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

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<ul><li> Slide 1 </li> <li> Quantitative Languages Krishnendu Chatterjee, UCSC Laurent Doyen, EPFL Tom Henzinger, EPFL CSL 2008 </li> <li> Slide 2 </li> <li> Languages L(A) A language can be viewed as a boolean function: L A : {0,1} </li> <li> Slide 3 </li> <li> 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 </li> <li> Slide 4 </li> <li> Automata &amp; Languages Input: finite automata A and B Question: is L(A) L(B) ? Model-checking as language inclusion A B ? </li> <li> Slide 5 </li> <li> Automata &amp; 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 </li> <li> Slide 6 </li> <li> 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. </li> <li> Slide 7 </li> <li> 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: </li> <li> Slide 8 </li> <li> Outline Motivation Weighted automata Decision problems Expressive power </li> <li> Slide 9 </li> <li> Automata Boolean languages are generated by finite automata. Nondeterministic Bchi automaton </li> <li> Slide 10 </li> <li> 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 </li> <li> Slide 11 </li> <li> 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} </li> <li> Slide 12 </li> <li> 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} </li> <li> Slide 13 </li> <li> Weighted automata Quantitative languages are generated by weighted automata. Weight function </li> <li> Slide 14 </li> <li> 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 </li> <li> Slide 15 </li> <li> Some value functions </li> <li> Slide 16 </li> <li> Slide 17 </li> <li> Outline Motivation Weighted automata Decision problems Expressive power </li> <li> Slide 18 </li> <li> 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 </li> <li> Slide 19 </li> <li> 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. </li> <li> Slide 20 </li> <li> Language-inclusion game Language inclusion as a game Discounted-sum automata, =3/4 </li> <li> Slide 21 </li> <li> Language-inclusion game Tokens on the initial states Language inclusion as a game </li> <li> Slide 22 </li> <li> 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 ) &gt; Val(r 2 ). Language inclusion as a game </li> <li> Slide 23 </li> <li> Language-inclusion game Challenger: Simulator: Language inclusion as a game </li> <li> Slide 24 </li> <li> Language-inclusion game Challenger: Simulator: Language inclusion as a game </li> <li> Slide 25 </li> <li> Language-inclusion game Challenger: Simulator: Language inclusion as a game </li> <li> Slide 26 </li> <li> Language-inclusion game Challenger: Simulator: Language inclusion as a game </li> <li> Slide 27 </li> <li> Language-inclusion game Challenger: Simulator: Language inclusion as a game </li> <li> Slide 28 </li> <li> Language-inclusion game Challenger: Simulator: Language inclusion as a game </li> <li> Slide 29 </li> <li> Language-inclusion game Challenger: Simulator: Language inclusion as a game </li> <li> Slide 30 </li> <li> Language-inclusion game Challenger: Simulator: Challenger wins since 4&gt;2. Language inclusion as a game However, L A (w) L B (w) for all w. </li> <li> Slide 31 </li> <li> 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. </li> <li> Slide 32 </li> <li> 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. </li> <li> Slide 33 </li> <li> Simulation is decidable </li> <li> Slide 34 </li> <li> 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. </li> <li> Slide 35 </li> <li> Outline Motivation Weighted automata Decision problems Expressive power </li> <li> Slide 36 </li> <li> 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. </li> <li> Slide 37 </li> <li> 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) </li> <li> Slide 38 </li> <li> Some easy facts and cannot be reduced to and and can define quantitative languages with infinite range, and cannot. </li> <li> Slide 39 </li> <li> 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 </li> <li> Slide 40 </li> <li> 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. </li> <li> Slide 41 </li> <li> 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 </li> <li> Slide 42 </li> <li> Bchi does not reduce to LimAvg LetWe have where for sufficienly large infinitely many Deterministic Bchi automaton </li> <li> Slide 43 </li> <li> where for sufficienly large Hence, Bchi does not reduce to LimAvg LetWe have and A cannot exist ! infinitely many Deterministic Bchi automaton </li> <li> Slide 44 </li> <li> (co)Bchi and LimAvg cannot be reduced to By analogous arguments, cannot be reduced to finitely many Deterministic coBchi automaton </li> <li> Slide 45 </li> <li> (co)Bchi and LimAvg Det. coBchi automaton L 2 is defined by the following nondet. LimAvg automaton: </li> <li> Slide 46 </li> <li> (co)Bchi and LimAvg Det. coBchi automaton L 2 is defined by the following nondet. LimAvg automaton: Hence, cannot be determinized. </li> <li> Slide 47 </li> <li> Reducibility relations </li> <li> Slide 48 </li> <li> Slide 49 </li> <li> Slide 50 </li> <li> What about Discounted Sum ? </li> <li> Slide 51 </li> <li> Last result cannot be determinized. =3/4 </li> <li> Slide 52 </li> <li> Disc cannot be determinized Value of a word : disc. sum of s </li> <li> Slide 53 </li> <li> Disc cannot be determinized Let </li> <li> Slide 54 </li> <li> Disc cannot be determinized Let </li> <li> Slide 55 </li> <li> Disc cannot be determinized Let </li> <li> Slide 56 </li> <li> Disc cannot be determinized Let </li> <li> Slide 57 </li> <li> Disc cannot be determinized Let </li> <li> Slide 58 </li> <li> Disc cannot be determinized Let </li> <li> Slide 59 </li> <li> Disc cannot be determinized Let </li> <li> Slide 60 </li> <li> Disc cannot be determinized Let </li> <li> Slide 61 </li> <li> Disc cannot be determinized Let </li> <li> Slide 62 </li> <li> Disc cannot be determinized Let If then </li> <li> Slide 63 </li> <li> Disc cannot be determinized Let If then </li> <li> Slide 64 </li> <li> Disc cannot be determinized </li> <li> Slide 65 </li> <li> Slide 66 </li> <li> How many different values can take ? </li> <li> Slide 67 </li> <li> Disc cannot be determinized How many different values can take ? infinitely many if for all </li> <li> Slide 68 </li> <li> Disc cannot be determinized How many different values can take ? infinitely many if for all By a careful analysis of the shape of the family of equations, it can be proven that no rational can be a solution. </li> <li> Slide 69 </li> <li> Last result cannot be determinized. =3/4 </li> <li> Slide 70 </li> <li> Reducibility relations </li> <li> Slide 71 </li> <li> Conclusion Quantitative generalization of languages to model programs/systems more accurately. LimAvg and Disc : deciding language inclusion is open; Simulation is a decidable over-approximation. Expressive power classification: DBW and LimAvg are incomparable; LimAvg and Disc cannot be determinized. </li> <li> Slide 72 </li> <li> Thank you ! Questions ? The end </li> <li> Slide 73 </li> </ul>