A journey through negatively-weighted timed games: undecidability, decidability, approximability

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(1)A journey through negatively-weighted timed games: undecidability, decidability, approximability Benjamin Monmege, Aix-Marseille Université MoRe 2018, Oxford.

(2) Motivation: quantitative aspects of real-time synthesis Environment. k. Controller??. |=. Spec. 2/33.

(3) Motivation: quantitative aspects of real-time synthesis Environment. k. Controller??. |=. Spec. Real-time requirements/environment =⇒ real-time controller. 2/33.

(4) Motivation: quantitative aspects of real-time synthesis Environment. k. Controller??. |=. Spec. Real-time requirements/environment =⇒ real-time controller. Among all valid controllers, choose a cheap/efficient one. 2/33.

(5) Motivation: quantitative aspects of real-time synthesis Environment. k. Controller??. |=. Spec. Two-player game Real-time requirements/environment =⇒ real-time controller. Among all valid controllers, choose a cheap/efficient one. 2/33.

(6) Motivation: quantitative aspects of real-time synthesis Environment. k. Controller??. |=. Spec. Two-player game Real-time requirements/environment =⇒ real-time controller Two-player timed game Among all valid controllers, choose a cheap/efficient one. 2/33.

(7) Motivation: quantitative aspects of real-time synthesis Environment. k. Controller??. |=. Spec. Two-player game Real-time requirements/environment =⇒ real-time controller Two-player timed game Among all valid controllers, choose a cheap/efficient one Two-player weighted timed game. 2/33.

(8) Motivation: quantitative aspects of real-time synthesis Environment. k. Controller??. |=. Spec. Two-player game Real-time requirements/environment =⇒ real-time controller Two-player timed game Among all valid controllers, choose a cheap/efficient one Two-player weighted timed game Additional difficulty: negative weights =⇒ to model production/consumption of resources. 2/33.

(9) Modelling via weighted timed games Peak-hour rate:. 15 ce/kWh total power × 15 ce/h. Offpeak-hour 12 ce/kWh total power × 12 ce/h. states to record which device is on/off: computation of the total power. 3/33.

(10) Modelling via weighted timed games Peak-hour rate:. 15 ce/kWh total power × 15 ce/h. Offpeak-hour 12 ce/kWh total power × 12 ce/h. states to record which device is on/off: computation of the total power Power consumption: I. 100W (1.5 ce/h in peak-hour, 1.2 ce/h in offpeak-hour). I. 2500W (37.5 ce/h in peak-hour, 30 ce/h in offpeak-hour). I. 2000W (24 ce/h in offpeak-hour). 3/33.

(11) Modelling via weighted timed games Peak-hour rate:. 15 ce/kWh total power × 15 ce/h. Offpeak-hour. Solar panels. 12 ce/kWh total power × 12 ce/h. Reselling: 20 ce/kWh −0.5 × 20 ce/h. states to record which device is on/off: computation of the total power. 3/33.

(12) Modelling via weighted timed games Peak-hour rate:. 15 ce/kWh total power × 15 ce/h. Offpeak-hour. Solar panels. 12 ce/kWh total power × 12 ce/h. Reselling: 20 ce/kWh −0.5 × 20 ce/h. states to record which device is on/off: computation of the total power Environment: user profile, weather profile / Controller: chooses contract (discrete cost for the monthly subscription) and exact consumption (what, when...). 3/33.

(13) Modelling via weighted timed games Peak-hour rate:. 15 ce/kWh total power × 15 ce/h. Offpeak-hour. Solar panels. 12 ce/kWh total power × 12 ce/h. Reselling: 20 ce/kWh −0.5 × 20 ce/h. states to record which device is on/off: computation of the total power Environment: user profile, weather profile / Controller: chooses contract (discrete cost for the monthly subscription) and exact consumption (what, when...) Goal: optimise the energy consumption based on the cost. 3/33.

(14) Modelling via weighted timed games Peak-hour rate:. 15 ce/kWh total power × 15 ce/h. Offpeak-hour. Solar panels. 12 ce/kWh total power × 12 ce/h. Reselling: 20 ce/kWh −0.5 × 20 ce/h. states to record which device is on/off: computation of the total power Environment: user profile, weather profile / Controller: chooses contract (discrete cost for the monthly subscription) and exact consumption (what, when...) Goal: optimise the energy consumption based on the cost Solution 1 : discretisation of time, resolution via a weighted game Solution 2 : thin time behaviours, resolution via a weighted timed game. 3/33.

(15) Weighted games 0 v2. v3. 0. 0. 1. v1 1. v4. 0. v5. v6. Weighted graph with vertices partition between 2 players + reachability objective. 2. 1. Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 4/33.

(16) Weighted games 0 v2. v3. 0. 0. 1. v1 1. v4. 0. v5. v6. Weighted graph with vertices partition between 2 players + reachability objective. 2. 1. v1. Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 4/33.

(17) Weighted games 0 v2. v3. 0. 0. 1. v1 1 &. v4. 0. v5. v6. Weighted graph with vertices partition between 2 players + reachability objective. 2. 1. v1 −→v4. Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 4/33.

(18) Weighted games 0 v2. v3. 0. 0. 1. v1 1 &. →. v4. 0. v5. v6. Weighted graph with vertices partition between 2 players + reachability objective. 2. 1. v1 −→v4 −→v5. Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 4/33.

(19) Weighted games 0 v2. v3. 0. 0. 1. v1 1 &. →. v4. ←. v5. 0 →. 1. v6. Weighted graph with vertices partition between 2 players + reachability objective. 2. %. v1 −→v4 −→v5 −→v4 −→v5 −→. Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 4/33.

(20) Weighted games 0 v2. v3. 0. 0. 1. v1 1 &. →. v4. ←. v5. 0 →. 1. v6. Weighted graph with vertices partition between 2 players + reachability objective. 2. %. v1 −→v4 −→v5 −→v4 −→v5 −→ 1 +1 +2. =4. Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 4/33.

(21) Weighted games 0 v2. v3. 0. 0. 1. v1 1 &. →. %. →. v4. ←. v5. 0 →. 1. ( Weight of a path:. 2. %. v1 −→v4 −→v5 −→v4 −→v5 −→ 1 +1 +2 v1 −→v2 −→v3 −→v3 −→v3. v6. Weighted graph with vertices partition between 2 players + reachability objective. ··· ···. =4 = +∞( not reached). +∞ total weight until . Benjamin Monmege (Aix-Marseille Université). if not reached otherwise. Min = #, Max = 2. 4/33.

(22) Weighted timed games x <1 x := 0 s2 x >0 x := 0. s3. x 62. x >1 x >1 x := 0. s1 x 61 s4. x >1 s5. s6. Timed automaton with state partition between 2 players + reachability objective. x >1 x := 0. Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 5/33.

(23) Weighted timed games x <1 x := 0 s2 x >0 x := 0. 2 0. 0. 0 s3 −1 x >1 1. x >1 x := 0. 1 s1 x 61 1. x 62. x >1. s4. 0. s5. −3. x >1 x := 0. 1. 2. s6. Timed automaton with state partition between 2 players + reachability objective + linear rates on states + discrete weights on transitions. 0 (s1 , 0). Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 5/33.

(24) Weighted timed games x <1 x := 0 s2 x >0 x := 0. 2 0. 0. 0 s3 −1 x >1 1. x >1 x := 0. 1 s1 x 61 1. x 62. x >1. s4. 0. s5. −3. x >1 x := 0. 1. 2. s6. Timed automaton with state partition between 2 players + reachability objective + linear rates on states + discrete weights on transitions. 0 0.4,&. (s1 , 0)−−−−→(s4 , 0.4). Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 5/33.

