey, Randour, Zimmermann 7 / 26
AE games AE L games Multi-dim. extensions Conclusion Average-energy: illustration − 2 2 Two-player turn-based games with 0 integer weights. 2 − 2 Focus on two memoryless strategies. = ⇒ We look at the energy level ( EL ) along a play. Energy Energy 3 3 2 2 AE = 4 / 3 1 AE = 1 1 Step Step 0 0 0 1 2 3 4 5 6 0 1 2 3 4 5 6 Average-energy ( AE ) further refines TP: average EL along a play. = ⇒ Natural concept (cf. case study). Bounding Average-Energy Games Bouyer, Hofman, Markey, Randour, Zimmermann 7 / 26
AE games AE L games Multi-dim. extensions Conclusion Formal definitions We consider games G = ( S 0 , S 1 , E ) between players P 0 and P 1 , such that each edge e ∈ E has an integer weight w ( e ). For a prefix ρ = ( e i ) 1 ≤ i ≤ n , we define its energy level as EL ( ρ ) = � n i =1 w ( e i ); � n its mean-payoff as MP ( ρ ) = 1 i =1 w ( e i ) = 1 n EL ( ρ ); n � n its average-energy as AE( ρ ) = 1 i =1 EL ( ρ ≤ i ) . n Natural extensions to plays by taking the upper-limit, e.g., n 1 � AE( π ) = lim sup EL ( π ≤ i ) . n n →∞ i =1 Bounding Average-Energy Games Bouyer, Hofman, Markey, Randour, Zimmermann 8 / 26
AE games AE L games Multi-dim. extensions Conclusion Overview of known results Objective 1-player 2-player memory MP P [Kar78] NP ∩ coNP [ZP96] memoryless [EM79] TP P [FV97] NP ∩ coNP [GS09] memoryless [GZ04] P [BFL + 08] NP ∩ coNP [CdAHS03, BFL + 08] EG L memoryless [CdAHS03] EXPTIME-c. [BFL + 08] EG LU PSPACE-c. [FJ15] exponential AE P NP ∩ coNP memoryless AE LU PSPACE-c. EXPTIME-c. exponential PSPACE-e./NP-h. open /EXPTIME-h. open ( ≥ exp.) AE L � Results without references are proved in [BMR + 16]. � The one-player AE L case is solved by reduction to an AE LU game for a sufficiently large upper bound U , obtained through results on one-counter automata that permit to bound the counter value along a path. = ⇒ Let’s first recall how we solve AE LU games. Bounding Average-Energy Games Bouyer, Hofman, Markey, Randour, Zimmermann 9 / 26
AE games AE L games Multi-dim. extensions Conclusion With energy constraints, memory is needed! AE LU � minimize AE while keeping EL ∈ [0 , 3] (init. EL = 0). Energy 0 1 3 2 a c b AE = 3 / 2 1 − 3 0 Step 0 2 1 2 3 4 5 6 7 8 (a) One-player AE LU game. (b) Play π 1 = ( acacacab ) ω . Energy Energy 3 3 2 2 AE = 8 / 5 1 1 AE = 1 Step Step 0 0 1 2 3 4 5 1 2 3 4 5 (c) Play π 2 = ( aacab ) ω . (d) Play π 3 = ( acaab ) ω . Minimal AE with π 3 : alternating between the +1, +2 and − 3 cycles. Bounding Average-Energy Games Bouyer, Hofman, Markey, Randour, Zimmermann 10 / 26
AE games AE L games Multi-dim. extensions Conclusion With energy constraints, memory is needed! AE LU � minimize AE while keeping EL ∈ [0 , 3] (init. EL = 0). Non-trivial behavior in general! ֒ → Need to choose carefully which cycles to play. The AE LU problem is EXPTIME-complete. ֒ → Reduction from AE LU to AE on pseudo-polynomial game ( = ⇒ AE LU ∈ NEXPTIME ∩ coNEXPTIME). ֒ → Reduction from this AE game to MP game + pseudo-poly. algorithm. Bounding Average-Energy Games Bouyer, Hofman, Markey, Randour, Zimmermann 10 / 26
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