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Robust Satisfaction of Signal Temporal Logics andApplications
Alexandre Donzé
Verimag, Grenoble
September 26th, 2011
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Overview
Design and analysis of hybrid systemse.g., embedded systems, mixed-signal circuits, biological systems
Simulation-based approaches for verification and parameter synthesisLightweight verification, as opposed to full-fledged Model-Checking
Hybrid System
x = fq(x, p) ||
Param. p, x0
Pok Pbad
q0
q1
q2
q0 → q1 → · · ·Simulation
(x, q) |= ϕ ?
Monitoring
ok
¬ ok
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Overview
Design and analysis of hybrid systemse.g., embedded systems, mixed-signal circuits, biological systems
Simulation-based approaches for verification and parameter synthesisLightweight verification, as opposed to full-fledged Model-Checking
Hybrid System
x = fq(x, p) ||
Param. p, x0
Pok Pbad
q0
q1
q2
q0 → q1 → · · ·Simulation
(x, q) |= ϕ ?
Monitoring
ok
¬ ok
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Overview
Design and analysis of hybrid systemse.g., embedded systems, mixed-signal circuits, biological systems
Simulation-based approaches for verification and parameter synthesisLightweight verification, as opposed to full-fledged Model-Checking
Hybrid System
x = fq(x, p) ||
Param. p, x0
Pok Pbad
q0
q1
q2
q0 → q1 → · · ·
x(t, p)
Simulation
(x, q) |= ϕ ?
Monitoring
ok
¬ ok
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Overview
Design and analysis of hybrid systemse.g., embedded systems, mixed-signal circuits, biological systems
Simulation-based approaches for verification and parameter synthesisLightweight verification, as opposed to full-fledged Model-Checking
Hybrid System
x = fq(x, p) ||
Param. p, x0
Pok Pbad
q0
q1
q2
q0 → q1 → · · ·
x(t, p)
Simulation
(x, q) |= ϕ ?
Monitoring
ok
¬ ok
Robust Satisfaction of STL Introduction DREAM Seminar 2 / 39
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Overview
Design and analysis of hybrid systemse.g., embedded systems, mixed-signal circuits, biological systems
Simulation-based approaches for verification and parameter synthesisLightweight verification, as opposed to full-fledged Model-Checking
Hybrid System
x = fq(x, p) ||
Param. p, x0
Pok Pbadq0
q1
q2
q0 → q1 → · · ·
x(t, p)
Simulation
(x, q) |= ϕ ?
Monitoring
ok
¬ ok
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Overview
I Signal Temporal Logic (STL): temporal specifications for continuousand hybrid systems
I Quantitative (Robust) satisfaction of STL adapted to deal withuncertainty
Hybrid System
x = fq(x, p) ||
Param. p, x0q0
q1
q2
q0 → q1 → · · ·
SimulationProperty ϕ ≡
alw[q0 → ev[0,1]
q2 U[0,.2] (x≥ .5)]
ok
¬ ok
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Overview
I Signal Temporal Logic (STL): temporal specifications for continuousand hybrid systems
I Quantitative (Robust) satisfaction of STL adapted to deal withuncertainty
Hybrid System
x = fq(x, p) ||
Param. p, x0q0
q1
q2
q0 → q1 → · · ·
x(t, p)
SimulationProperty ϕ ≡
alw[q0 → ev[0,1]
q2 U[0,.2] (x≥ .5)]
STL monitoring
ok
¬ ok
ok
¬ ok
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Overview
I Signal Temporal Logic (STL): temporal specifications for continuousand hybrid systems
I Quantitative (Robust) satisfaction of STL adapted to deal withuncertainty
Hybrid System
x = fq(x, p) ||
Param. p, x0q0
q1
q2
q0 → q1 → · · ·
x(t, p)± ε
SimulationProperty ϕ ≡
alw[q0 → ev[0,1]
q2 U[0,.2] (x≥ .5)]
STL monitoringok
¬ ok
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Overview
I Signal Temporal Logic (STL): temporal specifications for continuousand hybrid systems
I Quantitative (Robust) satisfaction of STL adapted to deal withuncertainty
Hybrid System
x = fq(x, p) ||
Param. p, x0q0
q1
q2
q0 → q1 → · · ·
x(t, p)± ε
SimulationProperty ϕ ≡
alw[q0 → ev[0,1]
q2 U[0,.2] (x≥ .5)]
STL monitoringok
¬ ok
ε
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Outline
1 Temporal Logics for Continuous Time and SpaceSignal Temporal LogicQuantitative Satisfaction of STL
2 An Implementation: The Breach ToolboxSimulation of Parametric Hybrid SystemsSpecifying STL Formulas
3 ApplicationsCase Study: Voltage Controlled OscillatorAn Example from Systems Biology
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Outline
1 Temporal Logics for Continuous Time and SpaceSignal Temporal LogicQuantitative Satisfaction of STL
2 An Implementation: The Breach ToolboxSimulation of Parametric Hybrid SystemsSpecifying STL Formulas
3 ApplicationsCase Study: Voltage Controlled OscillatorAn Example from Systems Biology
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Temporal logics in a nutshell
Temporal logics allow to specify patterns that timed behaviors of systems may ormay not satisfy. They come in many flavors.
The most intuitive is the Linear Temporal Logic (LTL), dealing with discretesequences of states.
Based on logic operators (¬, ∧, ∨) and temporal operators: “next”, “always”(alw), “eventually” (ev) and “until” (U)
Examples:
I ϕ ϕ ϕ ϕ · · · satisfies alw ϕ
I ψ ψ ψ ϕ ψ · · · satisfies ev ϕI ϕ ϕ ϕ ϕ ψ · · · satisfies ϕ U ψ
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From Discrete to Continuous
Temporal logics mostly developed for discrete systemsWhy not discretizing time and space and reuse existing logics and tools ?
Some reasons:I Discretization often leads to state-explosion problemI Specifications should not depend on the discretization used (e.g.,
“next” depends on time step)
Thus we need:I Temporal specifications involving dense-time intervalsI Constraints applying on variable in the continuous domain
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From Discrete to Continuous
Temporal logics mostly developed for discrete systemsWhy not discretizing time and space and reuse existing logics and tools ?
