Expressiveness and Closure Properties for Quantitative Languages Krishnendu Chatterjee, IST Austria Laurent Doyen, ULB Belgium Tom Henzinger, EPFL Switzerland

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<ul><li><p>Expressiveness and Closure Properties for Quantitative Languages Krishnendu Chatterjee, IST AustriaLaurent Doyen, ULB Belgium Tom Henzinger, EPFL SwitzerlandLICS 2009</p></li><li><p>Model-CheckingProperty/ SpecificationYes / NoSatisfaction RelationProgram/ System-perhaps a proof -perhaps some counterexamples </p></li><li><p>Model-CheckingProperty/ SpecificationYes/NoTrace inclusionProgram/ SystemFormulaEvery request is followed by a grantFinite automaton</p></li><li><p>Model-CheckingProperty/ SpecificationYes/NoTrace inclusionProgram/ SystemFinite automatonModel-checking is booleanFormulaEvery request is followed by a grant- a trace is either good or bad</p></li><li><p>Quantitative AnalysisProperty/ SpecificationValue (R)Quantitative AnalysisProgram/ SystemFinite automatonFormulaEvery request is followed by a grantMeasure of fit between system and spece.g. average number of requests immediately granted</p></li><li><p>Distance (R)Quantitative AnalysisQuantitative AnalysisProgram/ System #1Finite automaton Comparing two implementationse.g. cost or quality measureProgram/ System #2Every request is followed by a grant</p></li><li><p>Quantitative Model-checkingIs there a Quantitative Framework with - an appealing mathematical formulation, - useful expressive power, and - good algorithmic properties ?(Like the boolean theory of -regularity.)</p><p>Note: Quantitative is more than timed and probabilistic</p></li><li><p>A language is a boolean function:Quantitative languagesA quantitative language is a function:L(w) can be interpreted as:</p><p> 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).</p></li><li><p>Outline</p><p> Weighted automata</p><p> Expressive power</p><p> Closure properties</p></li><li><p>Weighted automataQuantitative languages are generated by weighted automata.</p></li><li><p>Weighted automataQuantitative 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)</p></li><li><p>Some value functions(reachability)(Bchi)(coBchi)(vi {0,1})</p></li><li><p>Some value functions(reachability)(Bchi)(coBchi)(vi {0,1})</p></li><li><p>Outline</p><p> Weighted automata</p><p> Expressive power</p><p> Closure properties</p></li><li><p>ReducibilityA class C of weighted automata can be reduced to a class C of weighted automata iffor all A C, there is A C such that LA = LA.</p></li><li><p>ReducibilityA class C of weighted automata can be reduced to a class C of weighted automata iffor all A C, there is A C such that LA = LA.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)</p></li><li><p> cannot be determinized. </p><p> cannot be determinized. </p><p>Some known facts (CSL08) cannot be reduced to cannot be reduced to </p></li><li><p>Reducibility relations</p></li><li><p>Cut-point languagesWords with value above some threshold:-regular for Sup, LimSup, LimInfcan be non--regular for LimAvg and Discounted</p></li><li><p>Cut-point languagesLimAvg:average number of as = 1is not -regular A deterministic automaton for would accept (anb) for some n</p></li><li><p>Cut-point languagesDisc:disc. sum of as 1is not -regular ambiguous word1p1p2</p></li><li><p>Cut-point languagesDisc:disc. sum of as 1is not -regular ambiguous word1From any two positions p1 and p2, there is a continuation accepted from p1 but not from p2 p1p2</p></li><li><p>Cut-point LanguagesCut-point languages for deterministic LimAvg-automata are studied in [Alur/Degorre/Maler/Weiss09] Cut-point languages of LimAvg and Discounted can be non--regularCut-point languages are not robust w.r.t. transition weights.