(25) Weighted timed games x <1 x := 0 s2 x >0 x := 0. 2 0. 0. 0 s3 −1 x >1 1. x >1 x := 0. 1 s1 x 61 1. x 62. x >1. s4. 0. s5. −3. x >1 x := 0. 1. 2. s6. Timed automaton with state partition between 2 players + reachability objective + linear rates on states + discrete weights on transitions. 0 0.4,&. 0.6,→. (s1 , 0)−−−−→(s4 , 0.4)−−−−→(s5 , 0). Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 5/33.

(26) Weighted timed games x <1 x := 0 s2 x >0 x := 0. 2 0. 0. 0 s3 −1 x >1 1. x >1 x := 0. 1 s1 x 61 1. x 62. x >1. s4. 0. s5. −3. x >1 x := 0. 1. s6. 2. Timed automaton with state partition between 2 players + reachability objective + linear rates on states + discrete weights on transitions. 0 0.4,&. 0.6,→. 1.5,←. 1.1,→. 2,%. (s1 , 0)−−−−→(s4 , 0.4)−−−−→(s5 , 0)−−−−→(s4 , 0)−−−−→(s5 , 0)−−−→(, 2). Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 5/33.

(27) Weighted timed games x <1 x := 0 s2 x >0 x := 0. 2 0. 0. 0 s3. x 61. Timed automaton with state partition between 2 players + reachability objective + linear rates on states + discrete weights on transitions. −1 x >1 1. x >1 x := 0. 1 s1 1. x 62. x >1. s4. 0. s5. −3. x >1 x := 0. 1. s6. 2. 0 0.4,&. 0.6,→. 1.5,←. 1.1,→. 2,%. 1×0.4+1. −3×0.6+0. +1×1.5+0. −3×1.1+0. +1×2+2. (s1 , 0)−−−−→(s4 , 0.4)−−−−→(s5 , 0)−−−−→(s4 , 0)−−−−→(s5 , 0)−−−→(, 2). Benjamin Monmege (Aix-Marseille Université). = 1.8. Min = #, Max = 2. 5/33.

(28) Weighted timed games x <1 x := 0 s2 x >0 x := 0. 2 0. 0. 0 s3. x 61. Timed automaton with state partition between 2 players + reachability objective + linear rates on states + discrete weights on transitions. −1 x >1 1. x >1 x := 0. 1 s1 1. x 62. x >1. s4. 0. s5. −3. x >1 x := 0. 1. s6. 2. 0 0.4,&. 0.6,→. 1.5,←. 1.1,→. 2,%. 1×0.4+1. −3×0.6+0. +1×1.5+0. −3×1.1+0. +1×2+2. (s1 , 0)−−−−→(s4 , 0.4)−−−−→(s5 , 0)−−−−→(s4 , 0)−−−−→(s5 , 0)−−−→(, 2) 0.2,%. 0.9,→. 1×0.2+0. +2×0.9+0. 0.2,. 0.9,. (s1 , 0)−−−−→(s2 , 0)−−−−→(s3 , 0.9)− −−− →(s3 , 0)− −−− →(s3 , 0) −1×0.2+0. −1×0.9+0. ( +∞ Weight of an execution : total weight until . Benjamin Monmege (Aix-Marseille Université). = 1.8. ··· ···. = +∞. if not reached otherwise. Min = #, Max = 2. 5/33.

(29) Strategies and objectives x <1 x := 0 s2 x >0 x := 0. 2 0. 0. 0 s3 −1 x >1 1. x >1 x := 0. 1 s1 x 61 1. x 62. x >1. s4. 0. s5. −3. x >1 x := 0. 1. s6. 2. 0. Strategy for a player: map finite executions to a delay and a transition. Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 6/33.

(30) Strategies and objectives x <1 x := 0 s2 x >0 x := 0. 2 0. 0. 0 s3 −1 x >1 1. x >1 x := 0. 1 s1 x 61 1. x 62. x >1. s4. 0. s5. −3. x >1 x := 0. 1. s6. 2. 0. Strategy for a player: map finite executions to a delay and a transition Objective of player #: reach and minimise the weight Objective of player 2: avoid or, if not possible, maximise the weight. Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 6/33.

(31) Strategies and objectives x <1 x := 0 s2 x >0 x := 0. 2 0. 0. 0 s3 −1 x >1 1. x >1 x := 0. 1 s1 x 61 1. x 62. x >1. s4. 0. s5. −3. x >1 x := 0. 1. s6. 2. 0. Strategy for a player: map finite executions to a delay and a transition Objective of player #: reach and minimise the weight Objective of player 2: avoid or, if not possible, maximise the weight Main object of interest: Val(s, ν) = inf. sup. σMin ∈StratMin σMax ∈StratMax. Weight(Exec(s, ν, σMin , σMax )) ∈ R. What weight can players guarantee? Following which strategies? Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 6/33.

(32) Part I : Weighted games.

(33) State of the art: weighted games (shortest-path objective) F6K : ∃ a strategy in the weighted game for player # reaching with a cost 6 K ? I one-player: shortest path in a weighted graph... polynomial algo. I two players, non-negative weights only: polynomial algo. "Dijkstra algorithm on 2 players games" (Khachiyan et al., 2008). Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 8/33.

(34) State of the art: weighted games (shortest-path objective) F6K : ∃ a strategy in the weighted game for player # reaching with a cost 6 K ? I one-player: shortest path in a weighted graph... polynomial algo. I two players, non-negative weights only: polynomial algo. "Dijkstra algorithm on 2 players games" (Khachiyan et al., 2008) I two players, arbitrary weights? −1 0 0. −W . Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 8/33.

(35) State of the art: weighted games (shortest-path objective) F6K : ∃ a strategy in the weighted game for player # reaching with a cost 6 K ? I one-player: shortest path in a weighted graph... polynomial algo. I two players, non-negative weights only: polynomial algo. "Dijkstra algorithm on 2 players games" (Khachiyan et al., 2008) I two players, arbitrary weights? −1 0 0. −W . # needs memory! Value −∞: detection is as hard as solving parity games (NP ∩ co-NP) Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 8/33.

(36) Pseudo-polynomial algorithm to solve weighted games Joint work with Thomas Brihaye, Gilles Geeraerts and Axel Haddad (Brihaye et al., 2016) i. Value iteration algorithm: compute F (+∞)...  Weight(e) + x v 0  min0 e=(v ,a,v )∈E F(x)v =  max Weight(e) + x v 0 0 e=(v ,a,v )∈E. −1. horizon 0:. 0 0. −W . if v ∈ VMin if v ∈ VMax 2. +∞. #. +∞.

(37) Pseudo-polynomial algorithm to solve weighted games Joint work with Thomas Brihaye, Gilles Geeraerts and Axel Haddad (Brihaye et al., 2016) i. Value iteration algorithm: compute F (+∞)...  Weight(e) + x v 0  min0 e=(v ,a,v )∈E F(x)v =  max Weight(e) + x v 0 0 e=(v ,a,v )∈E. −1. horizon 0: horizon 1:. 0 0. −W . if v ∈ VMin if v ∈ VMax 2. +∞ +∞. #. +∞ 0.

(38) Pseudo-polynomial algorithm to solve weighted games Joint work with Thomas Brihaye, Gilles Geeraerts and Axel Haddad (Brihaye et al., 2016) i. Value iteration algorithm: compute F (+∞)...  Weight(e) + x v 0  min0 e=(v ,a,v )∈E F(x)v =  max Weight(e) + x v 0 0 e=(v ,a,v )∈E. −1. horizon 0: horizon 1: horizon 2:. 0 0. −W . if v ∈ VMin if v ∈ VMax 2. +∞ +∞ −1. #. +∞ 0 0.

