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Timed Automata – lllllllllllDecidability Results
![Page 2: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/2.jpg)
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Decidability ?
Reachable?
a b
c
OBSTACLE:Uncountably infinite
state space
![Page 3: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/3.jpg)
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Stable Quotient
Reachable?
x
y
x
y
Partitioning
a b
c
![Page 7: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/7.jpg)
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Stable Quotient
Reachable?
x
y
x
y
Partitioning
a b
c
![Page 8: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/8.jpg)
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UCb
Stable Quotient
Reachable?
x
y
x
y
Partitioning
a b
c
![Page 9: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/9.jpg)
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UCb
Stable Quotient
Reachable?
x
y
x
y
Partitioning
a b
c
![Page 10: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/10.jpg)
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UCb
Stable Quotient
Reachable?
x
y
x
y
Partitioning
a b
c
![Page 11: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/11.jpg)
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Stable Quotient
Reachable?
x
y
x
y
Partitioning
a b
c
12
34
5
6
0
0 ε 1 a 2 ε 3 a 4 ε 5 c 6
![Page 12: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/12.jpg)
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RegionsFinite Partitioning of State Space
x
y
An equivalence class (i.e. a region)in fact there is only a finite number of regions!!
1 2 3
1
2
![Page 14: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/14.jpg)
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RegionsSuccessor Operation (wrt delay)
x
y
An equivalence class (i.e. a region)
Successor regions, Succ(r)
r
1 2 3
1
2
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RegionsReset Operation
x
y
An equivalence class (i.e. a region) r
{x}r
{y}r
r
Resetregions
1 2 3
1
2
![Page 18: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/18.jpg)
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An Example Region Graph
![Page 19: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/19.jpg)
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Modified light switch
)AFAG()AFAG(
)AG(
offonoffon
yx
9≤⇒⇒
≤
![Page 20: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/20.jpg)
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)AFAG()AFAG(
)AG(
offonoffon
yx
9≤⇒⇒
≤
Reachable partof region graph
Properties
![Page 21: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/21.jpg)
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Fundamental Results
Reachability ☺ Alur, Dill
Trace-inclusion Alur, Dill
Timed ; Untimed ☺
BisimulationTimed ☺ Cerans ; Untimed ☺
Model-checking ☺TCTL, Tmu, Lnu,...
PSPACE-c
PSPACE-c / EXPTIME-c
![Page 25: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/25.jpg)
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Updatable Timed Automata
Patricia Bouyer, Catherine Dufourd,Emmanuel Fleury, Antoine Petit
W Diagonals
W Diagonals
![Page 26: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/26.jpg)
TCTL:Timed Computational Tree
Logic
![Page 27: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/27.jpg)
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TCTL = CTL + Time
−−
∈∈
φα
inz
clocksformulaDznspropositioautomicAPp
,,,,
“freeze operator” introduces new formula clock z
E[ φ U φ ], A[ φ U φ ] - like in CTLNo EX φ
constraints over formula clocks and automata clocks
![Page 28: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/28.jpg)
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Derived Operators
Along any path φ holds continuously until within 7 time units ψ becomes valid.
=
=
The property φ becomes valid within 5 time units.
