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Quantitative Languages

Quantitative Languages. Krishnendu Chatterjee, UCSC Laurent Doyen, EPFL Tom Henzinger, EPFL. CSL 2008. Languages. A language. L(A)  Σ ω. can be viewed as a boolean function:. L A : Σ ω → {0,1}. Model-Checking. ?. B. ². A. Model-checking problem. Input:.

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Quantitative Languages

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  1. Quantitative Languages Krishnendu Chatterjee, UCSC Laurent Doyen, EPFL Tom Henzinger, EPFL CSL 2008

  2. Languages A language L(A)  Σω can be viewed as a boolean function: LA: Σω→ {0,1}

  3. Model-Checking ? B ² A Model-checking problem Input: Model A of the program Model B of the specification Question: does the program A satisfy the specification B ?

  4. Automata & Languages ? B ² A Model-checking as language inclusion Input: finite automata A and B Question: is L(A)  L(B) ?

  5. Automata & Languages Model-checking as language inclusion Input: finite automata A and B Question: is LA(w)  LB(w) for all words w ? Languages are boolean

  6. Quantitative languages A quantitative language (over infinite words) 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), • a probability, etc.

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

  8. Outline • Motivation • Weighted automata • Decision problems • Expressive power

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

  10. 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

  11. 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: max of {values of the runs r over w}

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

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

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

  15. Some value functions

  16. Some value functions

  17. Outline • Motivation • Weighted automata • Decision problems • Expressive power

  18. 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,

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

  20. Language-inclusion game Language inclusion as a game Discounted-sum automata, λ=3/4

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

  22. 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:

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

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

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

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

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

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

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

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

  31. 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.

  32. 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.

  33. Simulation is decidable

  34. Universality and Equivalence Universality problem: Given , is LA(w) ≥ for all words w ? Language equivalence problem: Is LA(w) = LB(w) for all words w ? Complexity/decidability: same situation as Language inclusion.

  35. Outline • Motivation • Weighted automata • Decision problems • Expressive power

  36. Reducibility A class C of weighted automata can be reduced to a class C’ of weighted automata if for all A  C, there is A’  C’ such that LA = LA’.

  37. Reducibility A class C of weighted automata can be reduced to a class C’ of weighted automata if for all A  C, there is A’  C’ such that LA = LA’. • E.g. for boolean languages: • Nondet. coBüchi can be reduced to nondet. Büchi • Nondet. Büchi cannot be reduced to det. Büchi • (nondet. Büchi cannot be determinized)

  38. Some easy facts and can define quantitative languages with infinite range, and cannot. and cannot be reduced to and

  39. Some easy facts For discounted-sum, prefixes provide good approximations of the value. For LimSup, LimInf and LimAvg, suffixes determine the value. cannot be reduced to and and cannot be reduced to

  40. Büchi does not reduce to LimAvg “infinitely many ’’ Deterministic Büchi automaton Assume that L is definable by a LimAvg automaton A. Then, all-cycles in A have average weight 0.

  41. Büchi does not reduce to LimAvg “infinitely many ’’ Deterministic Büchi automaton Hence, the maximal average weight of a run over any word in tends to (at most) 0 when

  42. Büchi does not reduce to LimAvg “infinitely many ’’ Deterministic Büchi automaton Let We have where for sufficienly large

  43. Büchi does not reduce to LimAvg “infinitely many ’’ Deterministic Büchi automaton Hence, Let We have where for sufficienly large and A cannot exist !

  44. (co)Büchi and LimAvg cannot be reduced to By analogous arguments, cannot be reduced to “finitely many ’’ Deterministic coBüchi automaton

  45. (co)Büchi and LimAvg Det. coBüchi automaton L2 is defined by the following nondet. LimAvg automaton:

  46. (co)Büchi and LimAvg Hence, cannot be determinized. Det. coBüchi automaton L2 is defined by the following nondet. LimAvg automaton:

  47. Reducibility relations

  48. Reducibility relations

  49. Reducibility relations

  50. Reducibility relations What about Discounted Sum ?

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