(39) Pseudo-polynomial algorithm to solve weighted games Joint work with Thomas Brihaye, Gilles Geeraerts and Axel Haddad (Brihaye et al., 2016) i. Value iteration algorithm: compute F (+∞)...  Weight(e) + x v 0  min0 e=(v ,a,v )∈E F(x)v =  max Weight(e) + x v 0 0 e=(v ,a,v )∈E. −1. horizon horizon horizon horizon. 0 0. −W . 0: 1: 2: 3:. if v ∈ VMin if v ∈ VMax 2. +∞ +∞ −1 −1. #. +∞ 0 0 −1.

(40) Pseudo-polynomial algorithm to solve weighted games Joint work with Thomas Brihaye, Gilles Geeraerts and Axel Haddad (Brihaye et al., 2016) i. Value iteration algorithm: compute F (+∞)...  Weight(e) + x v 0  min0 e=(v ,a,v )∈E F(x)v =  max Weight(e) + x v 0 0 e=(v ,a,v )∈E. −1 0 0. −W . horizon horizon horizon horizon horizon. 0: 1: 2: 3: 4:. if v ∈ VMin if v ∈ VMax 2. +∞ +∞ −1 −1 −2. #. +∞ 0 0 −1 −1.

(41) Pseudo-polynomial algorithm to solve weighted games Joint work with Thomas Brihaye, Gilles Geeraerts and Axel Haddad (Brihaye et al., 2016) i. Value iteration algorithm: compute F (+∞)...  Weight(e) + x v 0  min0 e=(v ,a,v )∈E F(x)v =  max Weight(e) + x v 0 0 e=(v ,a,v )∈E. −1 0 0. −W . horizon horizon horizon horizon horizon. 0: 1: 2: 3: 4:. horizon 2W + 1:. if v ∈ VMin if v ∈ VMax 2. +∞ +∞ −1 −1 −2 ... −W. #. +∞ 0 0 −1 −1 ... −W.

(42) Pseudo-polynomial algorithm to solve weighted games Joint work with Thomas Brihaye, Gilles Geeraerts and Axel Haddad (Brihaye et al., 2016) i. e=(v ,a,v )∈E. −1. horizon horizon horizon horizon horizon. 0 0. −W . 0: 1: 2: 3: 4:. horizon 2W + 1: horizon 2W + 2:. Benjamin Monmege (Aix-Marseille Université). if v ∈ VMin if v ∈ VMax 2. +∞ +∞ −1 −1 −2 ... −W −W. #. +∞ 0 0 −1 −1 ... −W −W. strategy of #. Value iteration algorithm: compute F (+∞)...  Weight(e) + x v 0  min0 e=(v ,a,v )∈E F(x)v =  max Weight(e) + x v 0 0. Min = #, Max = 2. 9/33.

(43) Pseudo-polynomial algorithm to solve weighted games Joint work with Thomas Brihaye, Gilles Geeraerts and Axel Haddad (Brihaye et al., 2016) i. e=(v ,a,v )∈E. −1. horizon horizon horizon horizon horizon. 0 0. −W. Theorem:. . 0: 1: 2: 3: 4:. horizon 2W + 1: horizon 2W + 2:. if v ∈ VMin if v ∈ VMax 2. +∞ +∞ −1 −1 −2 ... −W −W. #. +∞ 0 0 −1 −1 ... −W −W. strategy of #. Value iteration algorithm: compute F (+∞)...  Weight(e) + x v 0  min0 e=(v ,a,v )∈E F(x)v =  max Weight(e) + x v 0 0. We can compute in pseudo-polynomial time the value of a weighted game, as well as optimal strategies for both players: # may require (pseudopolynomial) memory to play optimally (but has counter strategies), 2 has optimal memoryless strategy. Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 9/33.

(44) Large polynomial fragment: divergent weighted games Joint work with Damien Busatto-Gaston and Pierre-Alain Reynier (Busatto-Gaston et al., 2017). Divergence property (in the underlying graph): Every cycle has total weight either 6 −1 or > 1. Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 10/33.

(45) Large polynomial fragment: divergent weighted games Joint work with Damien Busatto-Gaston and Pierre-Alain Reynier (Busatto-Gaston et al., 2017). Divergence property (in the underlying graph): Every cycle has total weight either 6 −1 or > 1. Theorem: We can compute in polynomial time the value of a divergent weighted game, as well as optimal strategies for both players.. Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 10/33.

(46) Large polynomial fragment: divergent weighted games Joint work with Damien Busatto-Gaston and Pierre-Alain Reynier (Busatto-Gaston et al., 2017). Divergence property (in the underlying graph): Every cycle has total weight either 6 −1 or > 1. Theorem: We can compute in polynomial time the value of a divergent weighted game, as well as optimal strategies for both players.. Theorem: Deciding if a weighted game is divergent is in PTIME.. Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 10/33.

(47) Divergent weighted games analysis divergence property. characterisation : p>1. −q 6 −1. Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 11/33.

(48) Divergent weighted games analysis divergence property. characterisation : All the simple cycles in a SCC have the same sign p>1. p>1 α β. −q 6 −1. −q 6 −1. Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 11/33.

(49) Divergent weighted games analysis. divergence property. characterisation : All the simple cycles in a SCC have the same sign. class decision. Benjamin Monmege (Aix-Marseille Université). value computation. Min = #, Max = 2. 11/33.

(50) Value computation in a divergent weighted game I Detect and remove +∞ vertices (outside of the attractor of player # toward ). Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 12/33.

(51) Value computation in a divergent weighted game I Detect and remove +∞ vertices (outside of the attractor of player # toward ) I SCC decomposition I Value computation SCC by SCC, bottom-up. Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 12/33.

(52) Value computation in a divergent weighted game I Detect and remove +∞ vertices (outside of the attractor of player # toward ) I SCC decomposition I Value computation SCC by SCC, bottom-up. positive SCC I The "value iteration" algorithm converges in linear time. Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 12/33.

(53) Value computation in a divergent weighted game I Detect and remove +∞ vertices (outside of the attractor of player # toward ) I SCC decomposition I Value computation SCC by SCC, bottom-up. positive SCC I The "value iteration" algorithm converges in linear time. negative SCC I Outside of the attractor of player 2 toward ⇒ −∞ I The "value iteration" algorithm converges in linear time with initialisation at −∞. Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 12/33.

(54) Example. −1 1. v4. v8 1. v6. −1 v1. v2 −1. v5 −10. 2. −1. . vf. v7. −1. v9 −1 v3. Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 13/33.

(55) Example. −1 1. +∞ v4. v8 1. v6. −1 v1. v2 −1. v5 −10. 2. −1. . vf. v7. −1. +∞ v9 −1 v3. Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 13/33.

(56) Example. −1 1. +∞ v4. v8 1. −1 v1. +∞. v6 v2 −1. v5 −10. 2. −1. . vf. v7. −1. +∞. +∞ v9. −1 v3. Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 13/33.

(57) Example. −1 1. +∞ v4. v8 1. −1 v1. 0. v6 v2 −1. v5 −10. 2. −1. . vf. v7. −1. +∞. +∞ v9. −1 v3. Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 13/33.

(58) Example. −1 1. +∞ v4. v8 1. −1 v1. 0. v6 v2 −1. v5 −10. −1. 2. . vf. v7. −1. +∞. 2 v9. −1 v3. Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 13/33.

(59) Example. −1 1. +∞ v4. v8 1. 1 −1 v1. 0. v6 v2 −1. v5 −10. −1. 2. . vf. v7. −1. +∞. 2 v9. −1 v3. Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 13/33.

