# 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

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