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Optimal strategies in weighted timed games: undecidability and approximation Nicolas Markey LSV, CNRS & ENS Cachan, France (joint work with Patricia Bouyer and Samy Jaziri) AVeRTS15 workshop Bangaluru, India December 19, 2015 Model

  1. Priced timed automata Definition ( [KPSY99,ALP01,BFH + 01] ) A priced timed automaton is made of a timed automaton; the price of each transition and location. Example − 3 +6 − 1 x :=0 − 6 +2 x =1 [KPSY99] Kesten, Pnueli, Sifakis, Yovine. Decidable Integration Graphs. Inf. & Comp., 1999. [ALP01] Alur, La Torre, Pappas. Optimal paths in weighted timed automata. HSCC, 2001. [BFH + 01] Behrmann et al. Minimum-cost reachability in priced timed automata. HSCC, 2001.

  2. Priced timed automata Definition ( [KPSY99,ALP01,BFH + 01] ) A priced timed automaton is made of a timed automaton; the price of each transition and location. Example 4 − 3 +6 3 2 − 1 x :=0 1 − 6 +2 0 x =1 0 1 − 3 [KPSY99] Kesten, Pnueli, Sifakis, Yovine. Decidable Integration Graphs. Inf. & Comp., 1999. [ALP01] Alur, La Torre, Pappas. Optimal paths in weighted timed automata. HSCC, 2001. [BFH + 01] Behrmann et al. Minimum-cost reachability in priced timed automata. HSCC, 2001.

  3. Priced timed automata Definition ( [KPSY99,ALP01,BFH + 01] ) A priced timed automaton is made of a timed automaton; the price of each transition and location. Example 4 − 3 +6 3 2 − 1 x :=0 1 − 6 +2 0 x =1 0 1 − 3 − 3 1 6 [KPSY99] Kesten, Pnueli, Sifakis, Yovine. Decidable Integration Graphs. Inf. & Comp., 1999. [ALP01] Alur, La Torre, Pappas. Optimal paths in weighted timed automata. HSCC, 2001. [BFH + 01] Behrmann et al. Minimum-cost reachability in priced timed automata. HSCC, 2001.

  4. Priced timed automata Definition ( [KPSY99,ALP01,BFH + 01] ) A priced timed automaton is made of a timed automaton; the price of each transition and location. Example 4 − 3 +6 3 2 − 1 x :=0 1 − 6 +2 0 x =1 0 1 − 3 − 3 +6 1 6 [KPSY99] Kesten, Pnueli, Sifakis, Yovine. Decidable Integration Graphs. Inf. & Comp., 1999. [ALP01] Alur, La Torre, Pappas. Optimal paths in weighted timed automata. HSCC, 2001. [BFH + 01] Behrmann et al. Minimum-cost reachability in priced timed automata. HSCC, 2001.

  5. Priced timed automata Definition ( [KPSY99,ALP01,BFH + 01] ) A priced timed automaton is made of a timed automaton; the price of each transition and location. Example 4 − 3 +6 3 2 − 1 x :=0 1 − 6 +2 0 x =1 0 1 − 3 − 3 +6 +6 1 1 6 2 [KPSY99] Kesten, Pnueli, Sifakis, Yovine. Decidable Integration Graphs. Inf. & Comp., 1999. [ALP01] Alur, La Torre, Pappas. Optimal paths in weighted timed automata. HSCC, 2001. [BFH + 01] Behrmann et al. Minimum-cost reachability in priced timed automata. HSCC, 2001.

  6. Priced timed automata Definition ( [KPSY99,ALP01,BFH + 01] ) A priced timed automaton is made of a timed automaton; the price of each transition and location. Example 4 − 3 +6 3 2 − 1 x :=0 1 − 6 +2 0 x =1 0 1 − 1 − 3 − 3 +6 +6 − 6 1 1 6 2 [KPSY99] Kesten, Pnueli, Sifakis, Yovine. Decidable Integration Graphs. Inf. & Comp., 1999. [ALP01] Alur, La Torre, Pappas. Optimal paths in weighted timed automata. HSCC, 2001. [BFH + 01] Behrmann et al. Minimum-cost reachability in priced timed automata. HSCC, 2001.

