Laurent Doyen LSV, ENS Cachan & CNRS

1. Quantitative Languages: Weighted Automata and Beyond Laurent DoyenLSV, ENS Cachan & CNRS

2. Credits This talk is based on joint work with… • K. Chatterjee, H. Edelsbrunner, T. A. Henzinger (IST Austria) • A. Degorre, J.-F. Raskin (ULB Brussels) • R. Gentilini (Uni. Perugia) • S. Torunczyk (Uni. Warsaw) • P. Rannou (ENS Cachan) (CSL’08, LICS’09, FCT’09, CSL’10, Concur’10)

3. Model-Checking Program/ System Property/ Specification Satisfaction Relation Yes / No -perhaps a proof -perhaps some counterexamples

4. Model-Checking Program/ System Property/ Specification Formula Every request is followed by a grant Finite-state machine Trace inclusion Yes / No

5. Model-Checking Program/ System Property/ Specification Formula Every request is followed by a grant Finite-state machine Trace inclusion Yes / No Model-checking is boolean • a trace is either good or bad • a system is either good or bad

6. Quantitative Analysis Program/ System Property/ Specification Formula Every request is followed by a grant Finite-state machine Quantitative Analysis Value (R) • Measure of “fit” between system and spec • e.g. average number of requests immediately granted

7. Quantitative Analysis Program/ System #1 Program/ System #2 Finite-state machine Every request is followed by a grant Quantitative Analysis Distance (R) • Comparing two implementations • e.g. cost or quality measure

8. Quantitative Model-checking Is there a Quantitative Framework with - an appealing mathematical formulation, - useful expressive power, robustness and - good algorithmic properties ? (Like the boolean theory of -regularity.) Note: “Quantitative” is more than “timed” and “probabilistic”

9. Quantitative languages A language is a boolean function:

10. Quantitative languages A language is a boolean function: A quantitative language is a function: • L(w) can be interpreted as: • the amount of some resource needed by thesystem to produce w (power, energy, time consumption), • a reliability measure (the average number of “faults” in w).

11. Quantitative languages Quantitative language inclusion Is LA(w)  LB(w) for all words w ? LA(w) peak resource consumption Long-run average response time LB(w) resource bound a Average response-time requirement

12. Quantitative languages Quantitative language inclusion Is LA(w)  LB(w) for all words w ? Distance: LA(w) peak resource consumption Long-run average response time LB(w) resource bound a Average response-time requirement

13. Outline • Motivation • Weighted automata • Decision problems • Expressive power • Closure properties • Beyond weighted automata

14. Automata Boolean languages are generated by finite automata. Nondeterministic Büchi automaton

15. Automata Boolean languages are generated by finite automata. Nondeterministic Büchi automaton Value of a run r: Val(r)= 1if an accepting state occurs -ly often in r 0 otherwise

16. Automata Boolean languages are generated by finite automata. Nondeterministic Büchi automaton Value of a run r: Val(r)= 1if an accepting state occurs -ly often in r Value of a word w: sup of {values of the runs r over w}

17. Automata Boolean languages are generated by finite automata. Nondeterministic Büchi automaton LA(w) = sup of {Val(r) | r is a run of A over w}

18. Weighted automata Weight function Quantitative languages are generated by weighted automata.

19. Weighted automata Weight function where is a value function Quantitative languages are generated by weighted automata. Value of a word w: sup of {values of the runs r over w} Value of a run r: Val(r)

20. Some value functions (vi {0,1}) (reachability) (Büchi) (coBüchi)

21. Some value functions (vi {0,1}) (reachability) (Büchi) (coBüchi)

22. Example: a motor Specification Deterministic LimAvg automaton

23. Example: a motor Implementation(refined spec.) Specification

24. Example: a motor Implementation(refined spec.) Specification

25. Example: a motor Implementation(refined spec.) Specification

26. Example: a motor Implementation(refined spec.) Specification

27. Example: a motor Implementation(refined spec.) Specification

28. Example: a motor Implementation(refined spec.) Specification In the long-run, implementation has average cost cheaper than specification (for all traces).

29. Outline • Motivation • Weighted automata • Decision problems • Expressive power • Closure properties • Beyond weighted automata

30. Emptiness • decidable in polynomial time for Given , is LA(w) ≥ for some word w ? • solved by finding the maximal value of an infinite path in the graph of A, • memoryless strategies exist in the corresponding quantitative 1-player games,

31. Language Inclusion Is LA(w)  LB(w) for all words w ? • PSPACE-complete for • equivalent to

32. Language Inclusion Is LA(w)  LB(w) for all words w ? • PSPACE-complete for • Solvable in polynomial-time when B is deterministic for and , • undecidable for nondeterministic automata. • open question for nondeterministic automata.

33. Language Inclusion

34. Language-inclusion game Language inclusion as a game Discounted-sum automata, λ=3/4 Is LA(w)  LB(w) for all words w ?

35. Language-inclusion game Language inclusion as a game Discounted-sum automata, λ=3/4 Yes ! Is LA(w)  LB(w) for all words w ?

36. Language-inclusion game Language inclusion as a game • turn-based, two players; • Challenger wins iff Is LA(w)  LB(w) for all words w ?

37. Language-inclusion game Language inclusion as a game • turn-based, two players; • Challenger wins iff winning strategy Is LA(w)  LB(w) for all words w ?

38. Language-inclusion game Language inclusion as a game Tokens on the initial states

39. Language-inclusion game Language inclusion as a game Challenger constructs a run r1 of A, Simulator constructs a run r2 of B. Challenger wins if Val(r1) > Val(r2). Challenger: Simulator:

40. Language-inclusion game Language inclusion as a game Challenger: Simulator:

41. Language-inclusion game Language inclusion as a game Challenger: Simulator:

42. Language-inclusion game Language inclusion as a game Challenger: Simulator:

43. Language-inclusion game Language inclusion as a game Challenger: Simulator:

44. Language-inclusion game Language inclusion as a game Challenger: Simulator:

45. Language-inclusion game Language inclusion as a game Challenger: Simulator:

46. Language-inclusion game Language inclusion as a game Challenger: Simulator:

47. Language-inclusion game Language inclusion as a game Challenger wins since 4>2. However, LA(w)  LB(w) for all w. Challenger: Simulator:

48. Language-inclusion game The game is blind if the Challenger cannot observethe state of the Simulator. Challenger has no winning strategy in the blind game if and only if LA(w)  LB(w) for all words w.

49. Language-inclusion game The game is blind if the Challenger cannot observe the state of the Simulator. Challenger has no winning strategy in the blind game if and only if LA(w)  LB(w) for all words w. When the game is not blind, we say that BsimulatesA if the Challenger has no winning strategy. Simulation implies language inclusion.

50. Simulation is decidable