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Logics & Preorders from logic to preorder – and back

Kim Guldstrand Larsen Paul Pettersson Mogens Nielsen BRICS@Aalborg BRICS@Aarhus

2UCb Petri Net, June 2000 Kim G. Larsen, Mogens Nielsen, Paul Pettersson

Timed Logics .....

Real-time temporal logic (RTTL, Ostroff and Wonham 85) Metric Temporal Logic (Koymans, 1990) Explicit Clock Temporal Logic (Harel, Lichtenstein, Pnueli,

1990) Timed Propositional Logic (Alur, Henzinger, 1991)

Timed Computational Tree Logic (Alur, Dill, 1989) Timed Modal Mu-Calculus (Larsen, Laroussinie, Weise,

1995)

Duration Calculus (Chaochen, Hoare, Ravn, 1991)

3UCb Petri Net, June 2000 Kim G. Larsen, Mogens Nielsen, Paul Pettersson

Timed Modal Logic

FF FF Fa [a]F X p

:: F

2121

Atomic Prop

Recursion Variables

ActionModalities

Boolean Connectives

,.......

2

1

i

nn

22

11

F x

F x

F x

: E

n

Kozen’83

4UCb Petri Net, June 2000 Kim G. Larsen, Mogens Nielsen, Paul Pettersson

Timed Modal Logic

FF FF Fa [a]F X p

:: F

2121

Atomic Prop

Recursion Variables

ActionModalities

Boolean Connectives

F F F in x c

FormulaClockConstr

FormulaClockReset

DelayModalities

,,,,~ n~y- x n~ x:: c

,.......

2

1

i

nn

22

11

F x

F x

F x

: E

n

Larsen, Laroussine, Weise, 1995Larsen, Pettersson, Wang, 1995

Larsen, Holmer, Wang’91

5UCb Petri Net, June 2000 Kim G. Larsen, Mogens Nielsen, Paul Pettersson

Semantics

state of timed automata

timed asgnfor formula clocks

formula

Semantics

6UCb Petri Net, June 2000 Kim G. Larsen, Mogens Nielsen, Paul Pettersson

Derived Operatorsholds between l and u

Invariantly

Weak UNTIL

Bounded UNTIL

Timed Modal Mu-calculusis at least as expressive

as TCTL

7UCb Petri Net, June 2000 Kim G. Larsen, Mogens Nielsen, Paul Pettersson

Symbolic Semantics

location region over C and K

formula

Region-based Semantics

THEOREM

8UCb Petri Net, June 2000 Kim G. Larsen, Mogens Nielsen, Paul Pettersson

Fundamental Results

Given does there exist an automaton A satisfying ?

Given and given clock-set C and max constant M.

Does there exist an automaton A over C and M satisfying ?

UNDECIDABLE(strong conjecture)

Decidable

Given and automaton A does A satisfy ?

Decidable

EXPTIME-complete(Aceto,Laroussinie’99)

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Timed BimulationWang’91, Cerans’92

10UCb Petri Net, June 2000 Kim G. Larsen, Mogens Nielsen, Paul Pettersson

Timed Bisimulation

Del.Acta allfor

Rt's's'ss'.t't ii)

Rt's't'tt'.s's i)

:holds following

the thensRt whenever if onbisimulati timed a is R

aa

aa

0Rd:dDel

R. onbisimulati timed

somefor sRt whenever t s write We

Wang’91

11UCb Petri Net, June 2000 Kim G. Larsen, Mogens Nielsen, Paul Pettersson

Timed Simulation

Del.Acta allfor

Rt's't'tt'.s's i)

:holds following

the thensRt whenever if simulation timed a is R

aa

0Rd:dDel

R. simulation

timed somefor sRt ifft s write We

12UCb Petri Net, June 2000 Kim G. Larsen, Mogens Nielsen, Paul Pettersson

Examples

13UCb Petri Net, June 2000 Kim G. Larsen, Mogens Nielsen, Paul Pettersson

Towards Timed Bisimulation Algorithm

independent“product-construction”

Cerans’92

14UCb Petri Net, June 2000 Kim G. Larsen, Mogens Nielsen, Paul Pettersson

on.bisimulati-product timed

somefor Bswhenever TB(s) write We

B's' s.t. 's's' then s's if iii)

B's' s.t. 's's' then s's if ii)

Bs' then s's if i)

:holds following the then Bs

whenever iff onbisimulati-product timed a is B

12

21

aa

aa

d

on.bisimulati-product timed

somefor Bswhenever TB(s) write We

B's' s.t. 's's' then s's if iii)

B's' s.t. 's's' then s's if ii)