  7. Priced timed automata Definition ( [KPSY99,ALP01,BFH + 01] ) A priced timed automaton is made of a timed automaton; the price of each transition and location. Example 4 − 3 +6 3 2 − 1 x :=0 1 − 6 +2 0 x =1 0 1 − 1 − 3 − 3 +6 +6 − 6 − 6 1 1 1 6 2 3 [KPSY99] Kesten, Pnueli, Sifakis, Yovine. Decidable Integration Graphs. Inf. & Comp., 1999. [ALP01] Alur, La Torre, Pappas. Optimal paths in weighted timed automata. HSCC, 2001. [BFH + 01] Behrmann et al. Minimum-cost reachability in priced timed automata. HSCC, 2001.

  8. Priced timed automata Definition ( [KPSY99,ALP01,BFH + 01] ) A priced timed automaton is made of a timed automaton; the price of each transition and location. Example 4 − 3 +6 3 2 − 1 x :=0 1 − 6 +2 0 x =1 0 1 − 1 − 3 − 3 +6 +6 − 6 − 6 +2 1 1 1 6 2 3 [KPSY99] Kesten, Pnueli, Sifakis, Yovine. Decidable Integration Graphs. Inf. & Comp., 1999. [ALP01] Alur, La Torre, Pappas. Optimal paths in weighted timed automata. HSCC, 2001. [BFH + 01] Behrmann et al. Minimum-cost reachability in priced timed automata. HSCC, 2001.

  9. Priced timed automata Definition ( [KPSY99,ALP01,BFH + 01] ) A priced timed automaton is made of a timed automaton; the price of each transition and location. Example 4 − 3 +6 3 2 − 1 x :=0 1 − 6 +2 0 x =1 0 1 − 1 − 3 − 3 +6 +6 − 6 − 6 +2 1 1 1 6 2 3 [KPSY99] Kesten, Pnueli, Sifakis, Yovine. Decidable Integration Graphs. Inf. & Comp., 1999. [ALP01] Alur, La Torre, Pappas. Optimal paths in weighted timed automata. HSCC, 2001. [BFH + 01] Behrmann et al. Minimum-cost reachability in priced timed automata. HSCC, 2001.

  10. Example: task graph scheduling Compute D × ( C × ( A + B ))+( A + B )+( C × D ) using two processors: A B C D P 1 (fast): P 2 (slow): + × T 1 T 2 time time C + 2 picoseconds + 5 picoseconds + × 3 picoseconds 7 picoseconds × × T 3 T 4 energy energy D idle 10 Watt idle 20 Watts + × in use 90 Watts in use 30 Watts T 5 T 6 0 5 10 15 20 25 13 picoseconds P 1 T 2 T 3 T 5 T 6 Sch 1 1 . 37 nanojoules P 2 T 1 T 4 12 picoseconds P 1 T 1 T 3 T 5 T 4 T 6 Sch 2 1 . 39 nanojoules P 2 T 2 19 picoseconds P 1 T 1 T 3 T 4 1 . 32 nanojoules Sch 3 P 2 T 2 T 5 T 6

  11. Modelling the task graph scheduling problem Processors: x =2 x =3 + done 1 done 1 × + + idle × × add 1 mul 1 c =90 ˙ c =10 ˙ c =90 ˙ x ≤ 2 x ≤ 3 x :=0 x :=0

  12. Modelling the task graph scheduling problem Processors: x =2 x =3 + done 1 done 1 × + + idle × × add 1 mul 1 c =90 ˙ c =10 ˙ c =90 ˙ x ≤ 2 x ≤ 3 x :=0 x :=0 x =5 x =7 done 2 done 2 + + + × × × idle add 2 mul 2 c =30 ˙ c =20 ˙ c =30 ˙ x ≤ 5 x ≤ 7 x :=0 x :=0

  13. Modelling the task graph scheduling problem Processors: x =2 x =3 + done 1 done 1 × + + idle × × add 1 mul 1 c =90 ˙ c =10 ˙ c =90 ˙ x ≤ 2 x ≤ 3 x :=0 x :=0 x =5 x =7 done 2 done 2 + + + × × × idle add 2 mul 2 c =30 ˙ c =20 ˙ c =30 ˙ x ≤ 5 x ≤ 7 x :=0 x :=0 Tasks: add 1 done 1 t 4 :=1 t 1 ∧ t 2 T 4 F 4 t 1 ∧ t 2 t 4 :=1 add 2 done 2