![Page 29: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/29.jpg)
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Paths
Example:
push
pushclick
9≤y
...)9,0,()9),3(9,()3,3,(),0,(
),()0,()5.3,()0,(
)3(93
5.3
==⎯⎯→⎯=+−=
⎯⎯⎯ →⎯+==⎯→⎯==
⎯⎯→⎯==⎯→⎯==
⎯⎯→⎯==⎯→⎯==
+−
yxoffyxonyxonyxon
yxonyxonyxoffyxoff
click
push
push
π
ππ
ππ
π
![Page 30: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/30.jpg)
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Elapsed time in path
...)9,0,()9),3(9,()3,3,(),0,(
),()0,()5.3,()0,(
)3(93
5.3
==⎯⎯→⎯=+−=
⎯⎯⎯ →⎯+==⎯→⎯==
⎯⎯→⎯==⎯→⎯==
⎯⎯→⎯==⎯→⎯==
+−
yxoffyxonyxonyxon
yxonyxonyxoffyxoff
click
push
push
π
ππ
ππ
π
Example:
Δ(σ,1)=3.5, Δ(σ,6)=3.5+9=12.5
σ=
![Page 31: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/31.jpg)
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TCTL Semantics
s - locationw - formula clock valuationPM(s) - set of paths from sPos(σ) - positions in σΔ(σ,i) - elapsed time
∞
(i,d) <<(i’,d’) iff (i<j) or ((i=j) and (d<d’))
![Page 32: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/32.jpg)
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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)
![Page 33: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/33.jpg)
The UPPAALVerification Engine
![Page 34: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/34.jpg)
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Overview
Zones and DBMsMinimal Constraint FormClock Difference Diagrams
Distributed UPPAAL [CAV2000, STTT2004]
Unification & Sharing [FTRTFT2002, SPIN2003]
Acceleration [FORMATS2002]
Static Guard Analysis [TACAS2003,TACAS2004]
Storage-Strategies [CAV2003]
![Page 35: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/35.jpg)
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ZonesFrom infinite to finite
State(n, x=3.2, y=2.5 )
x
y
x
y
Symbolic state (set)
Zone:conjunction ofx-y<=n, x<=>n
(n, 1·x·4, 1·y· 3)
![Page 36: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/36.jpg)
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Symbolic Transitions
n
m
x>3
y:=0
delays to
conjuncts to
projects to
x
y1<=x<=41<=y<=3
x
y1<=x, 1<=y-2<=x-y<=3
x
y 3<x, 1<=y-2<=x-y<=3
3<x, y=0
x
y
Thus (n,1<=x<=4,1<=y<=3) =a => (m,3<x, y=0)Thus (n,1<=x<=4,1<=y<=3) =a => (m,3<x, y=0)
a
![Page 37: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/37.jpg)
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Zones = Conjuctive Constraints
A zone Z is a conjunctive formula:
g1 & g2 & ... & gn
where gi is a clock constraint xi ~ bi or xi-xj~bij
Use a zero-clock x0 (constant 0)
A zone can be re-written as a set:{xi-xj ~ bij | ~ is < or ≤, i,j≤n}
This can be represented as a matrix, DBM (Difference Bound Matrices)
![Page 38: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/38.jpg)
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Operations on ZonesFuture delay Z↑:
[Z↑] = {u+d| d ∈ R, u∈[Z]}
Past delay Z↓:[Z↓] = {u| u+d∈[Z] for some d∈R}
Reset: {x}Z or Z(x:=0)[{x}Z] = {u[0/x] | u ∈[Z]}
Conjunction[Z&g]= [Z]∩[g]
![Page 39: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/39.jpg)
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THEOREM
The set of zones is closed under all constraint operations.
That is, the result of the operations on a zone is a zone.
That is, there will be a zone (a finite object i.e a zone/constraints) to representthe sets: [Z↑], [Z↓], [{x}Z], [Z&g].
![Page 40: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/40.jpg)
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Symbolic Exploration
Reachable?
x
y
![Page 41: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/41.jpg)
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Symbolic Exploration
Reachable?
x
y
Delay
![Page 42: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/42.jpg)
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Symbolic Exploration
Reachable?
x
y
Left
![Page 43: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/43.jpg)
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Symbolic Exploration
Reachable?
x
y
Left
![Page 44: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/44.jpg)
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Symbolic Exploration
Reachable?
x
y
Delay
![Page 45: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/45.jpg)
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Symbolic Exploration
Reachable?
x
y
Left
![Page 46: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/46.jpg)
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Symbolic Exploration
Reachable?
x
y
Left
![Page 47: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/47.jpg)
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Symbolic Exploration
Reachable?
x
y
Delay
![Page 48: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/48.jpg)
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Symbolic Exploration
Reachable?