Some reasons:I Discretization often leads to state-explosion problemI Specifications should not depend on the discretization used (e.g.,
“next” depends on time step)
Thus we need:I Temporal specifications involving dense-time intervalsI Constraints applying on variable in the continuous domain
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From Discrete to Continuous
Temporal logics mostly developed for discrete systemsWhy not discretizing time and space and reuse existing logics and tools ?
Some reasons:I Discretization often leads to state-explosion problemI Specifications should not depend on the discretization used (e.g.,
“next” depends on time step)
Thus we need:I Temporal specifications involving dense-time intervalsI Constraints applying on variable in the continuous domain
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Formal DefinitionsDefinition (STL Syntax)
ϕ := µ | ¬ϕ | ϕ ∧ ψ | ϕ U[a,b] ψ
where µ is a predicate of the form µ : µ(x) > 0
Definition (STL Semantics)The validity of a formula ϕ with respect to a signal x at time t is
(x, t) |= µ ⇔ µ(x[t]) > 0(x, t) |= ϕ ∧ ψ ⇔ (x, t) |= ϕ ∧ (x, t) |= ψ(x, t) |= ¬ϕ ⇔ ¬((x, t) |= ϕ)(x, t) |= ϕ U[a,b) ψ ⇔ ∃t ′ ∈ [t + a, t + b] s.t. (x, t ′) |= ψ ∧
∀t ′′ ∈ [t, t ′], (x, t ′′) |= ϕ}
Additionally: ev[a,b]ϕ = > U[a,b) ϕ and alw[a,b]ϕ = ϕ U[a,b) ⊥.
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Formal DefinitionsDefinition (STL Syntax)
ϕ := µ | ¬ϕ | ϕ ∧ ψ | ϕ U[a,b] ψ
where µ is a predicate of the form µ : µ(x) > 0
Definition (STL Semantics)The validity of a formula ϕ with respect to a signal x at time t is
(x, t) |= µ ⇔ µ(x[t]) > 0(x, t) |= ϕ ∧ ψ ⇔ (x, t) |= ϕ ∧ (x, t) |= ψ(x, t) |= ¬ϕ ⇔ ¬((x, t) |= ϕ)(x, t) |= ϕ U[a,b) ψ ⇔ ∃t ′ ∈ [t + a, t + b] s.t. (x, t ′) |= ψ ∧
∀t ′′ ∈ [t, t ′], (x, t ′′) |= ϕ}
Additionally: ev[a,b]ϕ = > U[a,b) ϕ and alw[a,b]ϕ = ϕ U[a,b) ⊥.
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ExamplesConsider a simple piecewise affine signal:
x
t
>
⊥1 2 3 4 5 6
1
2
3
4
5
Truth value of :I ϕ = x > 2
II ϕ = alw[0.5,1.5](x > 2)
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ExamplesConsider a simple piecewise affine signal:
x
t
>
⊥1 2 3 4 5 6
1
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Truth value of :I ϕ = x > 2
II ϕ = alw[0.5,1.5](x > 2)
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ExamplesConsider a simple piecewise affine signal:
x
t
>
⊥1 2 3 4 5 6
1
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Truth value of :I ϕ = x > 2I ϕ = ev[0,∞](x > 2)
I ϕ = alw[0.5,1.5](x > 2)
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ExamplesConsider a simple piecewise affine signal:
x
t
>
⊥1 2 3 4 5 6
1
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Truth value of :I ϕ = x > 2I ϕ = ev[0,.5](x > 2)
I ϕ = alw[0.5,1.5](x > 2)
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ExamplesConsider a simple piecewise affine signal:
x
t
>
⊥1 2 3 4 5 6
1
2
3
4
5
Truth value of :I ϕ = x > 2
II ϕ = alw[0,∞](x > 2)
ϕ = alw[0.5,1.5](x > 2)
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ExamplesConsider a simple piecewise affine signal:
x
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⊥1 2 3 4 5 6
1
2
3
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5
Truth value of :I ϕ = x > 2
II ϕ = alw[0.5,1.5](x > 2)Robust Satisfaction of STL Temporal Logics for Continuous Time and Space DREAM Seminar 8 / 39
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Outline
1 Temporal Logics for Continuous Time and SpaceSignal Temporal LogicQuantitative Satisfaction of STL
2 An Implementation: The Breach ToolboxSimulation of Parametric Hybrid SystemsSpecifying STL Formulas
3 ApplicationsCase Study: Voltage Controlled OscillatorAn Example from Systems Biology
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From Semantics to Satisfaction FunctionsSTL semantics
(x, t) � µ ⇔ µ(x[t]) > 0(x, t) � ¬ϕ ⇔ (x, t) 2 ϕ(x, t) � ϕ1 ∧ ϕ2 ⇔ (x, t) � ϕ1 and (x, t) � ϕ2(x, t) � ϕ1U[a,b]ϕ2 ⇔ ∃t′ ∈ [t + a, t + b] s.t. (x, t′) � ϕ2
and ∀t′′ ∈ [t, t′], (x, t′′) � ϕ1