  14. Outline of the talk Introduction: timed automata and timed games 1 Measuring extra quantities in timed automata 2 Example: task graph scheduling Timed automata with observer variables Cost-optimal strategies 3 Optimal reachability in priced timed automata Optimal reachability in priced timed games Conclusions and future works 4 Advertisements 5

  15. Cost-optimal reachability in priced timed automata Example x ≥ 3 p =6 ˙ p +=1 x ≤ 2 p =5 ˙ � y :=0 y =0 x ≥ 3 p =3 ˙ p +=9

  16. Cost-optimal reachability in priced timed automata Example x ≥ 3 p =6 ˙ p +=1 x ≤ 2 p =5 ˙ � y :=0 y =0 x ≥ 3 p =3 ˙ p +=9 Minimal cost for reaching � :

  17. Cost-optimal reachability in priced timed automata Example x ≥ 3 p =6 ˙ p +=1 x ≤ 2 p =5 ˙ � y :=0 y =0 x ≥ 3 p =3 ˙ p +=9 Minimal cost for reaching � : 22 5 t + 6(3 − t ) + 1 20 18 0 2

  18. Cost-optimal reachability in priced timed automata Example x ≥ 3 p =6 ˙ p +=1 x ≤ 2 p =5 ˙ � y :=0 y =0 x ≥ 3 p =3 ˙ p +=9 Minimal cost for reaching � : 22 5 t + 6(3 − t ) + 1 5 t + 3(3 − t ) + 9 20 18 0 2

  19. Cost-optimal reachability in priced timed automata Example x ≥ 3 p =6 ˙ p +=1 x ≤ 2 p =5 ˙ � y :=0 y =0 x ≥ 3 p =3 ˙ p +=9 Minimal cost for reaching � : 22 � 5 t + 6(3 − t ) + 1 � min 5 t + 3(3 − t ) + 9 20 18 0 2

  20. Cost-optimal reachability in priced timed automata Example x ≥ 3 p =6 ˙ p +=1 x ≤ 2 p =5 ˙ � y :=0 y =0 x ≥ 3 p =3 ˙ p +=9 Minimal cost for reaching � : 22 � 5 t + 6(3 − t ) + 1 � 0 ≤ t ≤ 2 min inf 5 t + 3(3 − t ) + 9 20 18 0 2

  21. Cost-optimal reachability in priced timed automata Example x ≥ 3 p =6 ˙ p +=1 x ≤ 2 p =5 ˙ � y :=0 y =0 x ≥ 3 p =3 ˙ p +=9 Minimal cost for reaching � : 22 � 5 t + 6(3 − t ) + 1 � 0 ≤ t ≤ 2 min inf = 17 5 t + 3(3 − t ) + 9 20 18 0 2

  22. Cost-optimal reachability in priced timed automata Example x ≥ 3 p =6 ˙ p +=1 x ≤ 2 p =5 ˙ � y :=0 y =0 x ≥ 3 p =3 ˙ p +=9 Minimal cost for reaching � : 22 � 5 t + 6(3 − t ) + 1 � 0 ≤ t ≤ 2 min inf = 17 5 t + 3(3 − t ) + 9 20 18 The optimal schedule consists in waiting 2 time units in ; going through . 0 2

  23. Cost-optimal reachability in priced timed automata Theorem ( [BBBR07] ) Optimal reachability in priced timed automata is PSPACE -complete. [BBBR07] Bouyer, Brihaye, Bruy` ere, Raskin. On the optimal reachability problem. FMSD, 2007.

  24. Cost-optimal reachability in priced timed automata Theorem ( [BBBR07] ) Optimal reachability in priced timed automata is PSPACE -complete. Proof Regions are not precise enough; p +=2 x :=0 p =3 ˙ p =3 ˙ p =3 ˙ p =5 ˙ [BBBR07] Bouyer, Brihaye, Bruy` ere, Raskin. On the optimal reachability problem. FMSD, 2007.

