mm RobustSatisfactionofSignalTemporalLogicsand …Robust Satisfaction of STL DREAM Seminar 1 / 39....

40 60 80 100 120

Robust Satisfaction of Signal Temporal Logics andApplications

Alexandre Donzé

Verimag, Grenoble

September 26th, 2011

Robust Satisfaction of STL DREAM Seminar 1 / 39

40 60 80 100 120

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 → · · ·Simulation

(x, q) |= ϕ ?

Monitoring

Robust Satisfaction of STL Introduction DREAM Seminar 2 / 39

40 60 80 100 120

Overview

Hybrid System

x = fq(x, p) ||

Param. p, x0

Pok Pbad

q0 → q1 → · · ·Simulation

(x, q) |= ϕ ?

Monitoring

40 60 80 100 120

Overview

Hybrid System

x = fq(x, p) ||

Param. p, x0

Pok Pbad

q0 → q1 → · · ·

x(t, p)

Simulation

(x, q) |= ϕ ?

Monitoring

40 60 80 100 120

Overview

Hybrid System

x = fq(x, p) ||

Param. p, x0

Pok Pbad

q0 → q1 → · · ·

x(t, p)

Simulation

(x, q) |= ϕ ?

Monitoring

40 60 80 100 120

Overview

Hybrid System

x = fq(x, p) ||

Param. p, x0

Pok Pbadq0

q0 → q1 → · · ·

x(t, p)

Simulation

(x, q) |= ϕ ?

Monitoring

40 60 80 100 120

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

q0 → q1 → · · ·

SimulationProperty ϕ ≡

alw[q0 → ev[0,1]

q2 U[0,.2] (x≥ .5)]

40 60 80 100 120

Overview

Hybrid System

x = fq(x, p) ||

Param. p, x0q0

q0 → q1 → · · ·

x(t, p)

alw[q0 → ev[0,1]

q2 U[0,.2] (x≥ .5)]

STL monitoring

40 60 80 100 120

Overview

Hybrid System

x = fq(x, p) ||

Param. p, x0q0

q0 → q1 → · · ·

x(t, p)± ε

alw[q0 → ev[0,1]

q2 U[0,.2] (x≥ .5)]

STL monitoringok

40 60 80 100 120

Overview

Hybrid System

x = fq(x, p) ||

Param. p, x0q0

q0 → q1 → · · ·

x(t, p)± ε

alw[q0 → ev[0,1]

q2 U[0,.2] (x≥ .5)]

STL monitoringok

40 60 80 100 120

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 Temporal Logics for Continuous Time and Space DREAM Seminar 4 / 39

40 60 80 100 120

Outline

40 60 80 100 120

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

40 60 80 100 120

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

40 60 80 100 120

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) ⊥.

40 60 80 100 120

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) ⊥.

40 60 80 100 120

ExamplesConsider a simple piecewise affine signal:

⊥1 2 3 4 5 6

Truth value of :I ϕ = x > 2

II ϕ = alw[0.5,1.5](x > 2)

40 60 80 100 120

⊥1 2 3 4 5 6

II ϕ = alw[0.5,1.5](x > 2)

40 60 80 100 120

⊥1 2 3 4 5 6

Truth value of :I ϕ = x > 2I ϕ = ev[0,∞](x > 2)

I ϕ = alw[0.5,1.5](x > 2)

40 60 80 100 120

⊥1 2 3 4 5 6

Truth value of :I ϕ = x > 2I ϕ = ev[0,.5](x > 2)

I ϕ = alw[0.5,1.5](x > 2)

40 60 80 100 120

⊥1 2 3 4 5 6

II ϕ = alw[0,∞](x > 2)

ϕ = alw[0.5,1.5](x > 2)

40 60 80 100 120

⊥1 2 3 4 5 6

II ϕ = alw[0.5,1.5](x > 2)Robust Satisfaction of STL Temporal Logics for Continuous Time and Space DREAM Seminar 8 / 39

40 60 80 100 120

Outline

40 60 80 100 120

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) = +∞

40 60 80 100 120

and ∀t′′ ∈ [t, t′], (x, t′′) � ϕ1

s∈[t,τ ]χ(ϕ1, x, s))

40 60 80 100 120

and ∀t′′ ∈ [t, t′], (x, t′′) � ϕ1

s∈[t,τ ]χ(ϕ1, x, s))

40 60 80 100 120

and ∀t′′ ∈ [t, t′], (x, t′′) � ϕ1

s∈[t,τ ]χ(ϕ1, x, s))

40 60 80 100 120

and ∀t′′ ∈ [t, t′], (x, t′′) � ϕ1

s∈[t,τ ]χ(ϕ1, x, s))

40 60 80 100 120

and ∀t′′ ∈ [t, t′], (x, t′′) � ϕ1

s∈[t,τ ]χ(ϕ1, x, s))