</p></li><li><p>Cut-point Languagesisolated cut-pointIsolated cut-point languages are robustIsolated cut-point languages are -regular(for deterministic automata)</p></li><li><p>Cut-point LanguagesEach s.c.c. defines an interval of values.Make accepting those s.c.c. with interval above LimAvg: s.c.c. decomposition</p></li><li><p>Either value is , then accept or value is , the rejectCut-point LanguagesDisc: after sufficiently long prefix, decision can be taken</p></li><li><p> is reducible to .Expressive power of {0,1}-automata</p><p> is not reducible to .</p></li><li><p> is reducible to .Expressive power of {0,1}-automataStore the value AB</p></li><li><p> is reducible to .Expressive power of {0,1}-automataAB can take finitely many different values.Store the value </p></li><li><p> is reducible to .Expressive power of {0,1}-automataAB</p></li><li><p> is reducible to .Expressive power of {0,1}-automataAB</p></li><li><p> is reducible to .Expressive power of {0,1}-automataAB</p></li><li><p> is reducible to .Expressive power of {0,1}-automataBA</p></li><li><p>Therefore for</p><p> is reducible to .Expressive power of {0,1}-automataABfor all</p></li><li><p>Outline</p><p> Weighted automata</p><p> Expressive power</p><p> Closure properties</p></li><li><p>OperationsOperations on quantitative languages:</p><p> shift(L1,c)(w) = L1(w) + c scale(L1,c)(w) = cL1(w) (c&gt;0)</p></li><li><p>OperationsOperations on quantitative languages:</p><p> shift(L1,c)(w) = L1(w) + c scale(L1,c)(w) = cL1(w) (c&gt;0) max(L1,L2)(w) = max(L1(w),L2(w)) min(L1,L2)(w) = min(L1(w),L2(w)) complement(L1)(w) = 1-L1(w)</p></li><li><p>OperationsOperations on quantitative languages:</p><p> shift(L1,c)(w) = L1(w) + c scale(L1,c)(w) = cL1(w) (c&gt;0) max(L1,L2)(w) = max(L1(w),L2(w)) min(L1,L2)(w) = min(L1(w),L2(w)) complement(L1)(w) = 1-L1(w) sum(L1,L2)(w) = L1(w) + L2(w)</p></li><li><p>Closure propertiesAll classes of weighted automata are closed under shift and scale.All classes of nondeterministic weighted automata are closed under max.</p></li><li><p>Closure properties</p></li><li><p>Closure propertiesAnalogous results for boolean languages.</p></li><li><p>Closure propertiesThere is no nondeterministic LimAvg automaton for the language Lm = min(La,Lb).</p></li><li><p>Closure propertiesThere is no nondeterministic LimAvg automaton for the language Lm = min(La,Lb).Assume that L is definable by a LimAvg automaton C.</p></li><li><p>Closure propertiesThere is no nondeterministic LimAvg automaton for the language Lm = min(La,Lb).Assume that L is definable by a LimAvg automaton C.Then, some a-cycle or b-cycle in C has average weight &gt;0.(consider the word for large)</p></li><li><p>Closure propertiesAssume that L is definable by a LimAvg automaton C.Then, some a-cycle or b-cycle in C has average weight &gt;0.Then, some word gets value &gt;0There is no nondeterministic LimAvg automaton for the language Lm = min(La,Lb).</p></li><li><p>Closure propertiesThere is no nondeterministic LimAvg automaton for the language Lm = min(La,Lb).There is no nondeterministic Discounted automaton for the language Lm = min(La,Lb).Proof: analogous argument.</p></li><li><p>Closure properties</p></li><li><p>Closure propertiesmin(L1,L2) = 1-max(1-L1,1-L2)</p></li><li><p>Closure propertiesBy analogous arguments (analysis of cycles).</p></li><li><p>Conclusion Quantitative generalization of languages to model programs/systems more accurately.</p><p> Expressive power: Cut-point languages; {0,1} automata.</p><p> Closure properties.</p><p> Outlook: other/equivalent formalisms for quantitative specification ? </p></li><li><p>Thank you !</p><p>Questions ?The end</p></li><li><p>Say that Sup generalizes Reachability condition, while LimSup and LimInf generalize Buchi and coBuchi.LimAvg and Disc have no boolean counterpart.Say that Sup generalizes Reachability condition, while LimSup and LimInf generalize Buchi and coBuchi.LimAvg and Disc have no boolean counterpart.Upper boundsUpper boundsUpper boundsUpper boundsUpper bounds</p></li></ul>

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