(60) Example. −1 1. +∞ v4. v8 1. 1 −1 v1. 0. v6 v2 −1. v5 −10. −1. 2. 1. . vf. v7. −1. +∞. 2 v9. −1 v3. Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 13/33.

(61) Example. −1 1. +∞ v4. v8 1. 1 −1 v1 −∞. 0. v6 v2 −1. −∞. v5 −10. −1. 2. 1. . vf. v7. −1. +∞. 2 v9. −∞ −1 v3. Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 13/33.

(62) Example. −1 1. +∞ v4. v8 1. 1 −1 v1 −∞. 0. v6 v2 −1. −9. v5 −10. −1. 2. 1. . vf. v7. −1. +∞. 2 v9. −∞ −1 v3. Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 13/33.

(63) Example. −1 1. +∞ v4. v8 1. 1 −1 v1 −∞. 0. v6 v2 −1. −9. v5 −10. −1. 2. 1. . vf. v7. −1. +∞. 2 v9. −9 −1 v3. Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 13/33.

(64) Part II : Weighted timed games.

(65) State of the art F6K : ∃ a strategy in the WTG (weighted timed game) for player # reaching with a cost 6 K ?. Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 15/33.

(66) State of the art F6K : ∃ a strategy in the WTG (weighted timed game) for player # reaching with a cost 6 K ?. I One-player case (Weighted timed automata): optimal reachability problem is PSPACE-complete I Algorithm based on regions (Bouyer et al., 2004a, 2007); I and hardness shown for timed automata with at least 2 clocks (Fearnley and Jurdziński, 2013; Haase et al., 2012). Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 15/33.

(67) State of the art F6K : ∃ a strategy in the WTG (weighted timed game) for player # reaching with a cost 6 K ?. I One-player case (Weighted timed automata): optimal reachability problem is PSPACE-complete I Algorithm based on regions (Bouyer et al., 2004a, 2007); I and hardness shown for timed automata with at least 2 clocks (Fearnley and Jurdziński, 2013; Haase et al., 2012). I 2-player WTGs: undecidable (Brihaye et al., 2005; Bouyer et al., 2006a), even with only non-negative weights and 3 clocks (only 2 clocks needed with arbitrary weights (Brihaye et al., 2014)). Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 15/33.

(68) State of the art F6K : ∃ a strategy in the WTG (weighted timed game) for player # reaching with a cost 6 K ?. I One-player case (Weighted timed automata): optimal reachability problem is PSPACE-complete I Algorithm based on regions (Bouyer et al., 2004a, 2007); I and hardness shown for timed automata with at least 2 clocks (Fearnley and Jurdziński, 2013; Haase et al., 2012). I 2-player WTGs: undecidable (Brihaye et al., 2005; Bouyer et al., 2006a), even with only non-negative weights and 3 clocks (only 2 clocks needed with arbitrary weights (Brihaye et al., 2014)) I WTGs with non-negative weights and strictly non-Zeno weight cycles: 2-exponential algorithm (Bouyer et al., 2004b; Alur et al., 2004a). Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 15/33.

(69) State of the art F6K : ∃ a strategy in the WTG (weighted timed game) for player # reaching with a cost 6 K ?. I One-player case (Weighted timed automata): optimal reachability problem is PSPACE-complete I Algorithm based on regions (Bouyer et al., 2004a, 2007); I and hardness shown for timed automata with at least 2 clocks (Fearnley and Jurdziński, 2013; Haase et al., 2012). I 2-player WTGs: undecidable (Brihaye et al., 2005; Bouyer et al., 2006a), even with only non-negative weights and 3 clocks (only 2 clocks needed with arbitrary weights (Brihaye et al., 2014)) I WTGs with non-negative weights and strictly non-Zeno weight cycles: 2-exponential algorithm (Bouyer et al., 2004b; Alur et al., 2004a) I One-clock WTGs with non-negative weights: exponential algorithm (Bouyer et al., 2006b; Rutkowski, 2011; Hansen et al., 2013). Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 15/33.

(70) State of the art F6K : ∃ a strategy in the WTG (weighted timed game) for player # reaching with a cost 6 K ?. I One-player case (Weighted timed automata): optimal reachability problem is PSPACE-complete I Algorithm based on regions (Bouyer et al., 2004a, 2007); I and hardness shown for timed automata with at least 2 clocks (Fearnley and Jurdziński, 2013; Haase et al., 2012). I 2-player WTGs: undecidable (Brihaye et al., 2005; Bouyer et al., 2006a), even with only non-negative weights and 3 clocks (only 2 clocks needed with arbitrary weights (Brihaye et al., 2014)) I WTGs with non-negative weights and strictly non-Zeno weight cycles: 2-exponential algorithm (Bouyer et al., 2004b; Alur et al., 2004a) I One-clock WTGs with non-negative weights: exponential algorithm (Bouyer et al., 2006b; Rutkowski, 2011; Hansen et al., 2013) I Decidability results for WTGs with arbitrary weights? Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 15/33.

(71) – ν ≈ ν′; – ν̄i < ε iff ν̄i′ < ε for all i ∈ {1, . . . , n} with νi ≤ ci ; – 1 − ε < ν̄i iff 1 − ε < ν̄i′ for all i ∈ {1, . . . , n} with νi ≤ ci .. One-player case: weighted timed automata I Main tool: refinement of regions via corner point abstraction /. Fig. 6 indicates the partition induced by the ε-equivalence for the timed automaton of Fig. 2. ε-graph (Bouyer et al., 2004a, 2007) x2. x1 Fig. 6. The ε-equivalence ≈ε. The relation ≈ε is extended to the states of TA as done previously with ≈. An equivalence class is called -region. The ε-region to which a state q belongs is denoted [q]ε and the set of all ε-regions is denoted by Rε . In order to define the ε-region graph of a timed automaton A, we do not need all the ε-regions of Rε (contrarily he construction of RA ). Due to Lemma 3, we only need to consider the ε-regions [(l, ν)]ε whose clock values ν lose enough to n-tuples of integers (the dashed zones on Fig. 6).. Definition 12. Given a timed automaton A and ε ∈]0, 21 ], the set of acceptable ε-regions, denoted S ε , is defined b ! Benjamin Monmege (Aix-Marseille Université) Min = #, Max = 2" 16/33 ε ε.

(72) One-clock Bi-Valued WTGs (1BWTGs). Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 17/33.

(73) One-clock Bi-Valued WTGs (1BWTGs) Joint work with Thomas Brihaye, Gilles Geeraerts, Shankara Krishna Narayanan, Lakshmi Manasa and Ashutosh Trivedi (Brihaye et al., 2014). Assumption: rates of states {p − , p + } included in {0, +d, −d} (d ∈ N) (no assumption on costs of transitions) x < 1, x := 0, 0 s2 x >0 x := 0, 0. 1 [x 6 2]. x 6 2, 0. s3 −1 [x 6 2]. [x 6 1] 1 s1 x 6 1, 1. s4 −1 [x 6 2]. x >1 x := 0, 0 x >1 x := 0, 0. Benjamin Monmege (Aix-Marseille Université). x > 1, 1 s6. s5. x > 1, 2. 1 [x 6 2]. Min = #, Max = 2. 18/33.