40 60 80 100 120

and ∀t′′ ∈ [t, t′], (x, t′′) � ϕ1

s∈[t,τ ]χ(ϕ1, x, s))

40 60 80 100 120

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))

40 60 80 100 120

χ(µ, x, t) = sign(µ(x[t]))×∞

s∈[t,τ ]ρ(ϕ1, x, s))

40 60 80 100 120

χ(µ, x, t) = sign(µ(x[t]))×∞

s∈[t,τ ]ρ(ϕ1, x, s))

40 60 80 100 120

Robust Satisfaction, Examples

40 60 80 100 120

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}

40 60 80 100 120

Param. p ∈ P

p∗ = max {ρ(ϕ, p) | p ∈ P}

40 60 80 100 120

Param. p ∈ P

p∗ = max {ρ(ϕ, p) | p ∈ P}

40 60 80 100 120

Outline

Robust Satisfaction of STL An Implementation: The Breach Toolbox DREAM Seminar 14 / 39

40 60 80 100 120

Outline

40 60 80 100 120

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

40 60 80 100 120

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)))

40 60 80 100 120

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

40 60 80 100 120

Breach GUIs for trajectories exploration

40 60 80 100 120

Outline

40 60 80 100 120

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

40 60 80 100 120

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))

40 60 80 100 120

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

40 60 80 100 120

y[s])))

40 60 80 100 120

y[s])))

40 60 80 100 120

y[s])))

40 60 80 100 120

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

i time(s)1 0.347472 0.463353 0.605994 0.760675 0.892016 1.03761

input size time(s)31416 0.18402345566 0.40761659716 0.75508973866 1.092681288016 1.4587

40 60 80 100 120

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

i time(s)1 0.347472 0.463353 0.605994 0.760675 0.892016 1.03761

input size time(s)31416 0.18402345566 0.40761659716 0.75508973866 1.092681288016 1.4587

40 60 80 100 120

Outline

Robust Satisfaction of STL Applications DREAM Seminar 24 / 39

40 60 80 100 120

Outline

40 60 80 100 120

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 )

ids1 ids2

Vd1 Vd2C C

IL1 IL2

L R R L

40 60 80 100 120

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)

40 60 80 100 120

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)

40 60 80 100 120

Breach Interface

40 60 80 100 120

Breach Interface

40 60 80 100 120

Breach Interface

40 60 80 100 120

Result on a Single Trace

40 60 80 100 120

Result on a Single Trace

40 60 80 100 120

Partitioning the Parameter Region

40 60 80 100 120

Partitioning the Parameter Region

40 60 80 100 120

Satisfaction Functioni.e., the resulting cost function

40 60 80 100 120

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

40 60 80 100 120

It turned out those were perfectly valid oscillations ... of period T/4 andT/2

40 60 80 100 120

Outline

40 60 80 100 120

Outline

40 60 80 100 120

An Enzymatic Network Involved in AngiogenesisCollagen (C1) degradation by matrix metalloproteinase (M P

2 ) and membrane type1 metalloproteinase (MT1).

40 60 80 100 120

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.

0 20 40 60 80 100 120 140 160 180 2000

Initial concentrations of TIMP2 (nM)

% of activated MMP2

Activated MMP2 after a fixed time

12 hours

40 60 80 100 120

0 20 40 60 80 100 120 140 160 180 2000

% of activated MMP2

12 hours36 hours

40 60 80 100 120

0 20 40 60 80 100 120 140 160 180 2000

% of activated MMP2

12 hours36 hours100 hours

40 60 80 100 120

0 20 40 60 80 100 120 140 160 180 2000

% of activated MMP2

12 hours36 hours100 hoursSteady

40 60 80 100 120

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

40 60 80 100 120

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

40 60 80 100 120

Oscillations Map

0 2 4 6 8 10 12 14 16 18 200

1.5x 10

40 60 80 100 120

Oscillations Map

0 0.5 1 1.5 2 2.5 3

x 10−9

0 2 4 6 8 10 12 14 16 18 200

1.5x 10

40 60 80 100 120

Oscillations Map

0 0.5 1 1.5 2 2.5 3

x 10−9

0 2 4 6 8 10 12 14 16 18 200

1.5x 10

40 60 80 100 120

Oscillations Map

0 0.5 1 1.5 2 2.5 3

x 10−9

0 2 4 6 8 10 12 14 16 18 200

1.5x 10

40 60 80 100 120

Oscillation, Robustness

40 60 80 100 120

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

mm RobustSatisfactionofSignalTemporalLogicsand …Robust Satisfaction of STL DREAM Seminar 1 / 39....

Documents

Transcript of mm RobustSatisfactionofSignalTemporalLogicsand …Robust Satisfaction of STL DREAM Seminar 1 / 39....

