Asymptotic behaviour in temporal logic Aldric Degorre Introduction Asymptotic behaviour in temporal logic Motivation LTL Entropy PLTL PLTL Eugene Asarin 1 , Michel Bockelet 2 , Aldric Degorre 1 , PLTL and its fragments alin Dima 2 and Chunyan Mu 1 Problem and C˘ at˘ main result B¨ uAPC Definitions 1 LIAFA – Universit´ e de Paris-Diderot Building automata for 2 LACL – Universit´ e de Paris-Est Cr´ eteil PLTL Limits of B¨ uAPC+ EQINOCS Meeting Limits of 10/01/2014 at UPEM B¨ uAPC − Conclusion Asymptotic behaviour in temporal logic EQINOCS Meeting 1 / 27
Asymptotic behaviour in What? temporal logic Aldric Degorre Introduction Motivation LTL Entropy PLTL • Temporal logics are a major specification formalism in PLTL PLTL and its verification and synthesis. fragments Problem and main result • A formula specifies a language, the entropy of which can B¨ uAPC be studied. Definitions Building automata for • Here, we study entropy of some temporal logic with PLTL Limits of parametrized time bounds. B¨ uAPC+ Limits of B¨ uAPC − Conclusion Asymptotic behaviour in temporal logic EQINOCS Meeting 2 / 27
Asymptotic behaviour in Why? temporal logic Aldric Degorre Introduction Why parametrized time bounds: Motivation LTL Entropy • Real life appliances may implement time-unbounded PLTL properties as time-bounded behaviors. PLTL PLTL and its fragments • Actual observers/monitors do not have inifinite patience. Problem and main result • Can we still observe the desired behaviors, despite the B¨ uAPC Definitions above, at least for big enough time bounds? Building automata for PLTL Limits of B¨ uAPC+ Limits of B¨ uAPC − Conclusion Asymptotic behaviour in temporal logic EQINOCS Meeting 3 / 27
Asymptotic behaviour in Why? temporal logic Aldric Degorre Introduction Why parametrized time bounds: Motivation LTL Entropy • Real life appliances may implement time-unbounded PLTL properties as time-bounded behaviors. PLTL PLTL and its fragments • Actual observers/monitors do not have inifinite patience. Problem and main result • Can we still observe the desired behaviors, despite the B¨ uAPC Definitions above, at least for big enough time bounds? Building automata for PLTL Limits of Why study entropy in this context: B¨ uAPC+ Limits of As usual: rough assessment of the quality of the B¨ uAPC − approximations made above. Conclusion (Probabilities are too precise: a typical safety property has probability 0.) Asymptotic behaviour in temporal logic EQINOCS Meeting 3 / 27
Asymptotic behaviour in Reminder: LTL temporal logic Aldric Degorre [Pnueli Focs’77] Introduction Motivation LTL Entropy Temporal logic over boolean variables p ∈ AP, with following PLTL syntax: PLTL PLTL and its fragments Problem and ϕ ::= p | ¬ p | ϕ 1 ∧ ϕ 2 | ϕ 1 ∨ ϕ 2 | � ϕ 1 | ϕ 1 U ϕ 2 | ϕ 1 R ϕ 2 main result B¨ uAPC Definitions (and usual syntactic sugar: ⊤ , ⊥ , = ⇒ , � , ♦ , ... ) Building automata for 2 AP � ω . � Models: infinite words in PLTL Limits of B¨ uAPC+ Example Limits of p 0 1 1 0 0. . . B¨ uAPC − ⇒ � q ): A model of � ( p = q 1 0 1 1 0. . . Conclusion Asymptotic behaviour in temporal logic EQINOCS Meeting 4 / 27
Asymptotic behaviour in EQINOCS’ nails and hammer temporal logic Aldric Degorre ... or why this talk is not about LTL(1) Our problem: Introduction Motivation • “How many” behaviors satisfy a formula? LTL Entropy • I.e., for infinite behaviors, how many prefixes? PLTL PLTL PLTL and its fragments Problem and main result B¨ uAPC Definitions Building automata for PLTL Limits of B¨ uAPC+ Limits of B¨ uAPC − Conclusion Asymptotic behaviour in temporal logic EQINOCS Meeting 5 / 27
Asymptotic behaviour in EQINOCS’ nails and hammer temporal logic Aldric Degorre ... or why this talk is not about LTL(1) Our problem: Introduction Motivation • “How many” behaviors satisfy a formula? LTL Entropy • I.e., for infinite behaviors, how many prefixes? PLTL PLTL Our tool: entropy H . For an ω -language L : PLTL and its fragments 1 Problem and main result H ( L ) = lim sup n log # pref ( L , n ) B¨ uAPC n →∞ Definitions Building automata for PLTL Limits of B¨ uAPC+ Limits of B¨ uAPC − Conclusion Asymptotic behaviour in temporal logic EQINOCS Meeting 5 / 27