  25. Cost-optimal reachability in priced timed automata Theorem ( [BBBR07] ) Optimal reachability in priced timed automata is PSPACE -complete. Proof Regions are not precise enough; Use regions with corner-points: [BBBR07] Bouyer, Brihaye, Bruy` ere, Raskin. On the optimal reachability problem. FMSD, 2007.

  26. Cost-optimal reachability in priced timed automata Theorem ( [BBBR07] ) Optimal reachability in priced timed automata is PSPACE -complete. Proof Regions are not precise enough; Use regions with corner-points: p +=2 x :=0 p =3 ˙ p =3 ˙ p =3 ˙ p =3 ˙ p =5 ˙ [BBBR07] Bouyer, Brihaye, Bruy` ere, Raskin. On the optimal reachability problem. FMSD, 2007.

  27. Cost-optimal reachability in priced timed automata Theorem ( [BBBR07] ) Optimal reachability in priced timed automata is PSPACE -complete. Proof Regions are not precise enough; Use regions with corner-points: p +=0 p +=3 p +=0 p +=2 x :=0 p =3 ˙ p =3 ˙ p =3 ˙ p =3 ˙ p =5 ˙ [BBBR07] Bouyer, Brihaye, Bruy` ere, Raskin. On the optimal reachability problem. FMSD, 2007.

  28. Cost-optimal reachability in priced timed automata Theorem ( [BBBR07] ) Optimal reachability in priced timed automata is PSPACE -complete. Proof Regions are not precise enough; Use regions with corner-points: p +=0 p +=3 p +=0 p +=2 x :=0 p =3 ˙ p =3 ˙ p =3 ˙ p =3 ˙ p =5 ˙ p +=0 p +=0 p +=2 x :=0 p =3 ˙ p =3 ˙ p =3 ˙ p =5 ˙ [BBBR07] Bouyer, Brihaye, Bruy` ere, Raskin. On the optimal reachability problem. FMSD, 2007.

  29. Cost-optimal reachability in priced timed automata Theorem ( [BBBR07] ) Optimal reachability in priced timed automata is PSPACE -complete. Proof optimal schedule as a linear programming problem: t 1 t 2 t 3 t 4 t 5 [BBBR07] Bouyer, Brihaye, Bruy` ere, Raskin. On the optimal reachability problem. FMSD, 2007.

  30. Cost-optimal reachability in priced timed automata Theorem ( [BBBR07] ) Optimal reachability in priced timed automata is PSPACE -complete. Proof optimal schedule as a linear programming problem: t 1 t 2 t 3 t 4 t 5 t 1 + t 2 ≤ 2 x ≤ 2 [BBBR07] Bouyer, Brihaye, Bruy` ere, Raskin. On the optimal reachability problem. FMSD, 2007.

  31. Cost-optimal reachability in priced timed automata Theorem ( [BBBR07] ) Optimal reachability in priced timed automata is PSPACE -complete. Proof optimal schedule as a linear programming problem: t 1 t 2 t 3 t 4 t 5 t 1 + t 2 ≤ 2 y :=0 x ≤ 2 y ≥ 3 t 2 + t 3 + t 4 ≥ 3 [BBBR07] Bouyer, Brihaye, Bruy` ere, Raskin. On the optimal reachability problem. FMSD, 2007.

  32. Cost-optimal reachability in priced timed automata Theorem ( [BBBR07] ) Optimal reachability in priced timed automata is PSPACE -complete. Proof optimal schedule as a linear programming problem: Minimize � i c i · t i + C disc t 1 t 2 t 3 t 4 t 5 t 1 + t 2 ≤ 2 y :=0 x ≤ 2 y ≥ 3 t 2 + t 3 + t 4 ≥ 3 [BBBR07] Bouyer, Brihaye, Bruy` ere, Raskin. On the optimal reachability problem. FMSD, 2007.