BerufErziehungJungen Zukunft Hausarbeit 20 40 100 60 80 20 40 100 60 80 20 40 100 60 80 20 40 100 60 80 20 40 100 60 80.

RX 60-60 RX 60-70 RX 60-80 RX 60-80/ · PDF file6 mm 2697 2697 2697 2697 4.8 Seat height in terms of SIP h 7 mm 1719 1719 1719 1719 ... RX 60-80/900, can lift up to eight tonnes in

RX 60-60 RX 60-70 RX 60-80 RX 60-80/900 - IFOY

NM - 60 150 Dimension in mm. MH - 30 6 mm. 140 Dimension in mm. 750' MH ... · 2015. 9. 7. · NM - 60 150 Dimension in mm. MH - 30 6 mm. 140 Dimension in mm. 750' MH - 80 9 mm. æuztn.manlhnuo

On Signal Temporal Logic - Peoplesseshia/fmee/lectures/EECS294-98_Spring2014_STL...40 60 80 100 120 40 60 80 mm Outline 1 SignalTemporalLogic FromLTLtoSTL RobustSemantics 2 RobustMonitoringofSTL

Polyamid-Räder und -Rollen Polypropylen-Räder und -Rollen · PPN 60 - 200 mm 80 - 450 kg LE-PPN 60 - 200 mm 80 - 300 kg LER-PPN 60 - 200 mm 80 - 300 kg L-PPN 80 - 200 mm 120 - 400

POMOLI PER PORTE / DOOR KNOBS - · PDF filepomoli per porte / door knobs veronica | cod. 900 pt 080 elika crystal | cod. 991 pt zen mesh | cod. 1157 pt 80 mm 80 mm 65 mm 65 mm 60 mm

PALPADORES SENSORES · SENSORES Instrumentos de medición Instrumentos de medição. 2 113 mm 65 mm 50 mm 10 0 20 30 40 50 60 70 80 9010 20 70 60 30 40 90 80 0.01 mm Germany Made

松阪興産株式会社...A=450 600 600 : A=75012 600 (mm) 1Ð'

today - dormakaba Argus · 2020. 10. 21. · Argus 60, Argus 80 Argus 60 at a length of 1,650 mm and Argus 80 at a length of 1,660 mm achieve the maximum security level: The horizontal

Stamps document · 2015. 12. 20. · ALPHA H25 - MFH 14mm ALPHA H25 - MFH 15mm H25 Progressive Design Stamps 60 mm 65 mm 70 mm 75 mm 80 mm 60 mm 65 mm 70 mm …

KopfrechnenGeometrieGleichungen Prozente Vermischtes 20 40 100 60 80 20 40 100 60 80 20 40 100 60 80 20 40 100 60 80 20 40 100 60 80.

Entrada de Audio - sharp.es · 1,329.1 mm 1,538.9 mm 865.6 mm 1,771.2 mm 996.3 mm 80-pulgadas 70-pulgadas 60-pulgadas Nota: las cifras anteriores indican zonas activas de pantalla.

Knuth Machine Tools USA - Radial Drill Presses R 40 V • R 60 V • R … · Specifications R 40 V R 60 V R 80 V R 100 drilling capacity in steel 40 mm 62 mm 80 mm 100 mm tapping

w.lagl.o.ensio.tgnssio - Home | GENESIS · 2018. 9. 19. · w.lagl.o.ensio.tgnssio FR. Kockòn 105 Kockòn 105 1620 g 60 mm 200 mm 85 mm 58 16 20 020 KOCKÒN 80 CLASSIC 80 3, 7% QUASIMENT

PRONTUARIO - Arturo Mancini · 2,16* 2,78* 3,49* 4,26 5,11 7,03 9,26 11,90 16,30 * Disponibile anche zincato Spessore mm Peso kg/ml L mm H mm 789 11 13 60 70 80 100 120 60 70 80 100

hi!dea™ · Wine bag (1 bottle). 100 x 330 x 100 mm TRS - 60 x 80 mm Coaster. 100 x 100 x 5 mm LSR - 80 x 80 mm Wine set. ø144 x 42 mm LSR - Ø90 mm 22 ECO LOGIC Originating in

Die beliebte Fernseh-Quiz-Show mit Mäni Weber. 20 40 60 80 100 60 80 100 60 80 100 60 80 100 60 80 100 60 80 100 Personen Web 2.0AllerleiHardwareProgramme.

FARICH system for Super c t Factory. · 3⇥3mm,pitch3.2mm. Proximity focusing multilayer RICH X, mm −100−80 −60 −40 −20 0 20 40 60 80 100 Y, mm −100 −80 −60 −40

Part 3 Assembly - HORNBACH...24,5 mm 008_015 104_107 100_015 mm mm 0 10 20 30 40 50 60 70 80 0 10 20 30 40 0 0 10 10 20 20 30 30 40 40 50 60 70 80 Hardware 002_221 38 mm 002_221 2×