(74) One-clock Bi-Valued WTGs (1BWTGs) Joint work with Thomas Brihaye, Gilles Geeraerts, Shankara Krishna Narayanan, Lakshmi Manasa and Ashutosh Trivedi (Brihaye et al., 2014). Assumption: rates of states {p − , p + } included in {0, +d, −d} (d ∈ N) (no assumption on costs of transitions) x < 1, x := 0, 0 s2 x >0 x := 0, 0. 1 [x 6 2]. x 6 2, 0. s3 −1 [x 6 2]. [x 6 1] 1 s1 x 6 1, 1. s4 −1 [x 6 2]. x >1 x := 0, 0 x >1 x := 0, 0. x > 1, 1 s6. s5. x > 1, 2. 1 [x 6 2]. regions: {0}, (0, 1), {1}, (1, 2), {2}, (2, +∞) regions refined with corner information: {0}, (0, η), (1 − η, 1), {1}, (1, 1 + η), (2 − η, 2), {2}, (2, +∞). Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 18/33.

(75) One-clock Bi-Valued WTGs (1BWTGs) Joint work with Thomas Brihaye, Gilles Geeraerts, Shankara Krishna Narayanan, Lakshmi Manasa and Ashutosh Trivedi (Brihaye et al., 2014). Assumption: rates of states {p − , p + } included in {0, +d, −d} (d ∈ N) (no assumption on costs of transitions) 2 1. x < 1, x := 0, 0 s2 x >0 x := 0, 0. 1 [x 6 2]. x 6 2, 0. [x 6 1] 1 s1 x 6 1, 1. s4 −1 [x 6 2]. x >1 x := 0, 0 x >1 x := 0, 0. s5. 0. x > 1, 1. 1. 0. {0}. s3 −1 [x 6 2]. 0. [0, η]. [1−η, 1). 0. [1, 1+η]. [2−η, 2]. 1. 1. 1 0. s6 x > 1, 2. {0} 2 1 1. 1 [x 6 2]. {0}. 2. (0, η]. 3. [1−η, 1). {1}. . {0} 0 0 −1. −1 1. Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 18/33.

(76) One-clock Bi-Valued WTGs (1BWTGs) Joint work with Thomas Brihaye, Gilles Geeraerts, Shankara Krishna Narayanan, Lakshmi Manasa and Ashutosh Trivedi (Brihaye et al., 2014). Assumption: rates of states {p − , p + } included in {0, +d, −d} (d ∈ N) (no assumption on costs of transitions) 2 1. x < 1, x := 0, 0 s2 x >0 x := 0, 0. 1 [x 6 2]. x 6 2, 0. [x 6 1] 1 s1 x 6 1, 1. s4 −1 [x 6 2]. x >1 x := 0, 0 x >1 x := 0, 0. s5. 0. x > 1, 1. 1. 0. {0}. s3 −1 [x 6 2]. 0. [0, η]. [1−η, 1). 0. [1, 1+η]. [2−η, 2]. 1. 1. 1 0. s6 x > 1, 2. {0} 2 1 1. 1 [x 6 2]. {0}. 2. ]0, η]. 3. [1−η, 1). {1}. . {0} 0 0 −1. −1 1. Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 18/33.

(77) One-clock Bi-Valued WTGs (1BWTGs) Joint work with Thomas Brihaye, Gilles Geeraerts, Shankara Krishna Narayanan, Lakshmi Manasa and Ashutosh Trivedi (Brihaye et al., 2014). Assumption: rates of states {p − , p + } included in {0, +d, −d} (d ∈ N) (no assumption on costs of transitions) 2 1. x < 1, x := 0, 0 s2 x >0 x := 0, 0. 1 [x 6 2]. x 6 2, 0. [x 6 1] 1 s1 x 6 1, 1. s4 −1 [x 6 2]. x >1 x := 0, 0 x >1 x := 0, 0. s5. 0. x > 1, 1. 1. 0. {0}. s3 −1 [x 6 2]. 0. [0, η]. [1−η, 1). 0. [1, 1+η]. [2−η, 2]. 1. 1. 1 0. s6 x > 1, 2. {0} 2 1 1. 1 [x 6 2]. {0}. 2. ]0, η]. 3. [1−η, 1). {1}. . {0} 0 0 −1. −1 1. Theorem: Computation of the value functions Val(s, ·) of states of a 1BWTG and synthesis of ε-optimal strategies for # in pseudo-polynomial time I Only non-negative costs =⇒ polynomial time. Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 18/33.

(78) 1BWTG: maximal fragment for corner-point abstraction Generalisation by Engel Lefaucheux: two rates {p − , p + } included in {0, +d, −c} (d, c ∈ N) In more general settings, players may need to play far from corners... I With 3 weights in {−1, 0, +1}: value 1/2... x = 1, x := 0 0. x 61. −1. x =1. 1 x 61. −1 x = 1. Benjamin Monmege (Aix-Marseille Université). . Min = #, Max = 2. 19/33.

(79) 1BWTG: maximal fragment for corner-point abstraction Generalisation by Engel Lefaucheux: two rates {p − , p + } included in {0, +d, −c} (d, c ∈ N) In more general settings, players may need to play far from corners... I With 3 weights in {−1, 0, +1}: value 1/2... x = 1, x := 0 0. x 61. −1. x =1. 1 x 61. −1 x = 1. . I With 2 weights in {−1, 0, +1} but 2 clocks: value 1/2.... 0. x 6 1, y := 0. y =0. 1. x =1. 0. y =1. 0 y =0. 0. x =1. 1. y =1. . I How to push further the resolution of WTGs? Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 19/33.

(80) One-clock WTG... Almost!. Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 20/33.

(81) Related work: 1-clock, non-negative weights (Hansen et al., 2013):. strategy improvement algorithm iterative elimination of locations. (Bouyer et al., 2006b; Rutkowski, 2011):. I precomputation: polynomial-time cascade of simplification of 1-clock WTGs into simple 1-clock WTGs (SWTGs) I clock bounded by 1, no guards/invariants, no resets. Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 21/33.

(82) Related work: 1-clock, non-negative weights (Hansen et al., 2013):. strategy improvement algorithm iterative elimination of locations. (Bouyer et al., 2006b; Rutkowski, 2011):. I precomputation: polynomial-time cascade of simplification of 1-clock WTGs into simple 1-clock WTGs (SWTGs) I clock bounded by 1, no guards/invariants, no resets. I for SWTGs: compute value functions Val(s, ·) for all states s. r2 = 3 r2 = 3. r4 = 9 r4 = 9. 2. 2. 4. v2 (x) 9. v2 (x) 9. 4. v4 (x) v4 (x). 6 5. 6 5. 5 5. 3. 3. r =9. r1 =1 9. 1 1. c(3,1)c(3,1) =3=3. r5 = 0 3. 3. r3 = r63 = 6. 5. v1 (x) 9. 2 3. 1 3. 5 v1 (x) 9. ⊥. 1. 2 3. 1 3x. v3 (x). v3 (x). 8. 8. 6 5. 6 5. 3. 3. c(5,⊥) = 5 c(5,⊥) =5 ⊥. 3. 3. r5 = 0. 1 x. 1 3. 1 x. 2 3. 1 x. v5 (x). v5 (x). 5. 5. 1 x. 1 3. 2 3. 1 x. 2 3. 1 x. 2 3. 1 x. 1 x. 1 x. Figure 1: Example of an SPTG, showing value functions and an optimal strategy profile.. Figure 1: Example of an SPTG, showing value functions and an optimal strategy profile.. current state-of-the-art tools for solving PTGs or various special cases (e.g., such as those of UPPAAL, Benjaminhttp://uppaal.org Monmege (Aix-Marseille Université) Min #,such Max = 2of UPPAAL, current state-of-the-art tools for solving PTGs or various special cases= (e.g., as those or HyTech http://embedded.eecs.berkeley.edu/research/hytech/), which. 21/33.