x
y
Down
![Page 49: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/49.jpg)
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A1 B1 CS1V:=1 V=1
A2 B2 CS2V:=2 V=2
InitV=1
2´
VCriticial Section
Fischer’s Protocolanalysis using zones
Y<10
X:=0
Y:=0
X>10
Y>10
X<10
![Page 50: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/50.jpg)
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Fischers cont. B1 CS1V:=1 V=1
A2 B2 CS2V:=2 V=2Y<10
X:=0
Y:=0
X>10
Y>10
X<10
A1,A2,v=1 A1,B2,v=2 A1,CS2,v=2 B1,CS2,v=1 CS1,CS2,v=1
Untimed case
A1
![Page 51: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/51.jpg)
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Fischers cont. B1 CS1V:=1 V=1
A2 B2 CS2V:=2 V=2Y<10
X:=0
Y:=0
X>10
Y>10
X<10
A1,A2,v=1 A1,B2,v=2 A1,CS2,v=2 B1,CS2,v=1 CS1,CS2,v=1
Untimed case
Taking time into account
X
Y
A1
![Page 52: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/52.jpg)
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Fischers cont. B1 CS1V:=1 V=1
A2 B2 CS2V:=2 V=2Y<10
X:=0
Y:=0
X>10
Y>10
X<10
A1,A2,v=1 A1,B2,v=2 A1,CS2,v=2 B1,CS2,v=1 CS1,CS2,v=1
Untimed case
Taking time into account
X
Y
A1
10X
Y1010
![Page 53: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/53.jpg)
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Fischers cont. B1 CS1V:=1 V=1
A2 B2 CS2V:=2 V=2Y<10
X:=0
Y:=0
X>10
Y>10
X<10
A1,A2,v=1 A1,B2,v=2 A1,CS2,v=2 B1,CS2,v=1 CS1,CS2,v=1
Untimed case
Taking time into account
A1
10X
Y10
X
Y10
![Page 54: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/54.jpg)
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Fischers cont. B1 CS1V:=1 V=1
A2 B2 CS2V:=2 V=2Y<10
X:=0
Y:=0
X>10
Y>10
X<10
A1,A2,v=1 A1,B2,v=2 A1,CS2,v=2 B1,CS2,v=1 CS1,CS2,v=1
Untimed case
Taking time into account
A1
10X
Y10
X
Y10
10X
Y10
![Page 55: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/55.jpg)
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Fischers cont. B1 CS1V:=1 V=1
A2 B2 CS2V:=2 V=2Y<10
X:=0
Y:=0
X>10
Y>10
X<10
A1,A2,v=1 A1,B2,v=2 A1,CS2,v=2 B1,CS2,v=1 CS1,CS2,v=1
Untimed case
Taking time into account
A1
10X
Y10
X
Y10
10X
Y10
![Page 56: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/56.jpg)
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Forward Rechability
Passed
WaitingFinal
Init
INITIAL Passed := Ø;Waiting := {(n0,Z0)}
REPEAT- pick (n,Z) in Waiting- if for some Z’ Z
(n,Z’) in Passed then STOP- else (explore) add
{ (m,U) : (n,Z) => (m,U) }to Waiting;Add (n,Z) to Passed
UNTIL Waiting = ØorFinal is in Waiting
Init -> Final ?
![Page 57: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/57.jpg)
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Forward Rechability
Passed
Waiting Final
Init
n,Z
INITIAL Passed := Ø;Waiting := {(n0,Z0)}
REPEAT- pick (n,Z) in Waiting- if for some Z’ Z
(n,Z’) in Passed then STOP- else (explore) add
{ (m,U) : (n,Z) => (m,U) }to Waiting;Add (n,Z) to Passed
UNTIL Waiting = ØorFinal is in Waiting
⊇
n,Z’
Init -> Final ?
![Page 58: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/58.jpg)
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Forward Rechability
Passed
Waiting Final
Init
n,Z
INITIAL Passed := Ø;Waiting := {(n0,Z0)}
REPEAT- pick (n,Z) in Waiting- if for some Z’ Z
(n,Z’) in Passed then STOP- else /explore/ add
{ (m,U) : (n,Z) => (m,U) }to Waiting;Add (n,Z) to Passed
UNTIL Waiting = ØorFinal is in Waiting
⊇
n,Z’
m,U
Init -> Final ?
![Page 59: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/59.jpg)
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UCb
Forward Rechability
Passed
Waiting Final
Init
INITIAL Passed := Ø;Waiting := {(n0,Z0)}
REPEAT- pick (n,Z) in Waiting- if for some Z’ Z
(n,Z’) in Passed then STOP- else /explore/ add
{ (m,U) : (n,Z) => (m,U) }to Waiting;Add (n,Z) to Passed
UNTIL Waiting = ØorFinal is in Waiting
⊇
n,Z’
m,U
n,Z
Init -> Final ?
