quantitative languages krishnendu chatterjee, ucsc laurent doyen, epfl tom henzinger, epfl csl 2008

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  • Slide 1
  • Quantitative Languages Krishnendu Chatterjee, UCSC Laurent Doyen, EPFL Tom Henzinger, EPFL CSL 2008
  • Slide 2
  • Languages L(A) A language can be viewed as a boolean function: L A : {0,1}
  • Slide 3
  • Model-Checking Model A of the program Model B of the specification does the program A satisfy the specification B ? A B ? Question: Input: Model-checking problem
  • Slide 4
  • Automata & Languages Input: finite automata A and B Question: is L(A) L(B) ? Model-checking as language inclusion A B ?
  • Slide 5
  • Automata & Languages Input: finite automata A and B Question: is L A (w) L B (w) for all words w ? Languages are boolean Model-checking as language inclusion
  • Slide 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 the system to produce w (power, energy, time consumption), a reliability measure (the average number of faults in w), a probability, etc.
  • Slide 7
  • Is L A (w) L B (w) for all words w ? Quantitative languages Quantitative language inclusion L A (w) peak resource consumption Long-run average response time L B (w) resource bound a Average response-time requirement Example:
  • Slide 8
  • Outline Motivation Weighted automata Decision problems Expressive power
  • Slide 9
  • Automata Boolean languages are generated by finite automata. Nondeterministic Bchi automaton
  • Slide 10
  • Automata Boolean languages are generated by finite automata. Nondeterministic Bchi automaton Value of a run r: Val(r)=1 if an accepting state occurs - ly often in r
  • Slide 11
  • Automata Boolean languages are generated by finite automata. Nondeterministic Bchi automaton Value of a run r: Val(r)=1 if an accepting state occurs - ly often in r Value of a word w: max of {values of the runs r over w}
  • Slide 12
  • Automata Boolean languages are generated by finite automata. Nondeterministic Bchi automaton L A (w) = max of {Val(r) | r is a run of A over w}
  • Slide 13
  • Weighted automata Quantitative languages are generated by weighted automata. Weight function
  • Slide 14
  • Weighted automata Quantitative languages are generated by weighted automata. Weight function Value of a word w: max of {values of the runs r over w} Value of a run r: Val(r) where is a value function
  • Slide 15
  • Some value functions
  • Slide 16
  • Slide 17
  • Outline Motivation Weighted automata Decision problems Expressive power
  • Slide 18
  • Emptiness Given, is L A (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, decidable in polynomial time for
  • Slide 19
  • Language Inclusion Is L A (w) L B (w) for all words w ? PSPACE-complete for Solvable in polynomial-time when B is deterministic for and, open question for nondeterministic automata.
  • Slide 20
  • Language-inclusion game Language inclusion as a game Discounted-sum automata, =3/4
  • Slide 21
  • Language-inclusion game Tokens on the initial states Language inclusion as a game
  • Slide 22
  • Language-inclusion game Challenger: Simulator: Challenger constructs a run r 1 of A, Simulator constructs a run r 2 of B. Challenger wins if Val(r 1 ) > Val(r 2 ). Language inclusion as a game
  • Slide 23
  • Language-inclusion game Challenger: Simulator: Language inclusion as a game
  • Slide 24
  • Language-inclusion game Challenger: Simulator: Language inclusion as a game
  • Slide 25
  • Language-inclusion game Challenger: Simulator: Language inclusion as a game
  • Slide 26
  • Language-inclusion game Challenger: Simulator: Language inclusion as a game
  • Slide 27
  • Language-inclusion game Challenger: Simulator: Language inclusion as a game
  • Slide 28
  • Language-inclusion game Challenger: Simulator: Language inclusion as a game
  • Slide 29
  • Language-inclusion game Challenger: Simulator: Language inclusion as a game
  • Slide 30
  • Language-inclusion game Challenger: Simulator: Challenger wins since 4>2. Language inclusion as a game However, L A (w) L B (w) for all w.
  • Slide 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 L A (w) L B (w) for all words w.
  • Slide 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 L A (w) L B (w) for all words w. When the game is not blind, we say that B simulates A if the Challenger has no winning strategy. Simulation implies language inclusion.
  • Slide 33
  • Simulation is decidable
  • Slide 34
  • Universality and Equivalence Is L A (w) = L B (w) for all words w ? Language equivalence problem: Universality problem: Given, is L A (w) for all words w ? Complexity/decidability: same situation as Language inclusion.
  • Slide 35
  • Outline Motivation Weighted automata Decision problems Expressive power
  • Slide 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 L A = L A.
  • Slide 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 L A = L A. E.g. for boolean languages: Nondet. coBchi can be reduced to nondet. Bchi Nondet. Bchi cannot be reduced to det. Bchi (nondet. Bchi cannot be determinized)
  • Slide 38
  • Some easy facts and cannot be reduced to and and can define quantitative languages with infinite range, and cannot.
  • Slide 39
  • Some easy facts For discounted-sum, prefixes provide good approximations of the value. For LimSup, LimInf and LimAvg, suffixes determine the value. and cannot be reduced to cannot be reduced to and
  • Slide 40
  • Bchi does not reduce to LimAvg infinitely many Deterministic Bchi automaton Assume that L is definable by a LimAvg automaton A. Then, all -cycles in A have average weight 0.
  • Slide 41
  • Bchi does not reduce to LimAvg Hence, the maximal average weight of a run over any word in tends to (at most) 0 when infinitely many Deterministic Bchi automaton
  • Slide 42
  • Bchi does not reduce to LimAvg LetWe have where for sufficienly large infinitely many Deterministic Bchi automaton
  • Slide 43
  • where for sufficienly large Hence, Bchi does not reduce to LimAvg LetWe have and A cannot exist ! infinitely many Deterministic Bchi automaton
  • Slide 44
  • (co)Bchi and LimAvg cannot be reduced to By analogous arguments, cannot be reduced to finitely many Deterministic coBchi automaton
  • Slide 45
  • (co)Bchi and LimAvg Det. coBchi automaton L 2 is defined by the following nondet. LimAvg automaton:
  • Slide 46
  • (co)Bchi and LimAvg Det. coBchi automaton L 2 is defined by the following nondet. LimAvg automaton: Hence, cannot be determinized.
  • Slide 47
  • Reducibility relations
  • Slide 48
  • Slide 49
  • Slide 50
  • What about Discounted Sum ?
  • Slide 51
  • Last result cannot be determinized. =3/4
  • Slide 52
  • Disc cannot be determinized Value of a word : disc. sum of s
  • Slide 53
  • Disc cannot be determinized Let
  • Slide 54
  • Disc cannot be determinized Let
  • Slide 55
  • Disc cannot be determinized Let
  • Slide 56
  • Disc cannot be determinized Let
  • Slide 57
  • Disc cannot be determinized Let
  • Slide 58
  • Disc cannot be determinized Let
  • Slide 59
  • Disc cannot be determinized Let
  • Slide 60
  • Disc cannot be determinized Let
  • Slide 61
  • Disc cannot be determinized Let
  • Slide 62
  • Disc cannot be determinized Let If then
  • Slide 63
  • Disc cannot be determinized Let If then
  • Slide 64
  • Disc cannot be determinized
  • Slide 65
  • Slide 66
  • How many different values can take ?
  • Slide 67
  • Disc cannot be determinized How many different values can take ? infinitely many if for all
  • Slide 68
  • Disc cannot be determinized How many different values can take ? infinitely many if for all By a careful a