Average-Energy Games Patricia Bouyer 1 Nicolas Markey 2 Mickael - PowerPoint PPT Presentation

Average-Energy Games Patricia Bouyer 1 Nicolas Markey 2 Mickael Randour 3 Kim G. Larsen 4 Simon Laursen 4 1 LSV - CNRS & ENS Cachan 2 IRISA - CNRS Rennes 3 ULB - Universit´ e libre de Bruxelles 4 Aalborg University September 09, 2016 - Highlights 2016 - Brussels

Application Average-energy in a nutshell Conclusion Advertisement To appear in Acta Informatica [BMR + 16]. Full paper available on arXiv [BMR + 15a]: abs/1512.08106 Average-Energy Games Bouyer, Markey, Randour, Larsen, Laursen 1 / 8

Application Average-energy in a nutshell 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 Average-Energy Games Bouyer, Markey, Randour, Larsen, Laursen 2 / 8

Application Average-energy in a nutshell Conclusion Motivating example Hydac oil pump industrial case study [CJL + 09] (Quasimodo research project). Goals: Average-Energy Games Bouyer, Markey, Randour, Larsen, Laursen 3 / 8

Application Average-energy in a nutshell Conclusion Motivating example 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 Average-Energy Games Bouyer, Markey, Randour, Larsen, Laursen 3 / 8

Application Average-energy in a nutshell Conclusion Motivating example 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 Average-Energy Games Bouyer, Markey, Randour, Larsen, Laursen 3 / 8

Application Average-energy in a nutshell Conclusion Motivating example 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 Average-Energy Games Bouyer, Markey, Randour, Larsen, Laursen 3 / 8

Application Average-energy in a nutshell Conclusion Average-energy: illustration − 2 1 Two-player turn-based games with 0 integer weights. 2 0 Average-Energy Games Bouyer, Markey, Randour, Larsen, Laursen 4 / 8

Application Average-energy in a nutshell Conclusion Average-energy: illustration − 2 1 Two-player turn-based games with 0 integer weights. 2 0 Focus on two memoryless strategies. Average-Energy Games Bouyer, Markey, Randour, Larsen, Laursen 4 / 8

Application Average-energy in a nutshell 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 2 2 1 1 Step Step 0 0 0 1 2 3 4 5 6 0 1 2 3 4 5 6 Average-Energy Games Bouyer, Markey, Randour, Larsen, Laursen 4 / 8

Application Average-energy in a nutshell 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. Average-Energy Games Bouyer, Markey, Randour, Larsen, Laursen 4 / 8

Application Average-energy in a nutshell 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. Average-Energy Games Bouyer, Markey, Randour, Larsen, Laursen 4 / 8

Application Average-energy in a nutshell 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. Average-Energy Games Bouyer, Markey, Randour, Larsen, Laursen 4 / 8

Application Average-energy in a nutshell 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. Average-Energy Games Bouyer, Markey, Randour, Larsen, Laursen 4 / 8

Application Average-energy in a nutshell 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. Average-Energy Games Bouyer, Markey, Randour, Larsen, Laursen 4 / 8

Application Average-energy in a nutshell 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. Average-Energy Games Bouyer, Markey, Randour, Larsen, Laursen 4 / 8

Application Average-energy in a nutshell 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). Average-Energy Games Bouyer, Markey, Randour, Larsen, Laursen 4 / 8

Application Average-energy in a nutshell Conclusion Average-energy: overview 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] memoryless [CdAHS03] EG L EXPTIME-c. [BFL + 08] PSPACE-c. [FJ13] pseudo-polynomial EG LU AE P NP ∩ coNP memoryless � For all but EG LU , memoryless strategies suffice. Average-Energy Games Bouyer, Markey, Randour, Larsen, Laursen 5 / 8

Application Average-energy in a nutshell Conclusion Average-energy: overview 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] memoryless [CdAHS03] EG L EXPTIME-c. [BFL + 08] PSPACE-c. [FJ13] pseudo-polynomial EG LU AE P NP ∩ coNP memoryless � For all but EG LU , memoryless strategies suffice. Techniques: � Classical criteria cannot be applied out-of-the-box [EM79, BSV04, AR14, GZ04, Kop06]. Average-Energy Games Bouyer, Markey, Randour, Larsen, Laursen 5 / 8

Application Average-energy in a nutshell Conclusion Average-energy: overview 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] memoryless [CdAHS03] EG L EXPTIME-c. [BFL + 08] PSPACE-c. [FJ13] pseudo-polynomial EG LU AE P NP ∩ coNP memoryless � For all but EG LU , memoryless strategies suffice. Techniques: � Classical criteria cannot be applied out-of-the-box [EM79, BSV04, AR14, GZ04, Kop06]. � 1-player: memorylessness proof and polynomial-time LP-based algorithm. Average-Energy Games Bouyer, Markey, Randour, Larsen, Laursen 5 / 8