![Page 60: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/60.jpg)
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Bellman 1958, Dill 1989
x<=1y-x<=2z-y<=2z<=9
x<=1y-x<=2z-y<=2z<=9
x<=2y-x<=3y<=3z-y<=3z<=7
x<=2y-x<=3y<=3z-y<=3z<=7
D1
D2
Inclusion
0
x
y
z
1 2
29
0
x
y
z
2 3
37
3
? ?
Graph
Graph
⊆
Canonical Datastructures for ZonesDifference Bounded Matrices
![Page 61: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/61.jpg)
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x<=1y-x<=2z-y<=2z<=9
x<=1y-x<=2z-y<=2z<=9
x<=2y-x<=3y<=3z-y<=3z<=7
x<=2y-x<=3y<=3z-y<=3z<=7
D1
D2
Inclusion
0
x
y
z
1 2
29
ShortestPath
Closure
ShortestPath
Closure
0
x
y
z
1 2
25
0
x
y
z
2 3
37
0
x
y
z
2 3
36
3
3 3
Graph
Graph
? ? ⊆
4
6
Canonical Datastructures for ZonesDifference Bounded Matrices
![Page 62: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/62.jpg)
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x<=1y>=5y-x<=3
x<=1y>=5y-x<=3
D
Emptiness
0y
x1
3
-5
Negative Cycleiffempty solution set
Graph
Compact
Canonical Datastructures for ZonesDifference Bounded Matrices
![Page 63: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/63.jpg)
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1<= x <=41<= y <=3
1<= x <=41<= y <=3
D
Future
x
y
x
y
Future D
0
y
x4
-1
3
-1
ShortestPath
Closure
Removeupper
boundson clocks
1<=x, 1<=y-2<=x-y<=3
1<=x, 1<=y-2<=x-y<=3
y
x
-1
-1
3
2
0
y
x
-1
-1
3
2
0
4
3
Canonical Datastructures for ZonesDifference Bounded Matrices
![Page 64: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/64.jpg)
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x
y
D1<=x, 1<=y-2<=x-y<=3
1<=x, 1<=y-2<=x-y<=3
y
x
-1
-1
3
2
0
Remove allbounds
involving yand set y to 0
x
y
{y}D
y=0, 1<=xy=0, 1<=x
Reset
y
x
-1
0
00
Canonical Datastructures for ZonesDifference Bounded Matrices
![Page 65: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/65.jpg)
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x1-x2<=4x2-x1<=10x3-x1<=2x2-x3<=2x0-x1<=3x3-x0<=5
x1-x2<=4x2-x1<=10x3-x1<=2x2-x3<=2x0-x1<=3x3-x0<=5
x1 x2
x3x0
-4
10
22
5
3
x1 x2
x3x0
-4
4
22
5
3 3 -2 -2
1
ShortestPath
ClosureO(n^3)
Canonical Datastructures for ZonesDifference Bounded Matrices
![Page 66: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/66.jpg)
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x1-x2<=4x2-x1<=10x3-x1<=2x2-x3<=2x0-x1<=3x3-x0<=5
x1-x2<=4x2-x1<=10x3-x1<=2x2-x3<=2x0-x1<=3x3-x0<=5
x1 x2
x3x0
-4
10
22
5
3
x1 x2
x3x0
-4
4
22
5
3
x1 x2
x3x0
-4
22
3
3 -2 -2
1
ShortestPath
ClosureO(n^3)
ShortestPath
ReductionO(n^3) 3
Space worst O(n^2)practice O(n)
RTSS 1997
Canonical Datastructures for ZonesMinimal Constraint Form
![Page 67: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/67.jpg)
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SPACE PERFORMANCE
00,10,20,30,40,50,60,70,80,9
1
Audio
Audio
w Col
B&OBox
Sort
erM. P
lant
Fische
r 2Fisc
her 3
Fische
r 4Fisc
her 5
Train Cros
sing
Per
cent Minimal Constraint
Global ReductionCombination
![Page 68: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/68.jpg)
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TIME PERFORMANCE
0
0,5
1
1,5
2
2,5
Audio
Audio
w Col
B&OBox
Sort
erM. P
lant
Fische
r 2Fisc
her 3
Fische
r 4Fisc
her 5
Train Cros
sing
Perc
ent Minimal Constraint
Global ReductionCombination
![Page 69: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/69.jpg)
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v and w are both redundantRemoval of one depends on presence of other.
v and w are both redundantRemoval of one depends on presence of other.