A Boolean Satisfaction Function χ
Map {false, true} to {−∞,∞} and define the function χ : (x, t)→ {−∞,∞}:
χ(µ, x, t) = sign(µ(x[t]))×∞χ(¬ϕ, x, t) = − χ(ϕ, x, t)χ(ϕ1 ∧ ϕ2, x, t) = min(χ(ϕ1, x, t), χ(ϕ2, x, t))χ(ϕ1U[a,b]ϕ2, x, t) = max
τ∈t+[a,b](min(χ(ϕ2, x, τ), min
s∈[t,τ ]χ(ϕ1, x, s))
We can verify that (x, t) |= ϕ⇔ χ(ϕ, x, t) = +∞
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From Semantics to Satisfaction FunctionsSTL semantics
(x, t) � µ ⇔ µ(x[t]) > 0(x, t) � ¬ϕ ⇔ (x, t) 2 ϕ(x, t) � ϕ1 ∧ ϕ2 ⇔ (x, t) � ϕ1 and (x, t) � ϕ2(x, t) � ϕ1U[a,b]ϕ2 ⇔ ∃t′ ∈ [t + a, t + b] s.t. (x, t′) � ϕ2
and ∀t′′ ∈ [t, t′], (x, t′′) � ϕ1
A Boolean Satisfaction Function χ
Map {false, true} to {−∞,∞} and define the function χ : (x, t)→ {−∞,∞}:
χ(µ, x, t) = sign(µ(x[t]))×∞χ(¬ϕ, x, t) = − χ(ϕ, x, t)χ(ϕ1 ∧ ϕ2, x, t) = min(χ(ϕ1, x, t), χ(ϕ2, x, t))χ(ϕ1U[a,b]ϕ2, x, t) = max
τ∈t+[a,b](min(χ(ϕ2, x, τ), min
s∈[t,τ ]χ(ϕ1, x, s))
We can verify that (x, t) |= ϕ⇔ χ(ϕ, x, t) = +∞
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From Semantics to Satisfaction FunctionsSTL semantics
(x, t) � µ ⇔ µ(x[t]) > 0(x, t) � ¬ϕ ⇔ (x, t) 2 ϕ(x, t) � ϕ1 ∧ ϕ2 ⇔ (x, t) � ϕ1 and (x, t) � ϕ2(x, t) � ϕ1U[a,b]ϕ2 ⇔ ∃t′ ∈ [t + a, t + b] s.t. (x, t′) � ϕ2
and ∀t′′ ∈ [t, t′], (x, t′′) � ϕ1
A Boolean Satisfaction Function χ
Map {false, true} to {−∞,∞} and define the function χ : (x, t)→ {−∞,∞}:
χ(µ, x, t) = sign(µ(x[t]))×∞χ(¬ϕ, x, t) = − χ(ϕ, x, t)χ(ϕ1 ∧ ϕ2, x, t) = min(χ(ϕ1, x, t), χ(ϕ2, x, t))χ(ϕ1U[a,b]ϕ2, x, t) = max
τ∈t+[a,b](min(χ(ϕ2, x, τ), min
s∈[t,τ ]χ(ϕ1, x, s))
We can verify that (x, t) |= ϕ⇔ χ(ϕ, x, t) = +∞
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From Semantics to Satisfaction FunctionsSTL semantics
(x, t) � µ ⇔ µ(x[t]) > 0(x, t) � ¬ϕ ⇔ (x, t) 2 ϕ(x, t) � ϕ1 ∧ ϕ2 ⇔ (x, t) � ϕ1 and (x, t) � ϕ2(x, t) � ϕ1U[a,b]ϕ2 ⇔ ∃t′ ∈ [t + a, t + b] s.t. (x, t′) � ϕ2
and ∀t′′ ∈ [t, t′], (x, t′′) � ϕ1
A Boolean Satisfaction Function χ
Map {false, true} to {−∞,∞} and define the function χ : (x, t)→ {−∞,∞}:
χ(µ, x, t) = sign(µ(x[t]))×∞χ(¬ϕ, x, t) = − χ(ϕ, x, t)χ(ϕ1 ∧ ϕ2, x, t) = min(χ(ϕ1, x, t), χ(ϕ2, x, t))χ(ϕ1U[a,b]ϕ2, x, t) = max
τ∈t+[a,b](min(χ(ϕ2, x, τ), min
s∈[t,τ ]χ(ϕ1, x, s))
We can verify that (x, t) |= ϕ⇔ χ(ϕ, x, t) = +∞
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From Semantics to Satisfaction FunctionsSTL semantics
(x, t) � µ ⇔ µ(x[t]) > 0(x, t) � ¬ϕ ⇔ (x, t) 2 ϕ(x, t) � ϕ1 ∧ ϕ2 ⇔ (x, t) � ϕ1 and (x, t) � ϕ2(x, t) � ϕ1U[a,b]ϕ2 ⇔ ∃t′ ∈ [t + a, t + b] s.t. (x, t′) � ϕ2
and ∀t′′ ∈ [t, t′], (x, t′′) � ϕ1
A Boolean Satisfaction Function χ
Map {false, true} to {−∞,∞} and define the function χ : (x, t)→ {−∞,∞}:
χ(µ, x, t) = sign(µ(x[t]))×∞χ(¬ϕ, x, t) = − χ(ϕ, x, t)χ(ϕ1 ∧ ϕ2, x, t) = min(χ(ϕ1, x, t), χ(ϕ2, x, t))χ(ϕ1U[a,b]ϕ2, x, t) = max
τ∈t+[a,b](min(χ(ϕ2, x, τ), min
s∈[t,τ ]χ(ϕ1, x, s))
We can verify that (x, t) |= ϕ⇔ χ(ϕ, x, t) = +∞
Robust Satisfaction of STL Temporal Logics for Continuous Time and Space DREAM Seminar 10 / 39
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From Semantics to Satisfaction FunctionsSTL semantics
(x, t) � µ ⇔ µ(x[t]) > 0(x, t) � ¬ϕ ⇔ (x, t) 2 ϕ(x, t) � ϕ1 ∧ ϕ2 ⇔ (x, t) � ϕ1 and (x, t) � ϕ2(x, t) � ϕ1U[a,b]ϕ2 ⇔ ∃t′ ∈ [t + a, t + b] s.t. (x, t′) � ϕ2
and ∀t′′ ∈ [t, t′], (x, t′′) � ϕ1
A Boolean Satisfaction Function χ
Map {false, true} to {−∞,∞} and define the function χ : (x, t)→ {−∞,∞}:
χ(µ, x, t) = sign(µ(x[t]))×∞χ(¬ϕ, x, t) = − χ(ϕ, x, t)χ(ϕ1 ∧ ϕ2, x, t) = min(χ(ϕ1, x, t), χ(ϕ2, x, t))χ(ϕ1U[a,b]ϕ2, x, t) = max
τ∈t+[a,b](min(χ(ϕ2, x, τ), min
s∈[t,τ ]χ(ϕ1, x, s))
We can verify that (x, t) |= ϕ⇔ χ(ϕ, x, t) = +∞
Robust Satisfaction of STL Temporal Logics for Continuous Time and Space DREAM Seminar 10 / 39
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From Semantics to Satisfaction FunctionsSTL semantics
(x, t) � µ ⇔ µ(x[t]) > 0(x, t) � ¬ϕ ⇔ (x, t) 2 ϕ(x, t) � ϕ1 ∧ ϕ2 ⇔ (x, t) � ϕ1 and (x, t) � ϕ2(x, t) � ϕ1U[a,b]ϕ2 ⇔ ∃t′ ∈ [t + a, t + b] s.t. (x, t′) � ϕ2
and ∀t′′ ∈ [t, t′], (x, t′′) � ϕ1
A Boolean Satisfaction Function χ