  33. Cost-optimal reachability in priced timed automata Theorem ( [BBBR07] ) Optimal reachability in priced timed automata is PSPACE -complete. Proof optimal schedule as a linear programming problem: Minimize � i c i · t i + C disc t 1 t 2 t 3 t 4 t 5 t 1 + t 2 ≤ 2 y :=0 x ≤ 2 y ≥ 3 t 2 + t 3 + t 4 ≥ 3 � infimum over bounded zone reached at a point on the frontier, with integer coordinates. [BBBR07] Bouyer, Brihaye, Bruy` ere, Raskin. On the optimal reachability problem. FMSD, 2007.

  34. Cost-optimal reachability in priced timed automata Theorem ( [BBBR07] ) Optimal reachability in priced timed automata is PSPACE -complete. Proof optimal schedule as a linear programming problem: ∀ π. ∃ π cp . cost ( π cp ) ≤ cost ( π ). [BBBR07] Bouyer, Brihaye, Bruy` ere, Raskin. On the optimal reachability problem. FMSD, 2007.

  35. Cost-optimal reachability in priced timed automata Theorem ( [BBBR07] ) Optimal reachability in priced timed automata is PSPACE -complete. Proof optimal schedule as a linear programming problem: ∀ π. ∃ π cp . cost ( π cp ) ≤ cost ( π ). approximate path in corner-point abstraction by a real run: ∀ π cp . ∃ π. cost ( π ) ≤ cost ( π cp ) + ǫ . [BBBR07] Bouyer, Brihaye, Bruy` ere, Raskin. On the optimal reachability problem. FMSD, 2007.

  36. Outline of the talk Introduction: timed automata and timed games 1 Measuring extra quantities in timed automata 2 Example: task graph scheduling Timed automata with observer variables Cost-optimal strategies 3 Optimal reachability in priced timed automata Optimal reachability in priced timed games Conclusions and future works 4 Advertisements 5

  37. Example: task graph scheduling Compute D × ( C × ( A + B ))+( A + B )+( C × D ) using two processors: A B C D P 1 (fast): P 2 (slow): + × T 1 T 2 time time C + 2 picoseconds + 5 picoseconds + × 3 picoseconds 7 picoseconds × × T 3 T 4 energy energy D idle 10 Watt idle 20 Watts + × in use 90 Watts in use 30 Watts T 5 T 6 0 5 10 15 20 25 13 picoseconds P 1 T 2 T 3 T 5 T 6 Sch 1 1 . 37 nanojoules P 2 T 1 T 4 12 picoseconds P 1 T 1 T 3 T 5 T 4 T 6 Sch 2 1 . 39 nanojoules P 2 T 2 19 picoseconds P 1 T 1 T 3 T 4 1 . 32 nanojoules Sch 3 P 2 T 2 T 5 T 6

  38. Cost-optimal reachability in priced timed games Using games to model uncertainty over delays Processors with exact delays: x =2 x =3 done 1 done 1 + + + × × × idle add 1 mul 1 c =90 ˙ c =10 ˙ c =90 ˙ x ≤ 2 x ≤ 3 x :=0 x :=0

  39. Cost-optimal reachability in priced timed games Using games to model uncertainty over delays Processors with exact delays: x =2 x =3 done 1 done 1 + + + × × × idle add 1 mul 1 c =90 ˙ c =10 ˙ c =90 ˙ x ≤ 2 x ≤ 3 x :=0 x :=0 Processors with approximate delays: x ≥ 2 x ≥ 3 + done 1 done 1 × + + idle × × add 1 mul 1 c =90 ˙ c =10 ˙ c =90 ˙ x ≤ 3 x ≤ 4 x :=0 x :=0

  40. Cost-optimal reachability in priced timed games Example x ≥ 3 p =6 ˙ p +=1 x ≤ 2 p =5 ˙ � y :=0 y =0 x ≥ 3 p =3 ˙ p +=9

  41. Cost-optimal reachability in priced timed games Example x ≥ 3 p =6 ˙ p +=1 x ≤ 2 p =5 ˙ � y :=0 y =0 x ≥ 3 p =3 ˙ p +=9 Minimal cost for reaching � :

  42. Cost-optimal reachability in priced timed games Example x ≥ 3 p =6 ˙ p +=1 x ≤ 2 p =5 ˙ � y :=0 y =0 x ≥ 3 p =3 ˙ p +=9 Minimal cost for reaching � : 22 20 5 t + 6(3 − t ) + 1 18 0 2

