Untimed systems ?????(??????)

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??????????????

• ??????????
• ???
• 2006.9.15

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??

• ???????
• Untimed systems ?????(??????)
• Untimed systems ?????(????LTL)
• ?????????(?????)
• ?????????(????)
• ?????????(????)
• ??????UPPAAL??
• ?????????????

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?? ??????????

• ?????
• ????????????????????????????????????
• 1. ????????????????(???????????????)
• 2. ?????????????(??????)
• 3. ?????????????

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Untimed systems ????(1)

• ????(Transition Systems)
• (Q,Q0, T ),
• ??Q?????,Q0?Q??????,
• T ? Q?Q?????.
• ????
• (??)????q0,q1,q2,.., ??
• (1)q0? Q0
• (2) ????i?0?, (qi, q(i1))?T .

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??1 ????

• (Q,Q0,T )
• Qon, off
• Q0 on
• T(on,on),(on,off), (off,on),(off,off)
• ????
• On,on,off,..,
• On, off,.
• ???on??,???on?off?????.

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Untimed systems ????(2)

• ??????(Labeled Transition Systems, LTS )
• (Q, Q0, ?, T ) ??
• Q ?????, Q0?Q??????,
• S ?????? ????(???), ???????????????,
• T? Q?S ?Q?????.

(??)????
??q0? Q0, ????????i ? (qi, ai , q(i1))?T
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??2 ??????

• (Q,Q0,?,T )
• Qon, off, Q0 on,
• ?a, b (??a ??????, b ??????)
• T(on,a,on),(on,b,off), (off,a,on),(off,b,off)
• ????

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Untimed systems ????(3)

• ?-automata (????????????)
• Büchi automata(Q,Q0,?,T,F), ??
• Q???????,
• Q0 ?Q??????,
• S ????????,
• T? Q ? S ? Q?????
• F ? Q?????.
• ???? F ???????????????.

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??3 Büchi automata

• (Q,Q0,?,T,F)
• Qon, off, Q0 on, ?a, b,
• T(on,a,on),(on,b,off), (off,a,on),(off,b,off)
• Foff
• ???? (??off??????)
• ??
• ???

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Untimed systems ????

• ????(Temporal Logic )
• ?????????????
• ??????(LTL, Linear Temporal Logic )
• ?????? CTL(Computation Tree Logic), (??)
• ?-?? (??)

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LTL ??

• ?P??????,LTL?????????
• ? p (?) (??/\ ?) ??(??\/
  ?)??????????????????(????)??(????) (O ?)
• ??,p?P?
• ???????? always, forever
• ?????? eventually, sometimes
• O ?????nexttime

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LTL ??

• ????
• ltgtp p ?????????.
• p ??????, p????.
• Op p???????

p
p
p
p
p
p
p
p
p
p
p
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?LTL????????

• ??? p
• ??? ltgtp
• ??? (p ? ltgtq)
• (?)??? ltgtp ? ltgtq
• (?)??? ltgt p ? ltgtq
• 
  
• ltgtp --- p??????
• ltgtp --- ??????, p?????

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?LTL???????? ?
?????????,?????? 1).????????,???????
(door_open /\ moving) 2). ??i?????????.
(elevator_atj /\ request_fori /\ i?j ? ltgt
(elevator_ati /\door_open))
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??????
?M???(??)??, ? ???LTL?? ?M ????????????,
????M?????, ?? M ? .
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?????????

• ?????????
• ?????(Timed Automata)
• Timed transition systems,
• Timed CSP, ?
• ????
• ???? TCTL(Timed Computation Tree Logic,
  MTL(Metric Temporal Logic), DC(Duration
  Calculus)?
• ????
• ????????????

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?????(Timed Automata)

• Timed Automata ?-automata clocks
• Clocks ????, ????????????,?????????
• ????
• 1. ???? ?????????
• 2. ???? ?????????

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??1???

• ???2???????

??????
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??2????

• ??????off??,??????,???light??,???????????brigh
  t??.
• ??????? 3????????.

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????? ????

• Locations
• Initial locations
• Clocks x, y, z,

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????? ????

• Clock interpretations

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????? ????

• Clock constraints
• Location invariants
• States

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????? ????

• Transitions

g ?????, a ?????, ? ?????????????.
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????
n
xlt5
xlt5 ? ygt3
a
x 0
1. Discrete transitions
m
a
( n , x2.4 , y3.5 ) ( m , x0 ,
y3.5 )
ylt10
2. Delay transitions
1.1
( n , x2.4 , y3. 5 ) ( n , x3.5
, y4.6 )
?????
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?????(TA) ??

• ??. A timed automaton A is a tuple ltL,L0,?,
  X,I,Egt, where
• 1). L is a finite set of locations,
• 2). L0 ? L is a set of initial locations,
• 3). ? is a finite set of synchronization
  labels,
• 4). X is a finite set of clocks,
• 5). I L?C(X) is a mapping from locations to
  clock constraints, and
• 6). E?L?C(X) ?? ?P (X) ? L is the set of
  transitions.

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?????

• Loff,on, L0off, ?press, Xx
• Ioff ? true, on?xlt2
• E(off,true,press,x,on),(on,x2, ?,?,off)

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????? (????)??

