real-time systems, dtu, feb 15, 2000 paul pettersson, brics, aalborg, denmark. timed automata and...

Real-Time Systems, DTU, Feb 15, 2000Paul Pettersson, BRICS, Aalborg, Denmark.

Timed Automata and

Timed Computation Tree Logic

Paul PetterssonBRICS@Aalborg


CTL Models = Kripke Structures


Computation Tree Logic, CTLClarke & Emerson 1980

Syntax


CTL, Derived Operators

. . .

. . .

. . .

. . .

p

p p

AF p

. . .

. . .

. . .

. . .

p

EF p

possible

inevitable

”exists eventually”

or ”reachable””exists globally”


CTL, Derived Operators

p p

p

. . .

. . .

. . .

. . .

AG p

p p p p

p

p

. . .

. . .

. . .

. . .

EG p

p

always

potentially always

for all paths next

”forall globally”

or ”invariantly”

”forall eventually”


Exercise 18


Exercise 22


Exercise 22

{}

)(. yEXpypEF


Exercise 22

},,,{,{}

)(.

4321 ssss

yEXpypEF


Exercise 22

},,,,{},,,,{,{}

)(.

432104321 sssssssss

yEXpypEF


Exercise 22

},,,,{

)(.

][

43210 sssss

yAXpypAG

pAGEF


Exercise 22

},,,{},,,,,{

)(.

][

432143210 sssssssss

yAXpypAG

pAGEF


Exercise 22

},,,{},,,,,{

)(.

][

432143210 sssssssss

yAXpypAG

pAGEF

},,,{

)(.

4321 ssss

yEXpypEF


Exercise 22

},,,{},,,,,{

)(.

][

432143210 sssssssss

yAXpypAG

pAGEF

},,,,{},,,,{{},

)(.

432104321 sssssssss

yEXpypEF


Off Light Brightpress? Press?

press?

Press?

WANT: if press is issued twice quickly then the light will get brighter; otherwise the light is turned off.

Timed Automata

Intelligent Light Control


Timed Automata

Intelligent Light Control

Off Light Bright

Solution: Add real-valued clock x

X:=0X<=3

X>3

press? Press?

press?

Press?


Timed Automata

n

m

a

(Alur & Dill 1990)

Clocks: x, y

x<=5 & y>3

x := 0

Guard Boolean combination of comp withinteger bounds

ResetAction perfumed on clocks

Transitions

( n , x=2.4 , y=3.1415 ) ( n , x=3.5 , y=4.2415 )

e(1.1)

( n , x=2.4 , y=3.1415 ) ( m , x=0 , y=3.1415 )

a

State ( location , x=v , y=u ) where v,u are in R

Actionused

for synchronization


n

m

a

Clocks: x, y

x<=5 & y>3

x := 0

Transitions

( n , x=2.4 , y=3.1415 ) ( n , x=3.5 , y=4.2415 )

e(1.1)

( n , x=2.4 , y=3.1415 )

e(3.2)

x<=5

y<=10

LocationInvariants

g1g2 g3

g4

Invariants ensure progress!!

Timed Safety Automata = Timed Automata + Invariants

(Henzinger et al, 1992)


Clock Constraints


Timed (Safety) Automata


Timed Automata: Exampleguard

reset

location


Timed Automata: Example

3x


Timed Automata: Example


Light Switch

push

pushclick

9y


Light Switch

• Switch may be turned on whenever at least 2 time units has elapsed since last “turn off”

push

pushclick

9y


Light Switch

• Switch may be turned on whenever at least 2 time units has elapsed since last “turn off”

• Light automatically switches off after 9 time units.

push

pushclick


Semantics• clock valuations:

• state:

• Semantics of timed automata is a labeled transition systemwhere

• action transition

• delay Transition

)(),( CVvandLlwherevl

})(|),({ LlandCVvvlS

0:)( RCvCV

),( S

0')')((

),(),(

RddwheneverdvlInv

iffdvlvl d

g a rl l’

)')('(][')(

)','(),(

vlInvandrvvandvg

iffvlvl a


Semantics: Example

...)9,0,()9),3(9,(

)3,3,(),0,(

),()0,(

)5.3,()0,(

)3(93

5.3

yxoffyxon

yxonyxon

yxonyxon

yxoffyxoff

click

push

push

push

pushclick

9y


Networks of Timed Automata + Integer Variables + arrays ….

l1

l2

a!

x>=2i==3

x := 0i:=i+4

m1

m2

a?

y<=4

…………. Two-way synchronizationon complementary actions.

Closed Systems!

(l1, m1,………, x=2, y=3.5, i=3,…..) (l2,m2,……..,x=0, y=3.5, i=7,…..)

(l1,m1,………,x=2.2, y=3.7, I=3,…..)

0.2

tau

Example transitions

If a URGENT CHANNEL


Timed Automata in UPPAAL• Timed Safety Automata

+ urgent actions+ urgent locations (i.e. zero-delay locations)+ committed locations (i.e. zero-delay and atomic locations)+ data-variables (integers with bounded domains)+ arrays of data-variables+ guards and assignments over data-variables and arrays...


Urgent and Committed Locations

m

n

o

2x

!a

0:x

p

q

r

?a

)0,|(

)0,|(

)5.2,|(

)5.2,|(

)0,|(

xro

xqo

xqn

xpm

xpm

2.5

a )5.2,|( xrn

)5.2,|( dxqn

)5.2,|( dxqo

committed

urgent

d

d


TCTL = CTL + Time

inz

clocksformulaDz

nspropositioautomicAPp

,,

,,

constraints over formula clocks and automata clocks

“freeze operator” introduces new formula clock z

E[ U ], A[ U ] - like in CTL

No EX


Derived Operators

Along any path holds continuously until within 7 time units

becomes valid.

=

=

The property becomes valid within 5 time units.


Paths

Example:

push

pushclick

9y

...)9,0,()9),3(9,(

)3,3,(),0,(

),()0,(

)5.3,()0,(

)3(93

5.3

yxoffyxon

yxonyxon

yxonyxon

yxoffyxoff

click

push

push


Elapsed time in path

...)9,0,()9),3(9,(

)3,3,(),0,(

),()0,(

)5.3,()0,(

)3(93

5.3

yxoffyxon

yxonyxon

yxonyxon

yxoffyxoff

click

push

push

Example:


TCTL Semantics

s - location

w - formula clock valuation

PM(s) - set of paths from s

Pos() - positions in ,i) - elapsed time

(i,d) <<(i’,d’) iff (i<j) or ((i=j) and (d<d’))


Timeliness Properties

receive(m) occurs within 5 time units after send(m)

receive(m) occurs exactly 11 time units after send(m)

putbox occurs periodically (exactly) every 25 time units

(note: other putbox’s may occur in between)

real-time systems, dtu, feb 15, 2000 paul pettersson, brics, aalborg, denmark. timed automata and...

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