  43. Cost-optimal reachability in priced timed games Example x ≥ 3 p =6 ˙ p +=1 x ≤ 2 p =5 ˙ � y :=0 y =0 x ≥ 3 p =3 ˙ p +=9 Minimal cost for reaching � : 22 20 5 t + 6(3 − t ) + 1 5 t + 3(3 − t ) + 9 18 0 2

  44. Cost-optimal reachability in priced timed games Example x ≥ 3 p =6 ˙ p +=1 x ≤ 2 p =5 ˙ � y :=0 y =0 x ≥ 3 p =3 ˙ p +=9 Minimal cost for reaching � : 22 20 � 5 t + 6(3 − t ) + 1 � max 5 t + 3(3 − t ) + 9 18 0 2

  45. Cost-optimal reachability in priced timed games Example x ≥ 3 p =6 ˙ p +=1 x ≤ 2 p =5 ˙ � y :=0 y =0 x ≥ 3 p =3 ˙ p +=9 Minimal cost for reaching � : 22 20 � 5 t + 6(3 − t ) + 1 � 0 ≤ t ≤ 2 max inf 5 t + 3(3 − t ) + 9 18 0 2

  46. Cost-optimal reachability in priced timed games Example x ≥ 3 p =6 ˙ p +=1 x ≤ 2 p =5 ˙ � y :=0 y =0 x ≥ 3 p =3 ˙ p +=9 Minimal cost for reaching � : 22 20 � 5 t + 6(3 − t ) + 1 � 0 ≤ t ≤ 2 max inf = 18 . 66 5 t + 3(3 − t ) + 9 18 0 2

  47. Cost-optimal reachability in priced timed games Example x ≥ 3 p =6 ˙ p +=1 x ≤ 2 p =5 ˙ � y :=0 y =0 x ≥ 3 p =3 ˙ p +=9 Minimal cost for reaching � : 22 20 � 5 t + 6(3 − t ) + 1 � 0 ≤ t ≤ 2 max inf = 18 . 66 5 t + 3(3 − t ) + 9 18 (with t opt = 1 3 ) 0 2

  48. Looking for optimal strategies... Optimal strategies need not exist... � x =1 p =2 ˙ p =1 ˙ x =0

  49. Looking for optimal strategies... Optimal strategies need not exist... � x =1 p =2 ˙ p =1 ˙ x =0 Optimal strategies may need memory... x =1 p =2 ˙ � x > 0 x < 1 , x :=0 p =1 ˙

  50. Cost-optimal reachability in priced timed games Theorem ( [BBR05,BBM06] ) Optimal reachability in priced timed games is undecidable. [BBR05] Brihaye, Bruy` ere, Raskin. On optimal timed strategies. FORMATS, 2005.

  51. Cost-optimal reachability in priced timed games Theorem ( [BBR05,BBM06] ) Optimal reachability in priced timed games is undecidable. Proof Encode a two-counter machine as a priced timed game. [BBR05] Brihaye, Bruy` ere, Raskin. On optimal timed strategies. FORMATS, 2005.

  52. Cost-optimal reachability in priced timed games Theorem ( [BBR05,BBM06] ) Optimal reachability in priced timed games is undecidable. Proof Encode a two-counter machine as a priced timed game. add the value of clock x to the accumulated cost y =1 , y :=0 y =1 , y :=0 z =0 x =1 z =1 p =0 ˙ p =1 ˙ x :=0 z :=0 Add + ( x ) [BBR05] Brihaye, Bruy` ere, Raskin. On optimal timed strategies. FORMATS, 2005.

  53. Cost-optimal reachability in priced timed games Theorem ( [BBR05,BBM06] ) Optimal reachability in priced timed games is undecidable. Proof Encode a two-counter machine as a priced timed game. add the value of clock x to the accumulated cost add 1 − x to the accumulated cost y =1 , y :=0 y =1 , y :=0 z =0 x =1 z =1 p =1 ˙ p =0 ˙ x :=0 z :=0 Add + ( x ) [BBR05] Brihaye, Bruy` ere, Raskin. On optimal timed strategies. FORMATS, 2005.