Map {false, true} to {−∞,∞} and define the function χ : (x, t)→ {−∞,∞}:
χ(µ, x, t) = sign(µ(x[t]))×∞χ(¬ϕ, x, t) = − χ(ϕ, x, t)χ(ϕ1 ∧ ϕ2, x, t) = min(χ(ϕ1, x, t), χ(ϕ2, x, t))χ(ϕ1U[a,b]ϕ2, x, t) = max
τ∈t+[a,b](min(χ(ϕ2, x, τ), min
s∈[t,τ ]χ(ϕ1, x, s))
We can verify that (x, t) |= ϕ⇔ χ(ϕ, x, t) = +∞
Robust Satisfaction of STL Temporal Logics for Continuous Time and Space DREAM Seminar 10 / 39
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From Boolean to Quantitative Satisfaction FunctionFor atomic predicates:
χ(µ, x, t) = sign(µ(x[t]))×∞
The sign removes the quantitative information in µ to get a boolean signal
Simple ideaI Get rid of sign to get a quantitative satisfaction function ρI Keep the same inductive rules for the quantitative semantics:
ρ(µ, x, t) = µ(x[t])ρ(¬ϕ, x, t) = −ρ(ϕ, x, t)ρ(ϕ1 ∧ ϕ2, x, t) = min(ρ(ϕ1, x, t), ρ(ϕ2, x, t))ρ(ϕ1U[a,b]ϕ2, x, t) = max
τ∈t+[a,b](min(ρ(ϕ2, x, τ), min
s∈[t,τ ]ρ(ϕ1, x, s))
Robust Satisfaction of STL Temporal Logics for Continuous Time and Space DREAM Seminar 11 / 39
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From Boolean to Quantitative Satisfaction FunctionFor atomic predicates:
χ(µ, x, t) = sign(µ(x[t]))×∞
The sign removes the quantitative information in µ to get a boolean signal
Simple ideaI Get rid of sign to get a quantitative satisfaction function ρI Keep the same inductive rules for the quantitative semantics:
ρ(µ, x, t) = µ(x[t])ρ(¬ϕ, x, t) = −ρ(ϕ, x, t)ρ(ϕ1 ∧ ϕ2, x, t) = min(ρ(ϕ1, x, t), ρ(ϕ2, x, t))ρ(ϕ1U[a,b]ϕ2, x, t) = max
τ∈t+[a,b](min(ρ(ϕ2, x, τ), min
s∈[t,τ ]ρ(ϕ1, x, s))
Robust Satisfaction of STL Temporal Logics for Continuous Time and Space DREAM Seminar 11 / 39
ρ
![Page 35: mm RobustSatisfactionofSignalTemporalLogicsand …Robust Satisfaction of STL DREAM Seminar 1 / 39. 40 60 80 100 120 40 60 80 mm Overview](https://reader035.vdocuments.mx/reader035/viewer/2022071104/5fdd84bd74980c2298565ee8/html5/thumbnails/35.jpg)
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From Boolean to Quantitative Satisfaction FunctionFor atomic predicates:
χ(µ, x, t) = sign(µ(x[t]))×∞
The sign removes the quantitative information in µ to get a boolean signal
Simple ideaI Get rid of sign to get a quantitative satisfaction function ρI Keep the same inductive rules for the quantitative semantics:
ρ(µ, x, t) = µ(x[t])ρ(¬ϕ, x, t) = −ρ(ϕ, x, t)ρ(ϕ1 ∧ ϕ2, x, t) = min(ρ(ϕ1, x, t), ρ(ϕ2, x, t))ρ(ϕ1U[a,b]ϕ2, x, t) = max
τ∈t+[a,b](min(ρ(ϕ2, x, τ), min
s∈[t,τ ]ρ(ϕ1, x, s))
Robust Satisfaction of STL Temporal Logics for Continuous Time and Space DREAM Seminar 11 / 39
ρ
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Robust Satisfaction, Examples
Robust Satisfaction of STL Temporal Logics for Continuous Time and Space DREAM Seminar 12 / 39
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Robust Satisfaction, Examples
Robust Satisfaction of STL Temporal Logics for Continuous Time and Space DREAM Seminar 12 / 39
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Robust Satisfaction, Examples
Robust Satisfaction of STL Temporal Logics for Continuous Time and Space DREAM Seminar 12 / 39
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Robust Satisfaction, Examples
Robust Satisfaction of STL Temporal Logics for Continuous Time and Space DREAM Seminar 12 / 39
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Robust Satisfaction, Examples
Robust Satisfaction of STL Temporal Logics for Continuous Time and Space DREAM Seminar 12 / 39
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Robust Satisfaction, Examples
Robust Satisfaction of STL Temporal Logics for Continuous Time and Space DREAM Seminar 12 / 39
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Robust Satisfaction, ApplicationsAssume that x depends on p, we get the following oracle:
Param. p ∈ P
OracleModel +STL Monitor
STL Prop. ϕRobust Sat. ρ(ϕ, p)
Parameter synthesis can be solved by solving
p∗ = max {ρ(ϕ, p) | p ∈ P}
If ρ(ϕ, p∗) > 0 then parameter p∗ is such that (x, p∗) |= ϕ. Moreover, itmaximizes the robustness of satisfaction.