(83) SWTGs with arbitrary weights. Joint work with Thomas Brihaye, Gilles Geeraerts, Axel Haddad and Engel Lefaucheux (Brihaye et al., 2015). s6 −12. 1. −2 s1. 3 s4 4 s3. s5 8. −14 s2. Benjamin Monmege (Aix-Marseille Université). 2. −7 6. s7. . −16. Min = #, Max = 2. 22/33.

(84) SWTGs with arbitrary weights Joint work with Thomas Brihaye, Gilles Geeraerts, Axel Haddad and Engel Lefaucheux (Brihaye et al., 2015). s6 −12. 1. −2 s1. 3 s4 4 s3. s5 8. −14 s2. 2. −7 6. s7. . −16. Val(s4 , x ) = sup06t61−x 3t − 7 = 3(1 − x ) − 7 = −3x − 4. Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 22/33.

(85) SWTGs with arbitrary weights Joint work with Thomas Brihaye, Gilles Geeraerts, Axel Haddad and Engel Lefaucheux (Brihaye et al., 2015). s6 −12. 1. −2 s1. 3 s4 4 s3. s5 8. −14 s2 Val(s4 , x ) = −3x − 4,. Benjamin Monmege (Aix-Marseille Université). 2. −7 6. s7. . −16. Val(s7 , x ) = −16(1 − x ). Min = #, Max = 2. 22/33.

(86) SWTGs with arbitrary weights Joint work with Thomas Brihaye, Gilles Geeraerts, Axel Haddad and Engel Lefaucheux (Brihaye et al., 2015). 6 s6 −12. −4 −7 s5. 1. −2 s1. 3 s4 4 s3. 8. −14 s2. −10. 2. −7 6. s7. . −16. Val(s4 , x ) = −3x − 4, Val(s7 , x ) = −16(1 − x ), Val(s3 , x ) = inf 06t61−x [4t + min(−3(x + t) − 4, 6 − 16(1 − (x + t)))] = min(−3x − 4, 16x − 10). Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 22/33.

(87) Recursive elimination of states I Player # prefers to stay as long as possible in states with minimal rate → add a final state allowing him to stay until the end, and make the state urgent. Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 23/33.

(88) Recursive elimination of states I Player # prefers to stay as long as possible in states with minimal rate → add a final state allowing him to stay until the end, and make the state urgent I Player 2 prefers to leave as soon as possible in states with minimal rate → make the state urgent. Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 23/33.

(89) Recursive elimination of states I Player # prefers to stay as long as possible in states with minimal rate → add a final state allowing him to stay until the end, and make the state urgent I Player 2 prefers to leave as soon as possible in states with minimal rate → make the state urgent. Theorem: For every SWTG, all value functions are piecewise affine, with at most an exponential number of cutpoints (in number of states). For general 1-clock WTGs? I removing guards and invariants: previously used techniques work! I removing resets: previously, bound the number of resets... Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 23/33.

(90) Solving SWTGs with arbitrary weights 0. 1 4. 1 2. 1. 10 x. −6 −5.5 −7 −10 1. x. −5.5. 0 x. 3 4. 1. −9.5 Val(s2 ,x ) 0. 1 2. −2 −6. Val(s3 ,x ). 1 4. 1 4. 1 2. 3 4. 9 10. 1. −0.2 −2. x. −4. −9.5 Val(s1 ,x ). Val(s4 ,x ) 0. −5.5. −6. −7. 0. 1 x. −16 Val(s7 ,x ). Benjamin Monmege (Aix-Marseille Université). 10 −2. 3 4. 1 x. −14 Val(s5 ,x ). Min = #, Max = 2. 24/33.

(91) Bounding the number of resets needed is not possible x = 1, x := 0 −1. 0. x 61 1. W. x =1 . Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 25/33.

(92) Bounding the number of resets needed is not possible x = 1, x := 0 −1. 0. x 61 1. W. x =1 Player # can guarantee (i.e., ensure to be below) value ε for all ε > 0.... Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 25/33.

(93) Bounding the number of resets needed is not possible x = 1, x := 0 −1. 0. x 61 1. W. x =1 Player # can guarantee (i.e., ensure to be below) value ε for all ε > 0... ... but cannot obtain 0: hence, no optimal strategy.... Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 25/33.

(94) Bounding the number of resets needed is not possible x = 1, x := 0 −1. 0. x 61 1. W. x =1 Player # can guarantee (i.e., ensure to be below) value ε for all ε > 0... ... but cannot obtain 0: hence, no optimal strategy.... ... moreover, to obtain ε, # needs to loop at least W + d1/ log εe times before reaching ! Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 25/33.

(95) Bounding the number of resets needed is not possible x = 1, x := 0 −1. 0. x 61 1. W. x =1 Player # can guarantee (i.e., ensure to be below) value ε for all ε > 0... ... but cannot obtain 0: hence, no optimal strategy.... ... moreover, to obtain ε, # needs to loop at least W + d1/ log εe times before reaching ! Best we can do: exponential time algorithm for reset-acyclic 1-clock WTGs with arbitrary weights Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 25/33.

(96) Finally several clocks.... Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 26/33.

(97) More than one clock? non-negative weights and strictly non-Zeno-cost cycles: 2-exponential algorithm (Bouyer et al., 2004c; Alur et al., 2004b) Value iteration algorithm: compute F i (+∞).... F(x)(s,ν).    . sup d × Weight(s) + Weight(t) + x (s 0 ,ν 0 ) d,t 0 0 = (s,ν)−−→(s ,ν )  inf d × Weight(s) + Weight(t) + x (s 0 ,ν 0 )   d,t (s,ν)− −→(s 0 ,ν 0 ). if s ∈ SMax if s ∈ SMin. Stabilises after a number of iterations at most exponential in the size of the game (because of the number of regions). Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 27/33.

(98) Extension to negative weights. Joint work with Damien Busatto-Gaston and Pierre-Alain Reynier (Busatto-Gaston et al., 2017). Divergence property (of the underlying timed automaton): Every execution following a cycle of the region automaton has a total weight either 6 −1 or > 1. Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 28/33.

(99) Extension to negative weights. Joint work with Damien Busatto-Gaston and Pierre-Alain Reynier (Busatto-Gaston et al., 2017). Divergence property (of the underlying timed automaton): Every execution following a cycle of the region automaton has a total weight either 6 −1 or > 1. Theorem: The value problem on divergent weighted timed games is in 2-EXP, and is EXP-hard.. Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 28/33.

(100) Extension to negative weights. Joint work with Damien Busatto-Gaston and Pierre-Alain Reynier (Busatto-Gaston et al., 2017). Divergence property (of the underlying timed automaton): Every execution following a cycle of the region automaton has a total weight either 6 −1 or > 1. Theorem: The value problem on divergent weighted timed games is in 2-EXP, and is EXP-hard.. Theorem: Deciding if a weighted timed game is divergent is PSPACE-complete.. Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 28/33.

(101) Weighted timed games analysis >1. divergence property. 6 −1 characterisation :. Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 29/33.

(102) Weighted timed games analysis >1. divergence property. 6 −1 characterisation :. Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 29/33.

(103) Weighted timed games analysis >1. divergence property >1. 6 −1 characterisation :. Benjamin Monmege (Aix-Marseille Université). 6 −1. Min = #, Max = 2. 29/33.

(104) Weighted timed games analysis. divergence property. characterisation : All the simple cycles in a SCC have the same sign. Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 29/33.

(105) Weighted timed games analysis. divergence property. characterisation : All the simple cycles in a SCC have the same sign. class decision. Benjamin Monmege (Aix-Marseille Université). value computation. Min = #, Max = 2. 29/33.

(106) Value computation in divergent weighted timed games I Remove +∞ states I SCC decomposition I Value computation SCC after SCC, bottom-up. Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 30/33.

