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1 Declarative modelling for timing The real-time logic: Duration Calculus Michael R. Hansen Informatics and Mathematical Modelling Technical University of Denmark Declarative Modelling, December, 2007, Michael R. Hansen p. 1/22

2 Informal introduction to Duration Calculus A logic for declarative modelling of real-time properties Background A simple case study: Gas Burner A decidability result pointers to current focus Declarative Modelling, December, 2007, Michael R. Hansen p. 2/22

3 Background Provable Correct Systems (ProCoS, ESPRIT BRA 3104) Bjørner Langmaack Hoare Olderog Project case study: Gas Burner Sørensen Ravn Rischel Declarative Modelling, December, 2007, Michael R. Hansen p. 3/22

4 Background Provable Correct Systems (ProCoS, ESPRIT BRA 3104) Bjørner Langmaack Hoare Olderog Project case study: Gas Burner Sørensen Ravn Rischel Intervals properties Timed Automata, Real-time Logic, Metric Temporal Logic, Explicit Clock Temporal,..., Alur, Dill, Jahanian, Mok, Koymans, Harel, Lichtenstein, Pnueli, Declarative Modelling, December, 2007, Michael R. Hansen p. 3/22

5 Background Provable Correct Systems (ProCoS, ESPRIT BRA 3104) Bjørner Langmaack Hoare Olderog Project case study: Gas Burner Sørensen Ravn Rischel Intervals properties Timed Automata, Real-time Logic, Metric Temporal Logic, Explicit Clock Temporal,..., Alur, Dill, Jahanian, Mok, Koymans, Harel, Lichtenstein, Pnueli,... Duration of states Duration Calculus Zhou Hoare Ravn Declarative Modelling, December, 2007, Michael R. Hansen p. 3/22

6 Background Provable Correct Systems (ProCoS, ESPRIT BRA 3104) Bjørner Langmaack Hoare Olderog Project case study: Gas Burner Sørensen Ravn Rischel Intervals properties Timed Automata, Real-time Logic, Metric Temporal Logic, Explicit Clock Temporal,..., Alur, Dill, Jahanian, Mok, Koymans, Harel, Lichtenstein, Pnueli,... Duration of states Duration Calculus Zhou Hoare Ravn 91 an Interval Temporal Logic Halpern Moszkowski Manna Declarative Modelling, December, 2007, Michael R. Hansen p. 3/22

7 Background Provable Correct Systems (ProCoS, ESPRIT BRA 3104) Bjørner Langmaack Hoare Olderog Project case study: Gas Burner Sørensen Ravn Rischel Intervals properties Timed Automata, Real-time Logic, Metric Temporal Logic, Explicit Clock Temporal,..., Alur, Dill, Jahanian, Mok, Koymans, Harel, Lichtenstein, Pnueli,... Duration of states Duration Calculus Zhou Hoare Ravn 91 an Interval Temporal Logic Halpern Moszkowski Manna Logical Calculi, Applications, Mechanical Support Duration Calculus: A formal approach to real-time systems Zhou Chaochen and Michael R. Hansen Springer Declarative Modelling, December, 2007, Michael R. Hansen p. 3/22

8 Gas Burner example: Requirements State variables modelling Gas and Flame: G, F : Time {0, 1} State expression modelling that gas is Leaking L = G F Declarative Modelling, December, 2007, Michael R. Hansen p. 4/22

9 Gas Burner example: Requirements State variables modelling Gas and Flame: G, F : Time {0, 1} State expression modelling that gas is Leaking L = G F Requirement Gas must at most be leaking 1/20 of the elapsed time (e b) 60 s 20 e L(t)dt (e b) b Declarative Modelling, December, 2007, Michael R. Hansen p. 4/22

10 Gas Burner example: Design decisions Leaks are detectable and stoppable within 1s: where c,d : b c < d e.(l[c,d] (d c) 1 s) P[c,d] = d P(t) = (d c) > 0 c which reads P holds throughout [c, d] Declarative Modelling, December, 2007, Michael R. Hansen p. 5/22

11 Gas Burner example: Design decisions Leaks are detectable and stoppable within 1s: where c,d : b c < d e.(l[c,d] (d c) 1 s) P[c,d] = d P(t) = (d c) > 0 c which reads P holds throughout [c, d] At least 30s between leaks: c,d,r,s : b c < r < s < d e. (L[c,r] L[r,s] L[s,d]) (s r) 30 s Declarative Modelling, December, 2007, Michael R. Hansen p. 5/22

12 Interval Logic [Halpern Moszkowski Manna 83] Terms: θ ::= x v θ 1 + θ n... Temporal Variable Declarative Modelling, December, 2007, Michael R. Hansen p. 6/22