More generally, one can characterize the validity domain of ϕ, given byd(ϕ,P) = {p ∈ P | ρ(ϕ, p) > 0}
Robust Satisfaction of STL Temporal Logics for Continuous Time and Space DREAM Seminar 13 / 39
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Robust Satisfaction, ApplicationsAssume that x depends on p, we get the following oracle:
Param. p ∈ P
OracleModel +STL Monitor
STL Prop. ϕRobust Sat. ρ(ϕ, p)
Parameter synthesis can be solved by solving
p∗ = max {ρ(ϕ, p) | p ∈ P}
If ρ(ϕ, p∗) > 0 then parameter p∗ is such that (x, p∗) |= ϕ. Moreover, itmaximizes the robustness of satisfaction.
More generally, one can characterize the validity domain of ϕ, given byd(ϕ,P) = {p ∈ P | ρ(ϕ, p) > 0}
Robust Satisfaction of STL Temporal Logics for Continuous Time and Space DREAM Seminar 13 / 39
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Robust Satisfaction, ApplicationsAssume that x depends on p, we get the following oracle:
Param. p ∈ P
OracleModel +STL Monitor
STL Prop. ϕRobust Sat. ρ(ϕ, p)
Parameter synthesis can be solved by solving
p∗ = max {ρ(ϕ, p) | p ∈ P}
If ρ(ϕ, p∗) > 0 then parameter p∗ is such that (x, p∗) |= ϕ. Moreover, itmaximizes the robustness of satisfaction.
More generally, one can characterize the validity domain of ϕ, given byd(ϕ,P) = {p ∈ P | ρ(ϕ, p) > 0}
Robust Satisfaction of STL Temporal Logics for Continuous Time and Space DREAM Seminar 13 / 39
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Outline
1 Temporal Logics for Continuous Time and SpaceSignal Temporal LogicQuantitative Satisfaction of STL
2 An Implementation: The Breach ToolboxSimulation of Parametric Hybrid SystemsSpecifying STL Formulas
3 ApplicationsCase Study: Voltage Controlled OscillatorAn Example from Systems Biology
Robust Satisfaction of STL An Implementation: The Breach Toolbox DREAM Seminar 14 / 39
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Outline
1 Temporal Logics for Continuous Time and SpaceSignal Temporal LogicQuantitative Satisfaction of STL
2 An Implementation: The Breach ToolboxSimulation of Parametric Hybrid SystemsSpecifying STL Formulas
3 ApplicationsCase Study: Voltage Controlled OscillatorAn Example from Systems Biology
Robust Satisfaction of STL An Implementation: The Breach Toolbox DREAM Seminar 14 / 39
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Hybrid Model
Breach deals with piecewise-continuous models of the form x = f (q,x,p), x(0) = x0y = g(x)
q+ = e(q−,y), q(0) = q0
where x ∈ Rn is the state variable
q ∈ N is the discrete state,
p ∈ Rnp is the parameter vector,
g is the guard function and
e is the event or transition function, where q+ 6= q− only if g(x) = 0
Robust Satisfaction of STL An Implementation: The Breach Toolbox DREAM Seminar 15 / 39
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Simulation Algorithm
Discontinuity locking + Event detection by zero crossing detection
1. Let fk(x,p) = f (q(tk),x,p) (block switching between tk and tk+1)
2. Solve ODE x = fk(x,p) on [tk , tk + hk ]
3. If for all i, sign(gi(x)) = Constant on (tk , tk + hk ] then let tk+1 = tk + hk
4. Else find the minimum time τ > tk for which gi(x(τ)) = 0 and let tk+1 = τ
5. Return ξp(tk+1) and restart with q(t+k+1) = e(q(tk), λ(t−k+1)))
Robust Satisfaction of STL An Implementation: The Breach Toolbox DREAM Seminar 16 / 39
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Simulation and Sensitivity Analysis
Simulation based on a state-of-the-art ODE solver CVodesI Variable-steps variable order implicit methods, efficient for stiff and non-stiff
dynamicsI Builtin zero-crossing detection for guards.
Sensitivity functions sij(t) = ∂xi∂pj
(t) are also computed by CVodes solver
Breach implementation addsI the computation of sensitivity discontinuities at transitionsI an efficient Matlab-C interface:
I The solver and the dynamics are in CI Matlab manipulates arrays of parameters and externally computed
arrays of trajectories⇒ Much more efficient than Matlab native ODE solvers
Robust Satisfaction of STL An Implementation: The Breach Toolbox DREAM Seminar 17 / 39
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Simulation and Sensitivity Analysis
Simulation based on a state-of-the-art ODE solver CVodesI Variable-steps variable order implicit methods, efficient for stiff and non-stiff
dynamicsI Builtin zero-crossing detection for guards.
Sensitivity functions sij(t) = ∂xi∂pj
(t) are also computed by CVodes solver
Breach implementation addsI the computation of sensitivity discontinuities at transitionsI an efficient Matlab-C interface:
I The solver and the dynamics are in CI Matlab manipulates arrays of parameters and externally computed
arrays of trajectories⇒ Much more efficient than Matlab native ODE solvers
Robust Satisfaction of STL An Implementation: The Breach Toolbox DREAM Seminar 17 / 39
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Simulation and Sensitivity Analysis
Simulation based on a state-of-the-art ODE solver CVodesI Variable-steps variable order implicit methods, efficient for stiff and non-stiff
dynamicsI Builtin zero-crossing detection for guards.