(107) Value computation in divergent weighted timed games I Remove +∞ states I SCC decomposition I Value computation SCC after SCC, bottom-up. positive SCC I weighted timed games with non-negative weights and strictly non-Zeno-cost cycles (Bouyer et al., 2004c; Alur et al., 2004b) I The iterative algorithm converges in a number of steps linear with the region automaton’s size. Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 30/33.

(108) Value computation in divergent weighted timed games I Remove +∞ states I SCC decomposition I Value computation SCC after SCC, bottom-up. positive SCC I weighted timed games with non-negative weights and strictly non-Zeno-cost cycles (Bouyer et al., 2004c; Alur et al., 2004b) I The iterative algorithm converges in a number of steps linear with the region automaton’s size. negative SCC I Outside of the attractor of player 2 toward ⇒ −∞ I The iterative algorithm converges on the other states in a number of steps linear with the region automaton’s size, with −∞ initialisation. Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 30/33.

(109) What to do in case of undecidability? I Adding cycles of weight = 0 to divergent WTG =⇒ Undecidable!. Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 31/33.

(110) What to do in case of undecidability? I Adding cycles of weight = 0 to divergent WTG =⇒ Undecidable! I Already with only non-negative weights (Bouyer et al., 2015): but possible to approximate the value (with elementary complexity).... Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 31/33.

(111) Extension in the negative case? Ongoing work with Damien Busatto-Gaston and Pierre-Alain Reynier. Almost-divergent WTG: every SCC of the region automaton has all its cycles either (> 1 or = 0),. Benjamin Monmege (Aix-Marseille Université). or (6 −1 or = 0). Min = #, Max = 2. 32/33.

(112) Extension in the negative case? Ongoing work with Damien Busatto-Gaston and Pierre-Alain Reynier. Almost-divergent WTG: every SCC of the region automaton has all its cycles or (6 −1 or = 0). either (> 1 or = 0), s0 s. 1. 0 3. Ks 0. s. Ks 00. sf. sf. −3 −1. −3. 1. 2. 1. s stop leaf. 2. 4. 1 −3. Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 32/33.

(113) Extension in the negative case? Ongoing work with Damien Busatto-Gaston and Pierre-Alain Reynier. Almost-divergent WTG: every SCC of the region automaton has all its cycles or (6 −1 or = 0). either (> 1 or = 0), s0 s. 1. 0 3. Ks 0. s. Ks 00. sf. sf. −3 −1. −3. 1. 2. 1. s stop leaf. 2. 4. 1 −3. Theorem: I Approximation is decidable (in doubly exponential time complexity) for almost-divergent WTGs. I We also provide a (semi-)symbolic algorithm that does not rely on an a-priori discretisation of the regions with a fixed granularity 1/N. Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 32/33.

(114) Extension in the negative case? Ongoing work with Damien Busatto-Gaston and Pierre-Alain Reynier. Almost-divergent WTG: every SCC of the region automaton has all its cycles or (6 −1 or = 0). either (> 1 or = 0), s0 s. 1. 0 3. Ks 0. s. Ks 00. sf. sf. −3 −1. −3. 1. 2. 1. s stop leaf. 2. 4. 1 −3. Theorem: I Approximation is decidable (in doubly exponential time complexity) for almost-divergent WTGs. I We also provide a (semi-)symbolic algorithm that does not rely on an a-priori discretisation of the regions with a fixed granularity 1/N. I circumvent the need for an SCC decomposition? Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 32/33.

(115) Conclusion. 1BWTG poly / pseudo-poly (+) (-). Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 33/33.

(116) Conclusion. 1WTG reset-acyclic exp / exp poly-hard. 1BWTG poly / pseudo-poly (+) (-). Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 33/33.

(117) Conclusion. 1WTG reset-acyclic exp / exp poly-hard. 1BWTG poly / pseudo-poly (+) (-). Benjamin Monmege (Aix-Marseille Université). divergent WTG 2-exp / 2-exp exp-hard. Min = #, Max = 2. 33/33.

(118) Conclusion almost-divergent WTG approx / approx 2-exp. + symbolic algorithm. 1WTG reset-acyclic exp / exp poly-hard. 1BWTG poly / pseudo-poly (+) (-). Benjamin Monmege (Aix-Marseille Université). divergent WTG 2-exp / 2-exp exp-hard. Min = #, Max = 2. 33/33.

(119) Conclusion WTG undec / undec > 3 clocks / > 2 clocks. 1WTG reset-acyclic exp / exp poly-hard. almost-divergent WTG approx / approx 2-exp. + symbolic algorithm. 1BWTG poly / pseudo-poly (+) (-). Benjamin Monmege (Aix-Marseille Université). divergent WTG 2-exp / 2-exp exp-hard. Min = #, Max = 2. 33/33.

(120) Conclusion WTG undec / undec > 3 clocks / > 2 clocks. 1WTG reset-acyclic exp / exp poly-hard. almost-divergent WTG approx / approx 2-exp. + symbolic algorithm. 1BWTG poly / pseudo-poly (+) (-). divergent WTG 2-exp / 2-exp exp-hard. 1WTG?. Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 33/33.

(121) Conclusion WTG undec / undec > 3 clocks / > 2 clocks. 1WTG reset-acyclic exp / exp poly-hard. almost-divergent WTG approx / approx 2-exp. + symbolic algorithm. 1BWTG poly / pseudo-poly (+) (-). divergent WTG 2-exp / 2-exp exp-hard. gap? 1WTG?. Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 33/33.

(122) Conclusion WTG undec / undec > 3 clocks / > 2 clocks. almost-divergent WTG approx / approx 2-exp. + symbolic algorithm. 2 clocks?. 1WTG reset-acyclic exp / exp poly-hard. 1BWTG poly / pseudo-poly (+) (-). divergent WTG 2-exp / 2-exp exp-hard. gap? 1WTG?. Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 33/33.

(123) Conclusion WTG undec / undec > 3 clocks / > 2 clocks. almost-divergent WTG approx / approx 2-exp. + symbolic algorithm. 2 clocks?. 1WTG reset-acyclic exp / exp poly-hard. 1BWTG poly / pseudo-poly (+) (-). divergent WTG 2-exp / 2-exp exp-hard. gap? 1WTG? tool?. Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 33/33.

(124) Conclusion WTG undec / undec > 3 clocks / > 2 clocks. almost-divergent WTG approx / approx 2-exp. + symbolic algorithm. 2 clocks?. 1WTG reset-acyclic exp / exp poly-hard. 1BWTG poly / pseudo-poly (+) (-). divergent WTG 2-exp / 2-exp exp-hard. gap? 1WTG? tool?. Thank you! Benjamin Monmege (Aix-Marseille Université). Min = #, Max = 2. 33/33.

(125) References I Alur, R., Bernadsky, M., and Madhusudan, P. (2004a). Optimal reachability for weighted timed games. In Proceedings of the 31st International Colloquium on Automata, Languages and Programming (ICALP’04), volume 3142 of Lecture Notes in Computer Science, pages 122–133. Springer. Alur, R., Bernadsky, M., and Madhusudan, P. (2004b). Optimal reachability for weighted timed games. In Proceedings of the 31st International Colloquium on Automata, Languages and Programming (ICALP’04), volume 3142 of LNCS, pages 122–133. Springer. Bouyer, P., Brihaye, T., Bruyère, V., and Raskin, J.-F. (2007). On the optimal reachability problem of weighted timed automata. Formal Methods in System Design, 31(2):135–175. Bouyer, P., Brihaye, T., and Markey, N. (2006a). Improved undecidability results on weighted timed automata. Information Processing Letters, 98(5):188–194. Bouyer, P., Brinksma, E., and Larsen, K. G. (2004a). Staying alive as cheaply as possible. In Hybrid Systems: Computation and Control, pages 203–218. Springer. Bouyer, P., Cassez, F., Fleury, E., and Larsen, K. G. (2004b). Optimal strategies in priced timed game automata. In Proceedings of the 24th Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS’04), volume 3328 of Lecture Notes in Computer Science, pages 148–160. Springer..