Asymptotic behaviour in EQINOCS’ nails and hammer temporal logic Aldric Degorre ... or why this talk is not about LTL(1) Our problem: Introduction Motivation • “How many” behaviors satisfy a formula? LTL Entropy • I.e., for infinite behaviors, how many prefixes? PLTL PLTL Our tool: entropy H . For an ω -language L : PLTL and its fragments 1 Problem and main result H ( L ) = lim sup n log # pref ( L , n ) B¨ uAPC n →∞ Definitions Building automata for PLTL Example Limits of B¨ uAPC+ • H (( a + b ) ω ) = log 2 = 1; Limits of B¨ uAPC − ]) = log 2 | AP | = | AP | (no constraint most of the • H ([ [ �♦ p ] Conclusion time); • H ([ [ ♦� p ] ]) = | AP | (for any prefix, it is always possible to append p ). Asymptotic behaviour in temporal logic EQINOCS Meeting 5 / 27
Asymptotic behaviour in Entroy of LTL: either too hard... temporal logic Aldric Degorre or too sad Introduction ... or why this talk is not about LTL(2) Motivation LTL Entropy PLTL • Unfortunately, except for a few easy and obvious cases PLTL H ([ [ ϕ ] ]) is hard to guess. PLTL and its fragments Problem and Example main result B¨ uAPC One easy case, “liveness” formulas: H ([ [ ♦ ψ ] ]) = | AP | , where [ [ ψ ] ] � = ∅ . Definitions Building automata for PLTL • Nonetheless, ω -regular languages = ⇒ ∃ translation to Limits of B¨ uAPC+ (Generalized B¨ uchi) Automata [Couvreur]. Limits of B¨ uAPC − • The usual (but sad!) approach H = log ρ ( M ) works well Conclusion ( M : adjacency matrix of the determinization of some subautomaton). Asymptotic behaviour in temporal logic EQINOCS Meeting 6 / 27
Asymptotic behaviour in PLTL temporal logic Aldric Degorre [Alur, Etessami, LaTorre, Peled ICALP’99] Introduction Motivation LTL • PLTL: LTL with parameters. Entropy PLTL • 2 new parametrized modalities: U t and R t PLTL PLTL and its (or equivalently � t and ♦ t ). fragments Problem and main result • Model of a PLTL formula: parameter value + behavior. B¨ uAPC • Classical problem: what parameter values make the Definitions Building automata for formula satisfiable? PLTL Limits of B¨ uAPC+ Limits of B¨ uAPC − Conclusion Asymptotic behaviour in temporal logic EQINOCS Meeting 7 / 27
Asymptotic behaviour in PLTL temporal logic Aldric Degorre [Alur, Etessami, LaTorre, Peled ICALP’99] Introduction Motivation LTL • PLTL: LTL with parameters. Entropy PLTL • 2 new parametrized modalities: U t and R t PLTL PLTL and its (or equivalently � t and ♦ t ). fragments Problem and main result • Model of a PLTL formula: parameter value + behavior. B¨ uAPC • Classical problem: what parameter values make the Definitions Building automata for formula satisfiable? PLTL Limits of Our problem: B¨ uAPC+ Limits of • For a given parameter value, compute H ? B¨ uAPC − Conclusion Asymptotic behaviour in temporal logic EQINOCS Meeting 7 / 27
Asymptotic behaviour in PLTL temporal logic Aldric Degorre [Alur, Etessami, LaTorre, Peled ICALP’99] Introduction Motivation LTL • PLTL: LTL with parameters. Entropy PLTL • 2 new parametrized modalities: U t and R t PLTL PLTL and its (or equivalently � t and ♦ t ). fragments Problem and main result • Model of a PLTL formula: parameter value + behavior. B¨ uAPC • Classical problem: what parameter values make the Definitions Building automata for formula satisfiable? PLTL Limits of Our problem: B¨ uAPC+ Limits of • For a given parameter value, compute H → no! (it’s LTL) B¨ uAPC − • Look at H when parameter values go to ∞ and compare Conclusion with LTL → yes, let’s do this! Asymptotic behaviour in temporal logic EQINOCS Meeting 7 / 27
Asymptotic behaviour in Outline temporal logic Aldric Degorre Introduction Motivation 1 Introduction LTL Entropy PLTL 2 PLTL PLTL PLTL and its fragments Problem and main result 3 B¨ uAPC B¨ uAPC Definitions Building automata for 4 Limits of B¨ uAPC+ PLTL Limits of B¨ uAPC+ uAPC − 5 Limits of B¨ Limits of B¨ uAPC − Conclusion 6 Conclusion Asymptotic behaviour in temporal logic EQINOCS Meeting 8 / 27
Asymptotic behaviour in PLTL syntax temporal logic Aldric Degorre A PLTL formula ϕ in positive normal form is as follows: Introduction Motivation LTL Entropy ϕ ::= p | ¬ p | ϕ 1 ∧ ϕ 2 | ϕ 1 ∨ ϕ 2 propositional logic PLTL | � ϕ 1 | ϕ 1 U ϕ 2 | ϕ 1 R ϕ 2 PLTL time modalities PLTL and its fragments | ϕ 1 U t ϕ 2 | ϕ 1 R t ϕ 2 parametrized time modalities Problem and main result B¨ uAPC ( p ∈ AP: propositional variable; t ∈ t : formal parameter) Definitions Building automata for PLTL Expected syntatic sugar: � t ϕ ≡⊥ R t ϕ , ♦ t ϕ ≡ ⊤U t ϕ . Limits of B¨ uAPC+ Limits of The following fragments are defined : B¨ uAPC − Conclusion • PLTL ♦ : PLTL without R t , “positive fragment”. • PLTL � : PLTL without U t , “negative fragment”. Asymptotic behaviour in temporal logic EQINOCS Meeting 9 / 27
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