1 Logics & Preorders from logic to preorder – and back Kim Guldstrand Larsen Paul PetterssonMogens...
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Transcript of 1 Logics & Preorders from logic to preorder – and back Kim Guldstrand Larsen Paul PetterssonMogens...
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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)
9
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
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
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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END