Shortest Path Reduction1st attempt
Idea
Problem
w
<=wAn edge is REDUNDANT if there existsan alternative path of no greater weight
THUS Remove all redundant edges!
An edge is REDUNDANT if there existsan alternative path of no greater weight
THUS Remove all redundant edges!
w
v
Observation: If no zero- or negativecycles then SAFE to remove all redundancies.
Observation: If no zero- or negativecycles then SAFE to remove all redundancies.
![Page 70: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/70.jpg)
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Shortest Path ReductionSolution
G: weighted graph
![Page 71: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/71.jpg)
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Shortest Path ReductionSolution
1. Equivalence classes basedon 0-cycles.
G: weighted graph
![Page 72: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/72.jpg)
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Shortest Path ReductionSolution
1. Equivalence classes basedon 0-cycles.
2. Graph based onrepresentatives. Safe to remove redundant edges
G: weighted graph
![Page 73: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/73.jpg)
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Shortest Path ReductionSolution
1. Equivalence classes basedon 0-cycles.
2. Graph based onrepresentatives.Safe to remove redundant edges
3. Shortest Path Reduction=
One cycle pr. class+
Removal of redundant edgesbetween classes
G: weighted graph
Canonical given order of clocks
![Page 74: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/74.jpg)
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Earlier Termination
Passed
Waiting Final
Init
INITIAL Passed := Ø;Waiting := {(n0,Z0)}
REPEAT- pick (n,Z) in Waiting- if for some Z’ Z
(n,Z’) in Passed then STOP- else /explore/ add
{ (m,U) : (n,Z) => (m,U) }to Waiting;Add (n,Z) to Passed
UNTIL Waiting = ØorFinal is in Waiting
⊇
n,Z’
m,U
n,Z
Init -> Final ?
![Page 75: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/75.jpg)
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Earlier Termination
Passed
Waiting Final
Init
INITIAL Passed := Ø;Waiting := {(n0,Z0)}
REPEAT- pick (n,Z) in Waiting- if for some Z’ Z
(n,Z’) in Passed then STOP- else /explore/ add
{ (m,U) : (n,Z) => (m,U) }to Waiting;Add (n,Z) to Passed
UNTIL Waiting = ØorFinal is in Waiting
⊇
n,Z’
m,U
n,Z
Init -> Final ?
ZZ'⊇
![Page 76: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/76.jpg)
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INITIAL Passed := Ø;Waiting := {(n0,Z0)}
REPEAT- pick (n,Z) in Waiting- if for some (n,Z’) in Passed then STOP
- else /explore/ add{ (m,U) : (n,Z) => (m,U) }to Waiting;Add (n,Z) to Passed
UNTIL Waiting = ØorFinal is in Waiting
Earlier Termination
Passed
Waiting Final
Init
⊇
n,Zk
m,U
n,Z
Init -> Final ?
n,Z1
n,Z2 ZZii⊇U
ZZ'⊇
![Page 77: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/77.jpg)
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Clock Difference Diagrams= Binary Decision Diagrams + Difference Bounded Matrices
CDD-representationsCDD-representations
CAV99
Nodes labeled with differencesMaximal sharing of substructures (also across different CDDs)Maximal intervalsLinear-time algorithms for set-theoretic operations.
NDD’s Maler et. al
DDD’s Møller, Lichtenberg
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SPACE PERFORMANCE
0
0,5
1
1,5
2
2,5
3
3,5
4
4,5
Philips
Philps
col
B&O
BRPPow
erDown1
PowerD
own2Daca
poGea
rBox
Fische
r4Fisc
her5
Per
cent CDD
Reduced CDDCDD+BDD
![Page 79: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/79.jpg)
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rmat
ionst
eknolo
gi
UCb
TIME PERFORMANCE
0
1
2
3
4
5
6
Philips
Philps
col
B&OBRP
PowerD
own1Pow
erDown2
Dacapo
GearB
oxFisc
her4
Fische
r5
Per
cent CDD
Reduced CDDCDD+BDD
![Page 80: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/80.jpg)
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rmat
ionst
eknolo
gi
UCb
UPPAAL 1995 - 2001
Dec’96 Sep’98
Every 9 month10 times better performance!