13 Interval Logic [Halpern Moszkowski Manna 83] Terms: θ ::= x v θ 1 + θ n... Temporal Variable v : Intv R Formulas: φ ::= θ 1 = θ n φ φ ψ φ ψ ( x)φ... chop φ : Intv {tt,ff} Declarative Modelling, December, 2007, Michael R. Hansen p. 6/22

14 Interval Logic [Halpern Moszkowski Manna 83] Terms: θ ::= x v θ 1 + θ n... Temporal Variable v : Intv R Formulas: φ ::= θ 1 = θ n φ φ ψ φ ψ ( x)φ... chop φ : Intv {tt,ff} Chop: φ ψ {}}{ b m e } {{ }} {{ } φ ψ for some m : b m e Declarative Modelling, December, 2007, Michael R. Hansen p. 6/22

15 Interval Logic [Halpern Moszkowski Manna 83] Terms: θ ::= x v θ 1 + θ n... Temporal Variable v : Intv R Formulas: φ ::= θ 1 = θ n φ φ ψ φ ψ ( x)φ... chop Chop: φ ψ {}}{ b m e } {{ }} {{ } φ ψ φ : Intv {tt,ff} for some m : b m e In DC: Intv = { [a,b] a,b R a b} Declarative Modelling, December, 2007, Michael R. Hansen p. 6/22

16 Duration Calculus [Zhou Hoare Ravn 91] State variables P : Time {0, 1} Finite Variability State expressions S ::= 0 1 P S S 1 S 2 S : Time {0, 1} pointwise defined Declarative Modelling, December, 2007, Michael R. Hansen p. 7/22

17 Duration Calculus [Zhou Hoare Ravn 91] State variables P : Time {0, 1} Finite Variability State expressions S ::= 0 1 P S S 1 S 2 S : Time {0, 1} pointwise defined Durations S : Intv R defined on [b,e] by e b S(t)dt Temporal variables with a structure Declarative Modelling, December, 2007, Michael R. Hansen p. 7/22

18 Example: Gas Burner Requirement l L l Design decisions D 1 = ( L l 1) D 2 = (( L L L ) l 30) where l denotes the length of the interval, and φ = true φ true for some sub-interval: φ φ = φ for all sub-intervals: φ P = P = l l > 0 P holds throughout a non-point interval Declarative Modelling, December, 2007, Michael R. Hansen p. 8/22

19 Example: Gas Burner Requirement l L l Design decisions D 1 = ( L l 1) D 2 = (( L L L ) l 30) where l denotes the length of the interval, and φ = true φ true for some sub-interval: φ φ = φ for all sub-intervals: φ P = P = l l > 0 P holds throughout a non-point interval succinct formulation no interval endpoints Declarative Modelling, December, 2007, Michael R. Hansen p. 8/22

20 Decidability Declarative Modelling, December, 2007, Michael R. Hansen p. 9/22

21 Decidability What can t the computer do for me? Declarative Modelling, December, 2007, Michael R. Hansen p. 9/22

22 Overview [Zhou Hansen Sestoft 93] Restricted Duration Calculus : S φ, φ ψ, φ ψ Satisfiability is reduced to emptiness of regular languages Hence decidable for both discrete and continuous time Declarative Modelling, December, 2007, Michael R. Hansen p. 10/22

23 Overview [Zhou Hansen Sestoft 93] Restricted Duration Calculus : S φ, φ ψ, φ ψ Satisfiability is reduced to emptiness of regular languages Hence decidable for both discrete and continuous time Even small extensions give undecidable subsets RDC 1 (Cont. time) RDC 2 RDC 3 l = r, S φ,φ ψ, φ ψ S 1 = S 2 φ,φ ψ, φ ψ l = x, S φ,φ ψ, φ ψ ( x)φ Declarative Modelling, December, 2007, Michael R. Hansen p. 10/22

24 Overview [Zhou Hansen Sestoft 93] Restricted Duration Calculus : S φ, φ ψ, φ ψ Satisfiability is reduced to emptiness of regular languages Hence decidable for both discrete and continuous time Even small extensions give undecidable subsets RDC 1 (Cont. time) RDC 2 RDC 3 l = r, S φ,φ ψ, φ ψ S 1 = S 2 φ,φ ψ, φ ψ l = x, S φ,φ ψ, φ ψ ( x)φ How would you show such results? Declarative Modelling, December, 2007, Michael R. Hansen p. 10/22

25 Decidability of RDC for Discrete Time Satisfiability is reduced to emptiness of regular languages Declarative Modelling, December, 2007, Michael R. Hansen p. 11/22