Bs' then s's if i)

:holds following the then Bs

whenever iff onbisimulati-product timed a is B

12

21

aa

aa

d

Definition

21 ss TB(s) 21 ss TB(s) Theorem

Towards Timed Bisimulation Algorithm

15UCb Petri Net, June 2000 Kim G. Larsen, Mogens Nielsen, Paul Pettersson

Timed Bisimulation Algorithm = Checking for TB-ness using Regions

x

y

AX,R0

AX,R1

AX,R2

AY,R3

a2 a1

1

1

2

16UCb Petri Net, June 2000 Kim G. Larsen, Mogens Nielsen, Paul Pettersson

Characteristic Propertyfor finite state automata

a1

ak

n

m1

mk

Larsen, Ingolfsdottir, Sifakis, 1987Ingolfsdottir, Steffen, 1994

17UCb Petri Net, June 2000 Kim G. Larsen, Mogens Nielsen, Paul Pettersson

Characteristic Propertyfor finite state automata

a1

ak

n

m1

mk

ai.am

a

imi

n

i

i

i

a

a

ai.am

a

imi

n

i

i

i

a

a

n | l nl n | l nl

Larsen, Ingolfsdottir, Sifakis, 1987Ingolfsdottir, Steffen, 1994

18UCb Petri Net, June 2000 Kim G. Larsen, Mogens Nielsen, Paul Pettersson

Characteristic Propertyfor timed automata

a1

ak

n

m1

mk

g1

r1

gk

rk

Inv(n)

IDEA_ Automata clocks become formula clocks

Larsen, Laroussinie, Weise, 1995

19UCb Petri Net, June 2000 Kim G. Larsen, Mogens Nielsen, Paul Pettersson

Characteristic Propertyfor timed automata

a1

ak

n

m1

mk

boarder

ai.aimi

a

imii i

n

Inv(n)

]

g in r a

in rag

Inv(n) [

i

i

i

boarder

ai.aimi

a

imii i

n

Inv(n)

]

g in r a

in rag

Inv(n) [

i

i

i

g1

r1

gk

rk

Inv(n)

IDEA_ Automata clocks become formula clocks

n | vu),(l, v)(n,u)(l, n | vu),(l, v)(n,u)(l,

Larsen, Laroussinie, Weise, 1995

20UCb Petri Net, June 2000 Kim G. Larsen, Mogens Nielsen, Paul Pettersson

Timed Bisimulation as a formula

on.bisimulati-product timed

somefor Bswhenever TB(s) write We

B's' s.t. 's's' then s's if iii)

B's' s.t. 's's' then s's if ii)

Bs' then s's if i)

:holds following the then Bs

whenever iff onbisimulati-product timed a is B

12

21

aa

aa

d

on.bisimulati-product timed

somefor Bswhenever TB(s) write We

B's' s.t. 's's' then s's if iii)

B's' s.t. 's's' then s's if ii)

Bs' then s's if i)

:holds following the then Bs

whenever iff onbisimulati-product timed a is B

12

21

aa

aa

d

Zaa ZaaZ 122a

1 Zaa ZaaZ 122

a1

Z | v)(n,u),(l,

TBv)(n,u),(l,

v)(n,u)(l,

Z | v)(n,u),(l,

TBv)(n,u),(l,

v)(n,u)(l,

21UCb Petri Net, June 2000 Kim G. Larsen, Mogens Nielsen, Paul Pettersson

Timed Safety LogicBack to Zones

Fp/c F F [a]F X p

:: F

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Atomic Prop

Recursion Variables

ActionModalities

Boolean Connectives

F F in x c

FormulaClockConstr

FormulaClockReset

DelayModalities

,,,,~ n~y- x n~ x:: c

i

nn

22

11

F x

F x

F x

: E 2

1

n

.......

Larsen, Pettersson, Wang, 1995

22UCb Petri Net, June 2000 Kim G. Larsen, Mogens Nielsen, Paul Pettersson

Zone Semantics

locationzone

over C and K

formula

MC wrt Safety Logic

is PSPACE complete

23UCb Petri Net, June 2000 Kim G. Larsen, Mogens Nielsen, Paul Pettersson

Characteristic Property/Simulationfor deterministic timed automata

a

a

n

m1

mk

]a g

in r a g

Inv(n) [

ii

mii

i

n

i

false

]a g

in r a g

Inv(n) [

ii

mii

i

n

i

false

g1

r1

gk

rk

Inv(n)

n | vu),(l, v)(n,u)(l, n | vu),(l, v)(n,u)(l,

Aceto, Burgueno,Bouyer, Larsen, 1998

gi and gj = Ø

determinism

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