• ??. ? ???TA,
  A???????????LTS

T??????????
-- Delay transitions

-- Discrete transitions

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??-??
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??????????
?
(i1,2)????????,? ?
?????????????
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???????????
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?????????

• ????
• ????????p??,?q??2?????????
• ????????p??,?q?2???????????
• ..

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?????????

• ?????????????????
• ?????????
• ?
• ??(??)????T,??(???)??????
• ??freeze??
• ????????

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MTL??
MTL(Metric Temporal Logic) ????????
??p???????,I?????(? 1,2, (1,2, (1,2), (1,
?), 2, ?)??
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MITL??

• ??MTL??????

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????????

• ?? ??????M????? ? ,???? ?????
• ????(Model Checking)
• ????????(??????)?

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????
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????????

• ??????

??????
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Region equivalence

• ????????????????(regions)

y
x
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Region equivalence
Run (n,(0,0))? (n,(0.1,0.1)) ? (n,(1,1))
?(n,(1.2, 1.2)) ?(m,(1.2, 0)) ?(m,(1.5,0.3))
?(m,(2,0.8))?(m,(2.1,0.9))

• ????(Symbolic executions)

• (x0?y0)?(0ltxylt1 )
• ?(xy1)?(1ltxylt2)
• ?(1ltxlt2 ? y0 )
• ?(1ltxlt2?0ltylt1?1ltx-ylt2)
• ?(x2?0ltylt1?1ltx-ylt2)
• ?(xgt2 ? 0ltylt1?1ltx-ylt2)

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Region automata
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Region automata

• ???????TA???????????
• ?? Regions????,???????
• ???????????????
• Zones??xc,x-yc(??c???,?gt,lt, ,
  ?,?)????????,?regions??,??zone?????regions???

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Zones
DBMs (Difference Bound Matrices)
D(x,y)(?,c), means x-y?c D(x,y)(lt,c), means
x-yltc

• (xgt1,ylt2,0ltx-ylt1)

0 x y 0 (?,0)
(lt,-1) (?,0) x (lt,?) (?,0) (lt,1) y
(lt,2) (?,0) (?,0)
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??Zones?DBMs?????
1ltx, 1lty -2ltx-ylt3
1ltxlt4 1ltylt3
y
?? gt
n
x
x
xgt3
3ltx, 1lty -2ltx-ylt3
?? gt
a
x
y0
??y gt
3ltx, y0
m
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??DBM????????

• ????
• ???????? ??
• ??
• ?????

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??????????

• UPPAAL(UppsalaAalborg)
• http//www.uppaal.com/
• Kronos
• http//www-verimag.imag.fr/TEMPORISE/kronos/
• HyTech (The HYbrid TECHnology Tool)
• http//www-cad.eecs.berkeley.edu/tah/HyTech
• Red (Region Encoding Diagram)
• http//cc.ee.ntu.edu.tw/farn/red/
• PRISM (Probabilistic Symbolic Model Checker)
• http//www.cs.bham.ac.uk/dxp/prism

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UPPAAL??

• ????? urgent actions urgent locations
  committed locations data-variables (with
  bounded domains) arrays of data-variables
  guards and assignments over data-variables and
  arrays templates with local clocks,
  data-variables, and constants

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UPPAAL??

• Networks of Timed Automata

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UPPAAL????
clock x1, , xn int i1, , im chan a1, ,
ao const c1 n1, , cp np ? clock x, y int i,
J0 int0,1 k5 const delay 5, true 1, false 0
?????
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UPPAAL?????
????
location invariants
n
xlt5
?? ????
xlt5 , ygt3
a
clock guards
x 0
data guards
m
ylt10
g1
g3
g2
????
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Urgent Channels
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Urgent Locations
?urgent location?????????,????????.
(s00, R00, V)?(s01, R00 ,V) ? (s01, R01,V )
?(s02, R01, V) ??
(s00, R00, V)?(s01, R00 ,V) ? (s02, R00,V )
?(s02, R00, V0.1) ??
(s00, R00, V)?(s01, R00 ,V) ? (s01,
R00,V0.1 ) ?(s01, R01, V0.1 ) ???
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Committed Locations
?committed location ?????,???????????????co
mmitted location???????? ??????????
(s1,r11,r21)?(s2,r12,r21) ? (s3,r12,r22) ??
(s1,r11,r21)? (s2,r12,r21) ? (s2,r13,r21) ?
(s3,r13,r22) ???
????????
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UPPAAL??????

• A p (AG p)
• Eltgt p (EF p)
• E p (EG p)
• Altgt p (AF p)
• p--gtq AG(p imply AFq)
• p a.l gd gc p and p
• p or p not p p imply p
• ( p )

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??????

• ???????????????????????
• ??????????
• ??(???)????????
• ?????????????????
• ??????????

near
far
far
crossing
approach
exit
controller
lower
raise
gate
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??
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?????
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??
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?????????????????

• chan approach,exit,raise,lower
• clock waiting_time
• const a 5,b 3,c 2
• //const a 5,b 4,c 2
• //const a 5,b 3,c 3

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????

• A (train.crossing imply gate.down)
• A(not gate.up imply waiting_timelt10)

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?????????

• M????????, ? ???MTL??, ????????????M? ,?
• M ? ??? M M? ??
• ?M?M??????????, M M?
  ?????????????.

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• Question?

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• Thanks!