26 Decidability of RDC for Discrete Time Satisfiability is reduced to emptiness of regular languages Idea: a Σ describes a piece of an interpretation, e.g. P 1 P 2 P Declarative Modelling, December, 2007, Michael R. Hansen p. 11/22

27 Decidability of RDC for Discrete Time Satisfiability is reduced to emptiness of regular languages Idea: a Σ describes a piece of an interpretation, e.g. P 1 P 2 P 3 Discrete time one letter corresponds to one time unit L( S ) = (DNF(S)) + L(ϕ ψ) = L(ϕ) L(ψ) L( ϕ) = Σ \ L(ϕ) L(ϕ ψ) = L(ϕ) L(ψ) Declarative Modelling, December, 2007, Michael R. Hansen p. 11/22

28 Decidability of RDC for Discrete Time Satisfiability is reduced to emptiness of regular languages Idea: a Σ describes a piece of an interpretation, e.g. P 1 P 2 P 3 Discrete time one letter corresponds to one time unit L( S ) = (DNF(S)) + L(ϕ ψ) = L(ϕ) L(ψ) L( ϕ) = Σ \ L(ϕ) L(ϕ ψ) = L(ϕ) L(ψ) L(φ) is regular φ is satisfiable iff L(φ) Satisfiability problem for RDC is decidable non-elementary Declarative Modelling, December, 2007, Michael R. Hansen p. 11/22

29 Example Is the formula ( P P ) P valid for discrete time? Declarative Modelling, December, 2007, Michael R. Hansen p. 12/22

30 Example Is the formula ( P P ) P valid for discrete time? Σ = {{P }, {}} Declarative Modelling, December, 2007, Michael R. Hansen p. 12/22

31 Example Is the formula ( P P ) P valid for discrete time? Σ = {{P }, {}}. We have ( P P ) P is valid iff iff (( P P ) P ) is not satisfiable ( P P ) P is not satisfiable iff L 1 ( P P ) L 1 ( P ) = {} iff {{P } i i 2} (Σ \ {{P } i i 1}) = {} The last equality holds Declarative Modelling, December, 2007, Michael R. Hansen p. 12/22

32 Example Is the formula ( P P ) P valid for discrete time? Σ = {{P }, {}}. We have ( P P ) P is valid iff iff (( P P ) P ) is not satisfiable ( P P ) P is not satisfiable iff L 1 ( P P ) L 1 ( P ) = {} iff {{P } i i 2} (Σ \ {{P } i i 1}) = {} The last equality holds. Therefore, the formula is valid for discrete time Declarative Modelling, December, 2007, Michael R. Hansen p. 12/22

33 Hybrid Duration Calculus Bolander Hansen Hansen Improved expressivity at the same price Declarative Modelling, December, 2007, Michael R. Hansen p. 13/22

34 Hybrid DC Hybrid DC is Restricted Duration Calculus extended by: Nominals a names a specific interval Declarative Modelling, December, 2007, Michael R. Hansen p. 14/22

35 Hybrid DC Hybrid DC is Restricted Duration Calculus extended by: Nominals a names a specific interval G(a) = [t a,u a ] Satisfaction operator a : φ φ holds at a downarrow binder a.φ holds if φ holds under the assumption that a names the current interval. global modality Eφ holds if there is some interval where φ holds. I,G, [t,u] = a iff G(a) = [t,u] I,G, [t,u] = a : φ iff I,G,G(a) = φ I,G, [t,u] = Eφ iff for some interval [v,w]: I,G, [v,w] = φ I,G, [t,u] = a.φ iff I,G[a := [t,u] ], [t,u] = φ Declarative Modelling, December, 2007, Michael R. Hansen p. 14/22

36 Expressibility: Neighbourhood RDC Propositional neighbourhood logic: ZhouHansen98,BaruaRoyZhou00 I, [t,u] = l φ iff I, [s,t] = φ for some s t I, [t,u] = r φ iff I, [u,v] = φ for some v u r φ {}}{{}}{ t u v φ {}}{}}{ φ l φ s t u for some v u for some s t Declarative Modelling, December, 2007, Michael R. Hansen p. 15/22

37 Expressibility: Neighbourhood RDC Propositional neighbourhood logic: ZhouHansen98,BaruaRoyZhou00 I, [t,u] = l φ iff I, [s,t] = φ for some s t I, [t,u] = r φ iff I, [u,v] = φ for some v u r φ {}}{{}}{ t u v φ {}}{}}{ φ l φ s t u can be embedded in Hybrid DC: τ( l φ) = a.e(φ a) τ( r φ) = a.e(a φ) for some v u for some s t Declarative Modelling, December, 2007, Michael R. Hansen p. 15/22

