1 Logics & Preorders from logic to preorder – and back Kim Guldstrand Larsen Paul PetterssonMogens Nielsen BRICS@Aalborg BRICS@Aarhus

1

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)


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


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


Semantics

state of timed automata

timed asgnfor formula clocks

formula

Semantics


Derived Operatorsholds between l and u

Invariantly

Weak UNTIL

Bounded UNTIL

Timed Modal Mu-calculusis at least as expressive

as TCTL


Symbolic Semantics

location region over C and K

formula

Region-based Semantics

THEOREM


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)

9

Timed BimulationWang’91, Cerans’92


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


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


Examples


Towards Timed Bisimulation Algorithm

independent“product-construction”

Cerans’92


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





Bs' then s's if i)



12

21

aa

aa

d

Definition

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

Towards Timed Bisimulation Algorithm


Timed Bisimulation Algorithm = Checking for TB-ness using Regions

x

y

AX,R0

AX,R1

AX,R2

AY,R3

a2 a1

1

1

2


Characteristic Propertyfor finite state automata

a1

ak

n

m1

mk

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


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


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


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


Timed Bisimulation as a formula





Bs' then s's if i)



12

21

aa

aa

d





Bs' then s's if i)



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,


Timed Safety LogicBack to Zones

Fp/c F F [a]F X p

:: F

21

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


Zone Semantics

locationzone

over C and K

formula

MC wrt Safety Logic

is PSPACE complete


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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Documents

1 Logics & Preorders from logic to preorder – and back Kim Guldstrand Larsen Paul PetterssonMogens Nielsen BRICS@Aalborg BRICS@Aarhus