Sensitivity functions sij(t) = ∂xi∂pj
(t) are also computed by CVodes solver
Breach implementation addsI the computation of sensitivity discontinuities at transitionsI an efficient Matlab-C interface:
I The solver and the dynamics are in CI Matlab manipulates arrays of parameters and externally computed
arrays of trajectories⇒ Much more efficient than Matlab native ODE solvers
Robust Satisfaction of STL An Implementation: The Breach Toolbox DREAM Seminar 17 / 39
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Breach GUIs for trajectories exploration
Robust Satisfaction of STL An Implementation: The Breach Toolbox DREAM Seminar 18 / 39
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Breach GUIs for trajectories exploration
Robust Satisfaction of STL An Implementation: The Breach Toolbox DREAM Seminar 18 / 39
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Breach GUIs for trajectories exploration
Robust Satisfaction of STL An Implementation: The Breach Toolbox DREAM Seminar 18 / 39
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Breach GUIs for trajectories exploration
Robust Satisfaction of STL An Implementation: The Breach Toolbox DREAM Seminar 18 / 39
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Outline
1 Temporal Logics for Continuous Time and SpaceSignal Temporal LogicQuantitative Satisfaction of STL
2 An Implementation: The Breach ToolboxSimulation of Parametric Hybrid SystemsSpecifying STL Formulas
3 ApplicationsCase Study: Voltage Controlled OscillatorAn Example from Systems Biology
Robust Satisfaction of STL An Implementation: The Breach Toolbox DREAM Seminar 19 / 39
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Temporal logic formulas: atomic predicatesSTL Syntax: ϕ := µ | ¬ϕ | ϕ ∧ ϕ | ϕ U[a,b] ϕ
+ usual syntactic sugars for disjunction, eventually and always.
Predicates: General constraints on the variables: µ ≡ µ(x,p, t) ≥ 0
% distance to (p0,p1) is more than 2.(x0[t]-p0)^2 + (x1[t]-p1)^2) >= 4.
% the system reached steady state (very slow evolution)abs(ddt{x0}[t])+abs(ddt{x1}[t])) <= 1e-3
% x0 is sensitive to parameter p3abs(d{x0}{p3}[t]) >= 10*x0[t]/p3
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Temporal logic formulas: formulasSTL Syntax: ϕ := µ | ¬ϕ | ϕ ∧ ϕ | ϕ U[a,b] ϕ
+ usual syntactic sugars for disjunction, eventually and always.
% x0 will become more than -.9 whithin .5 sev_[0,.5] (x0[t]>-.9)
% the system will eventually remain close to 0ev (always (abs(x0)[t] < 1e-6))
% x0 remains low until x1 stabilizes before 10 seconds(x0[t] < 0.1) until_[0, 10] always ((abs(ddt{x1}[t]) < 1e-6))
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Computing the satisfaction functions
Breach computes the function ρ(ϕ, x, ·) by induction on the structure of ϕ.
This reduces to three subproblems: given two functions y, y′ : T→ R, and aninterval [a, b]
1. (operator ¬) compute ∀t, z[t] = −y[t];
2. (operator ∧) compute ∀t, z[t] = min(y[t], y′[t])
3. (operator U ) compute ∀t, z[t] = maxτ∈t+[a,b]
(min(y′[τ ], mins∈[t,τ ]
y[s])))
1. and 2. are reasonably trivials. 3., less (maybe for a min−max guru).
In practice, Breach implementation behaves linearly in the size of formulas andthe size of traces
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Computing the satisfaction functions
Breach computes the function ρ(ϕ, x, ·) by induction on the structure of ϕ.
This reduces to three subproblems: given two functions y, y′ : T→ R, and aninterval [a, b]
1. (operator ¬) compute ∀t, z[t] = −y[t];
2. (operator ∧) compute ∀t, z[t] = min(y[t], y′[t])
3. (operator U ) compute ∀t, z[t] = maxτ∈t+[a,b]
(min(y′[τ ], mins∈[t,τ ]
y[s])))
1. and 2. are reasonably trivials. 3., less (maybe for a min−max guru).
In practice, Breach implementation behaves linearly in the size of formulas andthe size of traces
Robust Satisfaction of STL An Implementation: The Breach Toolbox DREAM Seminar 22 / 39
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40 60 80 100 120
40
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Computing the satisfaction functions
Breach computes the function ρ(ϕ, x, ·) by induction on the structure of ϕ.
This reduces to three subproblems: given two functions y, y′ : T→ R, and aninterval [a, b]
1. (operator ¬) compute ∀t, z[t] = −y[t];
2. (operator ∧) compute ∀t, z[t] = min(y[t], y′[t])
3. (operator U ) compute ∀t, z[t] = maxτ∈t+[a,b]
(min(y′[τ ], mins∈[t,τ ]
y[s])))
1. and 2. are reasonably trivials. 3., less (maybe for a min−max guru).
In practice, Breach implementation behaves linearly in the size of formulas andthe size of traces
Robust Satisfaction of STL An Implementation: The Breach Toolbox DREAM Seminar 22 / 39
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40 60 80 100 120
40
60
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mm
Computing the satisfaction functions
Breach computes the function ρ(ϕ, x, ·) by induction on the structure of ϕ.
This reduces to three subproblems: given two functions y, y′ : T→ R, and aninterval [a, b]
1. (operator ¬) compute ∀t, z[t] = −y[t];
2. (operator ∧) compute ∀t, z[t] = min(y[t], y′[t])
3. (operator U ) compute ∀t, z[t] = maxτ∈t+[a,b]
(min(y′[τ ], mins∈[t,τ ]
y[s])))
1. and 2. are reasonably trivials. 3., less (maybe for a min−max guru).