38 Expressibility: Allen s binary relations All 13 (Allen) relations between two intervals are expressible. E.g. a precedes b a meets b a overlaps b a finished by b a contains b a starts b a a a a a a {}}{{}}{{}}{{}}{{}}{{}}{ }{{} b }{{} b }{{} b }{{} b }{{} b }{{} b a precedes b a : r ( π r b) a meets b a : r b a overlaps b E( c. π a : ( π c) b : (c π)) a finished by b a : ( π b) a contains b a : ( π b π) a starts b b : (a π) Declarative Modelling, December, 2007, Michael R. Hansen p. 16/22

39 Monadic second-order theory of order L < 2 We reduce satisfiability of Hybrid Duration Calculus to satisfiability of L < 2. (Discrete as well as continuous time.) Declarative Modelling, December, 2007, Michael R. Hansen p. 17/22

40 Monadic second-order theory of order L < 2 We reduce satisfiability of Hybrid Duration Calculus to satisfiability of L < 2. (Discrete as well as continuous time.) The formulas of L < 2 are constructed from: First-order variables ranged over by x,y,z,... Second-order variables ranged over by P,Q,X, Declarative Modelling, December, 2007, Michael R. Hansen p. 17/22

41 Monadic second-order theory of order L < 2 We reduce satisfiability of Hybrid Duration Calculus to satisfiability of L < 2. (Discrete as well as continuous time.) The formulas of L < 2 are constructed from: First-order variables ranged over by x,y,z,... Second-order variables ranged over by P,Q,X,... The formulas are generated from the following grammar: φ ::= x < y x P φ ψ φ xφ Pφ Declarative Modelling, December, 2007, Michael R. Hansen p. 17/22

42 Semantics of L < 2 A structure (A,B,<) consists of a set A partially ordered by < and a set B of Boolean-valued functions from A. An element b B can be considered a, possibly infinite, subset of A. An interpretation I associates a member P I of B to every second-order variable P. A valuation ν is a function assigning a member ν(x) of A to every first-order variable x. The semantic relation I,ν = φ is then defined by: I,ν = x < y iff ν(x) < ν(y) I,ν = x P iff ν(x) P I I,ν = φ iff I,ν = φ I,ν = φ ψ iff I,ν = φ or I,ν = ψ I,ν = xφ iff for some a A: I,ν[x := a] = φ I,ν = Pφ iff for some b B: I[P := b],ν = φ Declarative Modelling, December, 2007, Michael R. Hansen p. 18/22

43 Decidability results for L < 2 Let ω = (N, 2 N,<). L < 2 (ω) is decidable Büchi Declarative Modelling, December, 2007, Michael R. Hansen p. 19/22

44 From Hybrid DC to L < 2 (ω) discrete time each state variable P corresponds to a second-order variable denoted by P. Idea: i P iff P(t) = 1 in the interval ]i,i + 1[. each nominal a is associated with two variables x a and y a Declarative Modelling, December, 2007, Michael R. Hansen p. 20/22

45 From Hybrid DC to L < 2 (ω) discrete time each state variable P corresponds to a second-order variable denoted by P. Idea: i P iff P(t) = 1 in the interval ]i,i + 1[. each nominal a is associated with two variables x a and y a. T x,y (π) = x = y T x,y (P) = x < y z(x z < y z P) T x,y ( φ) = T x,y (φ) T x,y (φ ψ) = T x,y (φ) T x,y (ψ) T x,y (φ ψ) = z(t x,z (φ) T z,y (φ) x z z y) T x,y (a) = x = x a y = y a T x,y (a : φ) = T xa,y a (φ) T x,y (Eφ) = x y(x y T x,y (φ)) T x,y ( a.φ) = x a y a (x = x a y = y a T x,y (φ)) Declarative Modelling, December, 2007, Michael R. Hansen p. 20/22

46 Correctness of translation Discrete time: φ is satisfiable in discrete-time Hybrid DC iff T x,y (φ) x y a in φ x a y a is satisfiable in L < 2 (ω). The decision problem is non-elementary Declarative Modelling, December, 2007, Michael R. Hansen p. 21/22

47 Current focus Model checking and deciding an interval logic with durations, aiming at verification of durational properties like: Per day, the telephone network is down for at most 15 seconds Down 15 Total delay of message delivery across the network is less than 235ms Σ i delay(mi ) 235 The lifetime of a system with two processors is at least k: c 1 (A1 A 2 ) + c 2 (A1 A 2 ) + c 3 ( A1 A 2 ) e l k + c 4 ( A1 A 2 ) Involves theory, applications and implementation Declarative Modelling, December, 2007, Michael R. Hansen p. 22/22

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