Chromatin Modifications Vered Fishbain Reading Group in Computational Molecular Biology 21/12/2006.
Dr. Vered Gafni 1 Modeling Real-Time Systems. Dr. Vered Gafni 2 Behavioral Model (Signature, Time)...
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Transcript of Dr. Vered Gafni 1 Modeling Real-Time Systems. Dr. Vered Gafni 2 Behavioral Model (Signature, Time)...
Dr. Vered Gafni 1
Modeling Real-Time Systems
Semantic Model
SystemOntology
SystemDesign
Formal SystemSpecification(properties)
Formal verification
NL Specification
ConsistencyCheck
Dr. Vered Gafni 2
Behavioral Model
(Signature, Time)
• Signature: v1:D1, v2:D2,…,vn:Dn
S = (D1… Dn) (states’ space)
• Time: (T, , 0)
Behavior: : T S
Discrete, Continuous
Discrete, Continuous
Dr. Vered Gafni 3
Signature: E - a finite set of events (env. & reactions)
Time domain: T={ (R+) | (t0t1t2…):
(a) t0 =0, ti ti+1, i=0,1,…
(b) tR+. i. t ti }
Timed trace over (E,T) is an -sequence:
T = (0,t0) (1,t1) (2,t2) … where: i2E, (t0t1t2…)T
Controller Behavioral Model: Timed Traces
Controller – a set of timed traces over E and T.
finite variability, non-Zenon
t0 t1 t2
0 1 2 Event – Instantaneous occurrence
Dr. Vered Gafni 4
Environment events
• Model : {P1, P2,…PN} where Pi : {R+ Di}, i=1..N
• Event – Boolean relation on Pi becomes true/false.
E.g., =def tr( pjK) then occur() = {t | pj(t)K becomes true}.
Environment & Controller Events
non-Zenon
Controller events
• Model – a set of (parallel) tasks (transition systems) over a set of variables
• Event –Assignment of a certain variable (write memory location).
time
temp'
50
30
TempHigh
TempHigh =def tr( Temp30)
Process properties:• TrainLocation : {0..100}Km• GatePosition : (0-90)• SemaphoreState : {pass, stop}
Process events:
• Tin =def tr(TrainLocation=xI)
• Tout =def tr(TrainLocation=xO)
• Close! =def tr(GatePosition=0°)
• Open! =def tr(GatePosition=90°)
Example: Railroad Crossing
Controller actions:
•close =def GatePort:=1
•open =def GatePort:=0
• pass =def LightsPort:=1
• stop =def LightsPort:=0xI xO
5
E={ Tin, Tout, close, close!, open, open!, stop, pass }
( , 2.7) ( , 2.8) ( , 8.0) ( , 19.1)…Tinclosestop
close!pass
Toutopen
Dr. Vered Gafni 6
General reactive software:
• controller comprises a set of (concurrent) tasks.
• reactive behavior of a task concerns:
initiation, synchronization, termination.
About Controller Computations & Events
Computations:
• Asynchronous – take observable duration (initiation<termination)
• Synchronous – instantaneous (initiation=termination)
Simplified reactive model:
computation does not synchronize during execution
only initiation, termination are observable events
Dr. Vered Gafni 7
Synchronous/Asynchronous Computations
• Formally, synchronous computation = executed in zero time,
• In practice, it is sufficient that the computation terminates before next environment event.
• Sequence of sync. computations is a sync. In practice, only finite
sequences that respect “next event” rule.
Finally, only the synchronous trace is of interest.
Dr. Vered Gafni 8
• System behavior:
– Time step – time advances + an event set.
– Reaction step – time freezes but new event set. A trace always starts with: (, 0) A reaction step may follow only trace elements: (, t)
• Super step – a sequence of reaction-steps (triggered actions)
that follow a time step until stabilization (=).
Thus, a trace looks like:
… (,2.0) ({p,q},3.1) ({r},3.1) (,3.1) ({q,r},3.8) …
Synchronous trace
time step reaction step reaction step time step
super step
tk tk+1 tk+1 tk+2
E1 E2, E3 E4 actions
Dr. Vered Gafni 9
• Activation by occurrence of events (may occur simultaneously)
t`t t``
{E2}{E1} {E1, E2, E3}
In practice, observations are taken w.r.t. to a finite precision
clock, henceDiscrete time (modeled by N),Nearby events may get same time record, still order is
preserved.
… ({p,q},53) ({r},53) ({u,w},62) …
So far: Event Driven Traces
Dr. Vered Gafni 10
• Global clock activation signature includes periodic event ‘tick’.
• Events during (ti-1,ti] considered at ti (order/repetition are lost).
titi-1 ti+1 ti+2
E2 E3E1 {E1, E2, E3}
• Time model N, but time-tag coincidences with index, hence represented by un-timed traces:
0 1 2… where i2E + ‘tick’ duration,
• Note: k may consist of the event ‘tick’ solely.
Clock Driven Execution Model
Dr. Vered Gafni 11
execution of T
asynchronousdata-processing-tasks executive
activation request: (T,t)
return (T.done,t’)
(Tdone,t’)deadline inspectionsynchronous
control executive
e (external event)
Asynchronous Computation in Clock Driven Model
In practice ….
12
• Clock driven synchronous model,
hence traces are untimed (time given by index)
• Clock + Event driven synchronous model, but external
events are tagged with last RTC
Dr. Vered Gafni 13
Hybrid Systems: continuous properties
• If 'pass' is accepted within 1 sec. then the speed remains steady
till it exits the crossing.
• If 'pass' has not been accepted within 1 second then the train
starts slowing down at a rate of 5m/s until 'pass' signal is accepted
or otherwise until its speed zeroes.
• When the expected pass signal is accepted, the train accelerates
again to 20m/s.
Dr. Vered Gafni 14
Hybrid Systems: Time model & Variables
Time model: T=[0, ) - non-negative continuous (physical) time
Variables
• Piecewise continuous (pwc)
Continuous range
Discrete range
• Events
Non Zeno
Dr. Vered Gafni 15
Hybrid models• V = Vd Vc (disjoint sets), S=DV (states, D unified domain)
E – set of events, =2E
flow={ | : R DVc } s.t.:
= (t) is defined on interval [0,t] or [0,t), where t>0.
= (t,v) is differentiable on (0,t) and limtt(t,v)DVc
• trace: w1w2…
s.t.: k either: wk=((sk-1,t),,(sk,t)) -- , sk-1,skS, tR
or: wk=((sk-1,t),,(sk,t’)) -- sk-1,skS, t,t’R , flow
s.t.: sdk-1=sd
k, t’tt