3.x
![Page 81: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/81.jpg)
Liveness Checking
![Page 82: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/82.jpg)
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rmat
ionst
eknolo
gi
UCb
Liveness Properties
F ::= E P |
A♦ P |
P Q
Possibly always P
Eventually Pis equivalent to (¬ E ¬ P)
P leads to Qis equivalent to
A ( P ⇒ A♦ Q)
in UPPAAL
Bouajjani, Tripakis, Yovine’97On-the-fly symbolic model checking of TCTL
![Page 83: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/83.jpg)
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ionst
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gi
UCb
in UPPAAL
LivenessE[]φ (A♦¬φ)
![Page 84: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/84.jpg)
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ionst
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gi
UCb
in UPPAALLivenessE[]φ (A♦¬φ)
(A♦¬φ)
Passed
STWS
Unexplored
![Page 85: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/85.jpg)
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ionst
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gi
UCb
in UPPAALLivenessE φ (A♦¬φ)
(A♦¬φ)
Passed
STWS
Unexplored
![Page 86: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/86.jpg)
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ionst
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gi
UCb
in UPPAALLivenessE φ (A♦¬φ)
(A♦¬φ)
Passed
STWS
Unexplored
![Page 87: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/87.jpg)
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ionst
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gi
UCb
in UPPAALLivenessE φ (A♦¬φ)
(A♦¬φ)
Passed
STWS
Unexplored
![Page 88: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/88.jpg)
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ionst
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gi
UCb
in UPPAALLivenessE φ (A♦¬φ)
(A♦¬φ)
Passed
STWS
Unexplored
?
![Page 89: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/89.jpg)
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ionst
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gi
UCb
in UPPAALLivenessE φ (A♦¬φ)
(A♦¬φ)
Passed
STWS
Unexplored
???
![Page 90: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/90.jpg)
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rmat
ionst
eknolo
gi
UCb
in UPPAALLivenessE φ (A♦¬φ)
(A♦¬φ)
Passed
STWS
Unexplored
[FORMATS05]Extensions allowing for automatic synthesis of
smallest bound t such that A♦· tφ holds
![Page 91: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/91.jpg)
Verification Options
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rmat
ionst
eknolo
gi
UCb
Verification OptionsSearch Order
Depth FirstBreadth First
State Space ReductionNoneConservativeAggressive
State Space RepresentationDBMCompact FormUnder ApproximationOver Approximation
Diagnostic TraceSomeShortestFastest
ExtrapolationHash Table sizeReuse
![Page 93: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/93.jpg)
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ionst
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gi
UCbNo Cycles: Passed list not needed for termination
However,Passed list useful forefficiency
State Space Reduction
![Page 94: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/94.jpg)
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ionst
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gi
UCb
Cycles:Only symbolic statesinvolving loop-entry points need to be saved on Passed list
State Space Reduction
![Page 95: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/95.jpg)
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ionst
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gi
UCb
To Store or Not To Store
Audio Protocol
117 statestotal→
81 statesentrypoint→
9 states
Behrmann, Larsen, Pelanek 2003
Time OHless than 10%
![Page 96: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/96.jpg)
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rmat
ionst
eknolo
gi
UCb
To Store or Not to StoreBehrmann, Larsen,
Pelanek 2003
![Page 97: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/97.jpg)
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rmat
ionst
eknolo
gi
UCb
Over-approximationConvex Hull
x
y
Convex Hull
1 3 5
1
3
5
TACAS04: An EXACT method performingas well as Convex Hull has been developed based on abstractions taking max constants into account
distinguishing between clocks, locations and · & ≥
![Page 98: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/98.jpg)
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rmat
ionst
eknolo
gi
UCb
Under-approximationBitstate Hashing
Passed
Waiting Final
Init
n,Z’
m,U
n,Z
![Page 99: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/99.jpg)
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rmat
ionst
eknolo
gi
UCb
Passed
Waiting Final
Init
n,Z’
m,U
n,Z
Passed=Bitarray
1
0
1
0
0
1
UPPAAL8 Mbits
HashfunctionF
Under-approximationBitstate Hashing
![Page 100: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/100.jpg)
ModellingPatterns
![Page 101: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/101.jpg)
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rmat
ionst
eknolo
gi
UCb
Variable Reduction
Reduce size of state space by explicitelyresetting variables when they are not used!
Automatically performed for clock variables (active clock reduction)
![Page 102: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/102.jpg)
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ionst
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gi
UCb
x is only active in location S1
x>3x<5
x:=0
x:=0
S x is inactive at S if on all path fromS, x is always reset before beingtested.