  54. Cost-optimal reachability in priced timed games Theorem ( [BBR05,BBM06] ) Optimal reachability in priced timed games is undecidable. Proof Encode a two-counter machine as a priced timed game. add the value of clock x to the accumulated cost add 1 − x to the accumulated cost check that y = 2 x Add + ( x ) Add + ( x ) Add − ( y ) p +=2 z =0 z =0 p =0 ˙ p =0 ˙ z =0 p +=1 Add + ( y ) Add − ( x ) Add − ( x ) Test( y =2 x )

  55. Cost-optimal reachability in priced timed games Theorem ( [BBR05,BBM06] ) Optimal reachability in priced timed games is undecidable. Proof Encode a two-counter machine as a priced timed game. add the value of clock x to the accumulated cost add 1 − x to the accumulated cost check that y = 2 x Add + ( x ) Add + ( x ) Add − ( y ) p +=2 z =0 cost=3+(2 x − y ) z =0 p =0 ˙ p =0 ˙ cost=3+( y − 2 x ) z =0 p +=1 Add + ( y ) Add − ( x ) Add − ( x ) Test( y =2 x )

  56. Cost-optimal reachability in priced timed games Theorem ( [BBR05,BBM06] ) Optimal reachability in priced timed games is undecidable. Proof Encode a two-counter machine as a priced timed game. add the value of clock x to the accumulated cost add 1 − x to the accumulated cost check that y = 2 x divide clock x by 2 y :=0 z =0 x =1 z =1 z =0 p =0 ˙ p =0 ˙ p =0 ˙ p =0 ˙ x :=0 z :=0 z =0 Test( x =2 y ) Divide 2 ( x )

  57. Cost-optimal reachability in priced timed games Theorem ( [BBR05,BBM06] ) Optimal reachability in priced timed games is undecidable. Proof Encode a two-counter machine as a priced timed game. add the value of clock x to the accumulated cost add 1 − x to the accumulated cost check that y = 2 x divide clock x by 2 � We can use the following encoding: x 1 = 1 x 2 = 1 2 c 1 2 c 2

  58. Cost-optimal reachability in priced timed games Theorem ( [BBR05,BBM06] ) Optimal reachability in priced timed games is undecidable. Proof Encode a two-counter machine as a priced timed game. Test Test Instr. Instr. � Instr. Instr. Test Test Instr. Test Instr. q halt Test Instr. Test

  59. Cost-optimal reachability in priced timed games Theorem ( [BBR05,BBM06] ) Optimal reachability in priced timed games is undecidable. Proof Encode a two-counter machine as a priced timed game. Lemma The halting state is reachable if, and only if, there is an optimal strategy in the priced timed game. [BBR05] Brihaye, Bruy` ere, Raskin. On optimal timed strategies. FORMATS, 2005. [BBM06] Bouyer, Brihaye, Markey. Improved undecidability results on weighted timed automa. IPL, 2006.

  60. Cost-optimal reachability in priced timed games Theorem ( [BBR05,BBM06] ) Optimal reachability in priced timed games is undecidable. Proof Encode a two-counter machine as a priced timed game. Lemma The halting state is reachable if, and only if, there is an optimal strategy in the priced timed game. reach terminal location with total weight at most 3 [BBR05] Brihaye, Bruy` ere, Raskin. On optimal timed strategies. FORMATS, 2005. [BBM06] Bouyer, Brihaye, Markey. Improved undecidability results on weighted timed automa. IPL, 2006.

  61. Cost-optimal reachability in priced timed games Theorem ( [BBR05,BBM06] ) Optimal reachability in priced timed games is undecidable. Proof Encode a two-counter machine as a priced timed game. Lemma The halting state is reachable if, and only if, there is an optimal strategy in the priced timed game. reach terminal location with total weight at most 3 Wouldn’t almost-optimal strategies be sufficient? [BBR05] Brihaye, Bruy` ere, Raskin. On optimal timed strategies. FORMATS, 2005. [BBM06] Bouyer, Brihaye, Markey. Improved undecidability results on weighted timed automa. IPL, 2006.

  62. The value of a game Definition

  63. The value of a game Definition Cost of a path: cost( π ) = sum of costs of all transitions until target location

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