In practice, Breach implementation behaves linearly in the size of formulas andthe size of traces
Robust Satisfaction of STL An Implementation: The Breach Toolbox DREAM Seminar 22 / 39
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Computational Cost, Some Experiments
(a) Same signal, formula ϕ = (x > 0) U[0,1) (x > 0) U[0,1) (x > 0) . . .︸ ︷︷ ︸i times
(b) Same formula: ϕ = alw(x > 1.5⇒ ev(alw(x < .1))), different input sizes
(a)
i time(s)1 0.347472 0.463353 0.605994 0.760675 0.892016 1.03761
(b)
input size time(s)31416 0.18402345566 0.40761659716 0.75508973866 1.092681288016 1.4587
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Computational Cost, Some Experiments
(a) Same signal, formula ϕ = (x > 0) U[0,1) (x > 0) U[0,1) (x > 0) . . .︸ ︷︷ ︸i times
(b) Same formula: ϕ = alw(x > 1.5⇒ ev(alw(x < .1))), different input sizes
(a)
i time(s)1 0.347472 0.463353 0.605994 0.760675 0.892016 1.03761
(b)
input size time(s)31416 0.18402345566 0.40761659716 0.75508973866 1.092681288016 1.4587
Robust Satisfaction of STL An Implementation: The Breach Toolbox DREAM Seminar 23 / 39
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Outline
1 Temporal Logics for Continuous Time and SpaceSignal Temporal LogicQuantitative Satisfaction of STL
2 An Implementation: The Breach ToolboxSimulation of Parametric Hybrid SystemsSpecifying STL Formulas
3 ApplicationsCase Study: Voltage Controlled OscillatorAn Example from Systems Biology
Robust Satisfaction of STL Applications DREAM Seminar 24 / 39
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60
80
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Outline
1 Temporal Logics for Continuous Time and SpaceSignal Temporal LogicQuantitative Satisfaction of STL
2 An Implementation: The Breach ToolboxSimulation of Parametric Hybrid SystemsSpecifying STL Formulas
3 ApplicationsCase Study: Voltage Controlled OscillatorAn Example from Systems Biology
Robust Satisfaction of STL Applications DREAM Seminar 24 / 39
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Case Study: Voltage Controlled Oscillators
I Characterizing oscillations in aVoltage Controlled Oscillator
I Non linear circuit with 3 statevariables (IL1, VD1, VD2) andaround 10 parameters (C, Vctrl,L, R, etc )
Vdd
ids1 ids2
Vd1 Vd2C C
Vctrl
IL1 IL2
L R R L
Robust Satisfaction of STL Applications DREAM Seminar 25 / 39
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Specifying Oscillations, Predicates
We look for oscillations of period T and given minimum and maximumamplitudes around 0
% Above and below a minimum amplitudemu0: IL1[t] > Aminmu1: IL1[t] < -Amin
% Bounded by a maximum amplitudemu2: abs(IL1[t]) < Amax
% (almost) Strict periodicitymu3: (abs(IL1[t] - IL1[t-T]) < epsi)
Robust Satisfaction of STL Applications DREAM Seminar 26 / 39
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Specifying Oscillations, Formulas
% Alternating above and below a minimum amplitudephi0: (ev_[0,T] (IL1[t]>Amin)) and (ev_[0,T] (IL1[t]<-Amin))
% and holding for 4 periodsphi1: alw_[0,4*T] (phi0)
% Holding strict periodicityphi2: alw_[0,4*T] ( (IL1[t] - IL1[t-T])^2 ) < epsi)
% Bounding amplitude globallyphi3: alw (IL1[t]^2 < Amax)
% Final formula, the ev operator gets rids of transientphi: ev (phi1 and phi2 and phi3)
Robust Satisfaction of STL Applications DREAM Seminar 27 / 39
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Breach Interface
Robust Satisfaction of STL Applications DREAM Seminar 28 / 39
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Breach Interface
Robust Satisfaction of STL Applications DREAM Seminar 28 / 39
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Breach Interface
Robust Satisfaction of STL Applications DREAM Seminar 28 / 39
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Result on a Single Trace
Robust Satisfaction of STL Applications DREAM Seminar 29 / 39
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Result on a Single Trace
Robust Satisfaction of STL Applications DREAM Seminar 29 / 39
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Partitioning the Parameter Region
Robust Satisfaction of STL Applications DREAM Seminar 30 / 39
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Partitioning the Parameter Region
Robust Satisfaction of STL Applications DREAM Seminar 30 / 39
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Satisfaction Functioni.e., the resulting cost function
Robust Satisfaction of STL Applications DREAM Seminar 31 / 39
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Finding Oscillations
I We defined 10 uncertainparameters with given ranges
I and picked 5 starting pointsrandomly distributed in thisdomain
Using an implementation of the Nelder Mead optimization algorithm,Breach was able to find two parameter valuations satisfying the property in98 s of computation time.
It turned out those were perfectly valid oscillations
... of period T/4 andT/2
Robust Satisfaction of STL Applications DREAM Seminar 32 / 39
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40 60 80 100 120
40
60
80
mm
Finding Oscillations
I We defined 10 uncertainparameters with given ranges
I and picked 5 starting pointsrandomly distributed in thisdomain
Using an implementation of the Nelder Mead optimization algorithm,Breach was able to find two parameter valuations satisfying the property in98 s of computation time.
It turned out those were perfectly valid oscillations
... of period T/4 andT/2
Robust Satisfaction of STL Applications DREAM Seminar 32 / 39
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40 60 80 100 120
40
60
80
mm
Finding Oscillations
I We defined 10 uncertainparameters with given ranges
I and picked 5 starting pointsrandomly distributed in thisdomain
Using an implementation of the Nelder Mead optimization algorithm,Breach was able to find two parameter valuations satisfying the property in98 s of computation time.
It turned out those were perfectly valid oscillations
... of period T/4 andT/2
Robust Satisfaction of STL Applications DREAM Seminar 32 / 39
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40 60 80 100 120
40
60
80
mm
Finding Oscillations
I We defined 10 uncertainparameters with given ranges
I and picked 5 starting pointsrandomly distributed in thisdomain
Using an implementation of the Nelder Mead optimization algorithm,Breach was able to find two parameter valuations satisfying the property in98 s of computation time.
It turned out those were perfectly valid oscillations ... of period T/4 andT/2
Robust Satisfaction of STL Applications DREAM Seminar 32 / 39
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40 60 80 100 120
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Outline
1 Temporal Logics for Continuous Time and SpaceSignal Temporal LogicQuantitative Satisfaction of STL
2 An Implementation: The Breach ToolboxSimulation of Parametric Hybrid SystemsSpecifying STL Formulas
3 ApplicationsCase Study: Voltage Controlled OscillatorAn Example from Systems Biology
Robust Satisfaction of STL Applications DREAM Seminar 33 / 39
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40 60 80 100 120
40
60
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Outline
1 Temporal Logics for Continuous Time and SpaceSignal Temporal LogicQuantitative Satisfaction of STL
2 An Implementation: The Breach ToolboxSimulation of Parametric Hybrid SystemsSpecifying STL Formulas
3 ApplicationsCase Study: Voltage Controlled OscillatorAn Example from Systems Biology
Robust Satisfaction of STL Applications DREAM Seminar 33 / 39
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An Enzymatic Network Involved in AngiogenesisCollagen (C1) degradation by matrix metalloproteinase (M P
2 ) and membrane type1 metalloproteinase (MT1).