Definitionx<7
Clock Reduction
![Page 103: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/103.jpg)
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rmat
ionst
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gi
UCb
Synchronous Value Passing
![Page 104: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/104.jpg)
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rmat
ionst
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gi
UCb
Atomicity
To allow encoding of control structure (for-or while-loops, conditionals, etc.) without erroneous interleaving
To allow encoding of multicasting.
Heavy use of committed locations.
![Page 105: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/105.jpg)
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rmat
ionst
eknolo
gi
UCb
Bounded Liveness
Intent: Check for properties that are guaranteed to hold eventually within some upper (time) bound.
− Provide additional information (with a valid bound).
− More efficient verification.− φ leadsto≤t ψ reduced to A□(b⇒z ≤ t)
with bool b set to true and clock z reset when φ starts to hold. When ψ starts to hold, set bto false.
![Page 106: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/106.jpg)
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rmat
ionst
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gi
UCb
Bounded Liveness
The truth value of b indicates whether or not ψ should hold in the future.
φ
ψ
¬ψ
¬φ
b=truez=0
b=false
b true, check z ≤ t
b=false
A[] (b imply z≤t)b --> not b (for non zenoness)E<> b (for meaningful check)
![Page 107: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/107.jpg)
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rmat
ionst
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gi
UCb
Timers
Parametric timer:(re-)start(value)start! var=value
expired?active (bool)active go?(bool+urgent chan)time-out eventtimeout?
Declare ‘to’ with a tight range.
![Page 108: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/108.jpg)
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ionst
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gi
UCb
Zenoness
Problem: UPPAAL does not check for zenonessdirectly.
− A model has “zeno” behavior if it can take an infinite amount of actions in finite time.
− That is usually not a desirable behavior in practice.− Zeno models may wrongly conclude that some
properties hold though they logically should not.− Rarely taken into account.
Solution: Add an observer automata and check for non-zenoness, i.e., that time will always pass.
![Page 109: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/109.jpg)
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ionst
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gi
UCb
Zenoness
x≤1 x≤1x=0
ZenoOK Detect by•adding theobserver:
Constant (10) can be anything(>0), but choose it well w.r.t.your model for efficiency.Clocks ‘x’ are local.
•and check the propertyZenoCheck.A --> ZenoCheck.B
x ≥ 1x==1
![Page 110: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/110.jpg)
Compositionality &Abstraction
![Page 111: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/111.jpg)
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rmat
ionst
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gi
UCb
The State Explosion Problem
a
cb
a
cb
a
cb
a
cb
a
cb
a
cb
a
cb
a
cb
ϕ sat
Model-checking is either EXPTIME-complete or PSPACE-complete (for TA’s this is true even for a single TA)
Model-checking is either EXPTIME-complete or PSPACE-complete (for TA’s this is true even for a single TA)
Sys
![Page 112: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/112.jpg)
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ionst
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UCb
Abstraction
ϕϕ
satSys AbsSys satAbs ≤
a
cb
a
cb
a
cb
a
cb
a
cb
a
cb
a
cb
a
cb
ϕ sat
Sys
1 2
43ϕ sat
AbsREDUCE TO Preserving safety
properties
![Page 113: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/113.jpg)
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ionst
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UCb
Compositionality
AbsSysAbsAbs |Abs Abs Sys AbsSys
21
22
11
≤≤
≤≤
a
cb
a
cb
a
cb
a
cb
a
cb
a
cb
a
cb
a
cb
Sys
1 2
43
1 2
43
Sys1 Sys2
Abs1 Abs2
2121
22
11
Abs |AbsSys |Sys Abs Sys AbsSys
≤≤
≤
![Page 114: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/114.jpg)
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gi
UCb
Abstraction & Compositionality
dealing w stateexplosion
a
cb
a
cb
a
cb
a
cb
a
cb
a
cb
a
cb
a
cb
1 2
43
1 2
43
2121
2211AACC
ACACp
pp
Concrete Abstract
”trace” inclusion
![Page 115: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/115.jpg)
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UCb
Abstraction Example
a1 a2 a3 a4 a5
a b