(126) References II Bouyer, P., Cassez, F., Fleury, E., and Larsen, K. G. (2004c). Optimal strategies in priced timed game automata. In Proceedings of the 24th Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS’04), volume 3328 of LNCS, pages 148–160. Springer. Bouyer, P., Jaziri, S., and Markey, N. (2015). On the value problem in weighted timed games. In Proceedings of the 26th International Conference on Concurrency Theory (CONCUR’15), volume 42 of Leibniz International Proceedings in Informatics, pages 311–324. Leibniz-Zentrum für Informatik. Bouyer, P., Larsen, K. G., Markey, N., and Rasmussen, J. I. (2006b). Almost optimal strategies in one-clock priced timed games. In Proceedings of the 26th Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS’06), volume 4337 of Lecture Notes in Computer Science, pages 345–356. Springer. Brihaye, T., Bruyère, V., and Raskin, J.-F. (2005). On optimal timed strategies. In Proceedings of the Third international conference on Formal Modeling and Analysis of Timed Systems (FORMATS’05), volume 3829 of Lecture Notes in Computer Science, pages 49–64. Springer. Brihaye, T., Geeraerts, G., Haddad, A., Lefaucheux, E., and Monmege, B. (2015). Simple priced timed games are not that simple. In Proceedings of the 35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS’15), volume 45 of LIPIcs, pages 278–292. Schloss Dagstuhl–Leibniz-Zentrum für Informatik..

(127) References III Brihaye, T., Geeraerts, G., Haddad, A., and Monmege, B. (2016). Pseudopolynomial iterative algorithm to solve total-payoff games and min-cost reachability games. Acta Informatica. Brihaye, T., Geeraerts, G., Narayanan Krishna, S., Manasa, L., Monmege, B., and Trivedi, A. (2014). Adding negative prices to priced timed games. In Proceedings of the 25th International Conference on Concurrency Theory (CONCUR’14), volume 8704, pages 560–575. Springer. Busatto-Gaston, D., Monmege, B., and Reynier, P.-A. (2017). Optimal reachability in divergent weighted timed games. In Esparza, J. and Murawski, A. S., editors, Proceedings of the 20th International Conference on Foundations of Software Science and Computation Structures (FoSSaCS’17), volume 10203 of Lecture Notes in Computer Science, pages 162–178, Uppsala, Sweden. Springer. Fearnley, J. and Jurdziński, M. (2013). Reachability in two-clock timed automata is PSPACE-complete. In Proceedings of ICALP’13, volume 7966 of Lecture Notes in Computer Science, pages 212–223. Springer. Haase, C., Ouaknine, J., and Worrell, J. (2012). On the relationship between reachability problems in timed and counter automata. In Proceedings of RP’12, pages 54–65. Hansen, T. D., Ibsen-Jensen, R., and Miltersen, P. B. (2013). A faster algorithm for solving one-clock priced timed games. In Proceedings of the 24th International Conference on Concurrency Theory (CONCUR’13), volume 8052 of LNCS, pages 531–545. Springer..

(128) References IV. Khachiyan, L., Boros, E., Borys, K., Elbassioni, K., Gurvich, V., Rudolf, G., and Zhao, J. (2008). On short paths interdiction problems: Total and node-wise limited interdiction. Theory of Computing Systems, 43(2):204–233. Rutkowski, M. (2011). Two-player reachability-price games on single-clock timed automata. In Proceedings of the Ninth Workshop on Quantitative Aspects of Programming Languages (QAPL’11), volume 57 of Electronic Proceedings in Theoretical Computer Science, pages 31–46..

(129) Sketch of proof for 1BWTG. 1. Reduce the space of strategies in the 1BWTG I restrict to uniform strategies w.r.t. timed behaviours. 2. Build a finite weighted game G I based on a refinement of the region abstraction. 3. Study G 4. Lift results of G to the original 1BWTG.

(130) 1. Reduce the space of strategies Intuition: no need for both players to play far from borders of regions x < 1, x := 0, 0 s2 x >0 x := 0, 0. 1 [x 6 2]. x 6 2, 0. s3 −1 [x 6 2]. [x 6 1] 1 s1 x 6 1, 1. s4 −1 [x 6 2]. x >1 x := 0, 0 x >1 x := 0, 0. s5. x > 1, 1 s6. Regions: {0}, (0, 1), {1}, (1, 2), {2}, (2, +∞). x > 1, 2. 1 [x 6 2]. Player # wants to leave as soon as possible a state with rate p + , and wants to stay as long as possible in a state with rate p − : so, he will always play η-close to a border.... Lemma: Both players can play arbitrarily close to borders w.l.o.g.: for every η η. Valη (s, v ) 6 Val(s, v ) 6 Val(s, v ) 6 Val (s, v ).

(131) 2. Finite weighted game abstraction. x < 1, x := 0, 0 s2 x >0 x := 0, 0. 1 [x 6 2]. x 6 2, 0. s3 −1 [x 6 2]. [x 6 1] 1 s1 x 6 1, 1. s4 −1 [x 6 2]. x >1 x := 0, 0 x >1 x := 0, 0. s5. x > 1, 1 s6 x > 1, 2. 1 [x 6 2]. η-regions: {0}, (0, η), (1 − η, 1), {1}, (1, 1 + η), (2 − η, 2), {2}, (2, +∞).

(132) 2. Finite weighted game abstraction 2 1 1. 0. {0} 0. 0. [0, η]. [1−η, 1). 0. [1, 1+η]. 1. [2−η, 2]. 1. 1. 0 {0} 2 1 1 {0}. 2. (0, η]. 3. [1−η, 1). {1}. {0} 0 0 −1. −1 1. .

(133) 3. Study G:. values, optimal strategies of a min-cost reachability game. (Brihaye et al., 2016) 2 1 1. 0. {0} 0. 0. [0, η]. [1−η, 1). 0. [1, 1+η]. 1. [2−η, 2]. 1. 1. 0 {0} 2 1 1 {0}. 2. ]0, η]. 3. [1−η, 1). {1}. . {0} 0 0 −1. −1 1. Optimal value: ValG (s1 , {0}) = +2 (for both players).

(134) 4. Lift results to the original 1BWTG Reconstruct strategies in the 1BWTG from optimal strategies of G. Lemma: For all ε > 0, there exists η > 0 such that: η. ValG (s, {0}) − ε 6 Valη (s, 0) 6 Val(s, 0) 6 Val(s, 0) 6 Val (s, 0) 6 ValG (s, {0}) + ε.

(135) 4. Lift results to the original 1BWTG Reconstruct strategies in the 1BWTG from optimal strategies of G. Lemma: For all ε > 0, there exists η > 0 such that: η. ValG (s, {0}) − ε 6 Valη (s, 0) 6 Val(s, 0) 6 Val(s, 0) 6 Val (s, 0) 6 ValG (s, {0}) + ε. I So Val(s, 0) = Val(s, 0), i.e., determination I ε-optimal strategies for both players I Finite memory for player # (finite memory in finite weighted games) I Infinite memory for player 2 (even though memoryless in finite weighted games), because it needs to ensure convergence of its differences between the 1BWTG and G. I Overall complexity: pseudo-polynomial (polynomial if non-negative costs) in the size of G, which is polynomial in the 1BWTG (because 1 clock).

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