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Rigorous Steady State AnalysisIn [KP04], activation of M P
2 after 12h “Nearly steady state” for T2(0) between 0and 200 nM. It turned out that steady state was not reached for T2(0) > 20 nM.
Using ϕ ≡ ev alw (|M2(t)| < ε×M P2 (0)) we could guarantee the correct plot.
Robust Satisfaction of STL Applications DREAM Seminar 35 / 39
0 20 40 60 80 100 120 140 160 180 2000
10
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90
100
Initial concentrations of TIMP2 (nM)
% of activated MMP2
Activated MMP2 after a fixed time
12 hours
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40
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Rigorous Steady State AnalysisIn [KP04], activation of M P
2 after 12h “Nearly steady state” for T2(0) between 0and 200 nM. It turned out that steady state was not reached for T2(0) > 20 nM.
Using ϕ ≡ ev alw (|M2(t)| < ε×M P2 (0)) we could guarantee the correct plot.
Robust Satisfaction of STL Applications DREAM Seminar 35 / 39
0 20 40 60 80 100 120 140 160 180 2000
10
20
30
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50
60
70
80
90
100
Initial concentrations of TIMP2 (nM)
% of activated MMP2
Activated MMP2 after a fixed time
12 hours36 hours
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40 60 80 100 120
40
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mm
Rigorous Steady State AnalysisIn [KP04], activation of M P
2 after 12h “Nearly steady state” for T2(0) between 0and 200 nM. It turned out that steady state was not reached for T2(0) > 20 nM.
Using ϕ ≡ ev alw (|M2(t)| < ε×M P2 (0)) we could guarantee the correct plot.
Robust Satisfaction of STL Applications DREAM Seminar 35 / 39
0 20 40 60 80 100 120 140 160 180 2000
10
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30
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50
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90
100
Initial concentrations of TIMP2 (nM)
% of activated MMP2
Activated MMP2 after a fixed time
12 hours36 hours100 hours
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40 60 80 100 120
40
60
80
mm
Rigorous Steady State AnalysisIn [KP04], activation of M P
2 after 12h “Nearly steady state” for T2(0) between 0and 200 nM. It turned out that steady state was not reached for T2(0) > 20 nM.
Using ϕ ≡ ev alw (|M2(t)| < ε×M P2 (0)) we could guarantee the correct plot.
Robust Satisfaction of STL Applications DREAM Seminar 35 / 39
0 20 40 60 80 100 120 140 160 180 2000
10
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90
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Initial concentrations of TIMP2 (nM)
% of activated MMP2
Activated MMP2 after a fixed time
12 hours36 hours100 hoursSteady
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Open ModelWe extended the model by introducing production and degradation terms
More complex behaviors becomes possible, such as oscillatory regimesRobust Satisfaction of STL Applications DREAM Seminar 36 / 39
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Open ModelWe extended the model by introducing production and degradation terms
More complex behaviors becomes possible, such as oscillatory regimesRobust Satisfaction of STL Applications DREAM Seminar 36 / 39
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Oscillations Map
Robust Satisfaction of STL Applications DREAM Seminar 37 / 39
0 2 4 6 8 10 12 14 16 18 200
1
2x 10
−6
m2
p
0 2 4 6 8 10 12 14 16 18 200
2
4
6x 10
−7
t2
0 2 4 6 8 10 12 14 16 18 200
0.5
1
1.5x 10
−6
mt1
time
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Oscillations Map
Robust Satisfaction of STL Applications DREAM Seminar 37 / 39
0 0.5 1 1.5 2 2.5 3
x 10−9
0
1
2
3
4
5
6x 10
−9
pm
t1
pt2
0 2 4 6 8 10 12 14 16 18 200
1
2x 10
−6
m2
p
0 2 4 6 8 10 12 14 16 18 200
2
4
6x 10
−7
t2
0 2 4 6 8 10 12 14 16 18 200
0.5
1
1.5x 10
−6
mt1
time
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Oscillations Map
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0 0.5 1 1.5 2 2.5 3
x 10−9
0
1
2
3
4
5
6x 10
−9
pm
t1
pt2
0 2 4 6 8 10 12 14 16 18 200
1
2x 10
−6
m2
p
0 2 4 6 8 10 12 14 16 18 200
2
4
6x 10
−7
t2
0 2 4 6 8 10 12 14 16 18 200
0.5
1
1.5x 10
−6
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time
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Oscillations Map
Robust Satisfaction of STL Applications DREAM Seminar 37 / 39
0 0.5 1 1.5 2 2.5 3
x 10−9
0
1
2
3
4
5
6x 10
−9
pm
t1
pt2
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−6
m2
p
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2
4
6x 10
−7
t2
0 2 4 6 8 10 12 14 16 18 200
0.5
1
1.5x 10
−6
mt1
time
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Oscillation, Robustness
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Oscillation, Robustness
Robust Satisfaction of STL Applications DREAM Seminar 38 / 39
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Oscillation, Robustness
Robust Satisfaction of STL Applications DREAM Seminar 38 / 39
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Conclusion
SummaryI Specification language for hybrid systems behaviors, with a robust
semanticsI An implementation with advanced simulation and parameter
exploration featuresI Case studies of parameter synthesis problems
PerspectivesI Going further with global robustness and sensitivity analysis for
specificationsI Different optimization strategies for parameter synthesis/optimal
controlI From robust satisfaction to formal specifications
Robust Satisfaction of STL Conclusion DREAM Seminar 39 / 39