![Page 116: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/116.jpg)
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UCb
Example Continued
abstractedby
![Page 117: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/117.jpg)
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UCb
Proving abstractionsusing reachability
A[] not TestAbstPoP1.BAD
Recognizesall the BADcomputationsof PoP1
Henrik Ejersbo Jensen PhD Thesis 1999
Applied to
IEEE 1394a Root contention protocol(Simons, Stoelinga)
B&O Power Down Protocol(Ejersbo, Larsen, Skou, FTRTFT2k)
![Page 118: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/118.jpg)
Further Optimizations
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UCb
x1 x2
x3x0
-4
4
22
5
3 3 -2 -2
1
Datastructures for Zones
Difference Bounded Matrices (DBMs)
Minimal Constraint Form
[RTSS97]
Clock Difference Diagrams
[CAV99]
PW List [SPIN03]
![Page 120: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/120.jpg)
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ionst
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UCb
x1 x2
x3x0
-4
4
22
5
3 3 -2 -2
1
Datastructures for Zones
Difference Bounded Matrices (DBMs)
Minimal Constraint Form
[RTSS97]
Clock Difference Diagrams
[CAV99]
PW List [SPIN03]
Elegant RUBY bindings for easy implementations
Alexandre David
![Page 121: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/121.jpg)
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UCb
Zone Abstractions
Abstraction taking maximum constantinto account necessary for termination
Utilization of distinction between lower and upper bounds
Utilization of location-dependency
[TACAS03,TACAS04]
![Page 122: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/122.jpg)
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UCb
LU Abstraction
THEOREMFor any state in the LU- abstraction there is a state
in the original set simulating it
LU abstraction is exact wrt reachability
[TACAS04]
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Zone abstractions
Classical Loc. dep. Max Loc. dep. LU Convex Hull
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Symmetry Reduction
Exploitation of full symmetry may give factorial reduction
Many timed systems are inherently symmetric
Computation of canonical state representative using swaps.
[Formats 2003]
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Symmetry Reduction
SWAP: 1 2 ; 3 4
Exploitation of full symmetry may give factorial reduction
Many timed systems are inherently symmetric
Computation of canonical state representative using swaps.
[Formats 2003]
![Page 126: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/126.jpg)
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Symmetry Reduction
[Formats 2003]
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Symmetry Reduction
[Formats 2003]
10Gerd Behrmann
UPPAAL 3.6
Iterators for (i: int[0,4]) { }
Quantifiers forall (i: int[0,4]) a[i]==0
Selection select i: int[0,4]; guard...
Template sets process P[4](...) { }
Scalar set based symmetry reductionCompact state-space representationsPriorities
Martijn Henriks, Nijmegen U
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D-UPPAALGerd Behrmann
Distributed implementation of UPPAAL on PC-cluster [CAV'00, PDMC'02, STTT'03].WWW frontend availableApplications
− Synthesis of Dynamic Voltage Scaling strategies (CISS).
− Ad-hoc mobile real-time protocol (Leslie Lamport) - 25GB in 3 min!
Running on NorduGrid.Local cluster: 50 CPUs and 50GB of RAMTo be used as inspiration for verification GRID platform within ARTIST2 NoE.
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D-UPPAALGerd Behrman
Distributed implementation of UPPAAL on PC-cluster [CAV'00, PDMC'02, STTT'03].WWW frontend available.Applications
− Synthesis of Dynamic Voltage Scaling strategies (CISS).
− Ad-hoc mobile real-time protocol (Leslie Lamport) - 25GB in 3 min!
Running on NorduGrid.Local cluster: 50 CPUs and 50GB of RAMTo be used as inspiration for verification GRID platform within ARTIST2 NoE.
![Page 130: Timed Automata – lllllllllll Decidability Results · Informationsteknologi UCb Regions Finite Partitioning of State Space x y An equivalenceclass(i.e. a region) in factthereis onlya](https://reader030.vdocuments.site/reader030/viewer/2022041208/5d65f2d988c9936e7d8b509d/html5/thumbnails/130.jpg)
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D-UPPAALGerd Behrman
Distributed implementation of UPPAAL on PC-cluster [CAV'00, PDMC'02, STTT'03].Applications
− Synthesis of Dynamic Voltage Scaling strategies (CISS).
− Ad-hoc mobile real-time protocol (Leslie Lamport) - 25GB in 3 min!
Running on NorduGrid.Local cluster: 50 CPUs and 50GB of RAMTo be used as inspiration for verification GRID platform within ARTIST2 NoE.