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Compatibility between shared
variable valuations in
timed automaton network modelchecking
Zhao Jianhua, Zhou Xiuyi,
Li Xuandong, Zheng Guoliang
Presented by ZHAO Jianhua
Background (Time Automata)
♦ A timed automaton can be viewed as a

conventional finite state automaton plus some
clock variables , which are used to
constraint time distances between events.
Clocks: x, y
E1:x < 5, y := 0
x<5
A

y<8
B
E2: y < 8, x := 0
Background
(timed automaton network)
♦ A timed automaton network is a finite set of

timed automata which interact with each
other.
♦ These timed automata may interact with
each other through a finite set of shared
variables.
♦ For each timed automaton network, an
equivalent timed automaton can be built.
Background
(timed automaton network)
♦ An example:
Clocks: x

Clocks: y
E11:x < 5, x:=0

x<5
A

v:=1

E21:y < 8, y:=0

x<8
B

E12: x < 8, x := 0
v==0

v==1

1
y<8

y<3
2

E12: y < 3, y := 0
v:=0
Background
(reachability analysis 1)
♦ Many interesting properties (for example,

safety) can be expressed as reachability of
locations of timed automata.
♦ Because the state spaces of timed automata
are infinite, model checking techniques can
not be applied to timed automaton directly.
– Symbolic representation of states are used in
automatically reachability analysis.
Background
(Symbolic States)
♦ A symbolic state of a timed automaton network is

a tuple (l,s, D)

– l is the global location of the network.
– s is the valuation of the set of shared variables.
– D is a conjunction of formulas like x-y<c.

♦ A symbolic state (l,s, D ) represents a set of

concrete states (l,s,v), where v satisfies D.
♦ Given a symbolic state S, the set of concrete states
which are reachable from a concrete state in S
through a given transition t can also be represented
as a symbolic state. We call it as the successor of S
w.r.t. t.
Background (Basic reachability
analysis algorithm 1)
Wait = { S0}, Passed = {}, where S0 is the initial symbolic state
while (Wait != {} ) do
{ S = a symbolic state in Wait;
Wait = Wait – {S}
for each transition t leaving S do
{ S’ = successor of S w.r.t. t;
if (S’!= Φ and S’ is not contained by any state in Passed)
Wait = Wait + {S’}
if (the location of S’ is the target location)
return true;
}
Passed = Passed + {S}
}
Background (Basic reachability
analysis algorithm 2)
♦ The algorithm explores the state space by

generating successors of generated states
continuously.
♦ The algorithm will not generated the successors of
a generated symbolic state (l,s, D1 ) only if
– another symbolic state (l, s, D2 ) containing (l,s, D1 ) has
already been generated.
– a symbolic state S1 contains another one S2, if the set of
concrete states represented by S1 contains the one
represented by S2.
Compatibility between shared
variable valuations
♦ A shared variable valuations s1 is compatible with s2

on a tuple (l,D) if for each transition e leaving l,
one of the following conditions holds.

– s1 and s2 are identical.
– The conjunction of D and g is false, where g is the time
guard of e.
– Neither s1 nor s2 satisfies the shared variable guards of e.
– The variable guard of e is satisfied by s1, and the
transition e sets s1 and s2 to two compatible variable
valuations.
An example of Compatibility
Shared variables: v1, v2
Clocks: y

Clocks: x
e11 : x > 5; v2 = 3
x:=0, v1:=0

B

A
e12 : x < 3; v1 = 3
x:=0, v1:=v1+1

M

e21 : y < 10;
v1:=v2+1, y:= 0

N

C

♦ (v1 = 3;v2 = 3) is compatible with (v1

= 2;v2 = 3) on ((A,M), (x>3 ^y<10))
Compatibility contain
♦ Definition 3. Let (l,s1,D1) and (l,s2,D2) be

two symbolic states of a timed
automaton network. We say (l,s1,D1)
compatibility contains (l,s2,D2)
– if s1 is compatible with s2 on (l, D1) and
– D1 contains D2.
A lemma about the compatibility
contain

♦ Lemma
– Let S1,and S2 be two symbolic states of a timed
automaton network. We have that all the locations
reachable from S2 are also reachable from S1 if S1
compatibilitycontainsS2.
♦ Intuitively, (l,s1,D1) is more like to reach the

target location than (l,s2,D2) is.
♦ The algorithm can avoid generating successors of
a generated symbolic state (l, s, D1 ) if
– another symbolic state which compatibility-contains (l,
s, D) has already been generated.

♦ This condition is weaker than the basic one.
Find the compatible valuations
♦ During the reachability analysis, if a symbolic state (l,s,D)

is generated, an algorithm can be used to find valuations
with which s is compatible on (l,D).
♦ This algorithm uses a backward propagation method to
compute such valuations based on the definition of
compatibility.
♦ All these valuations are recorded in valuation sets attached
to the generated states.
♦ For each generated state (l, s’,D’), it is compatibility
contained by (l,s,D) if D’ is contained by D and s is found
to be compatible with s’.
A compact data structure
♦ Let v1, v2, …, vn be a set of shared variables.

We proved that the attached valuation sets
can be represented as Cartesian products
s1× s2 × … × sn
♦ This observation leads to a compact data

structure to record the compatible shared
variable valuations.
The optimization
♦ The algorithm is optimized as follows
– A shared variable valuation set is attached to
each generated state. (using the compact data
structure)
– Avoid generating successor of (l,s,D) if there
is another generated state (l, s’, D’) such that s
is in the attached set of (l, s’, D’) and D’
contains D
– During the reachability analysis, the attached
sets are continuously expanded by backward
propagation.
The performance (1)
(The bounded retransmission protocol)
The performance (2)
(the Bang&Olufsion audio protocol)
♦ The optimized algorithm uses only about

40% memories as the original one does.

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Zhao

  • 1. Compatibility between shared variable valuations in timed automaton network modelchecking Zhao Jianhua, Zhou Xiuyi, Li Xuandong, Zheng Guoliang Presented by ZHAO Jianhua
  • 2. Background (Time Automata) ♦ A timed automaton can be viewed as a conventional finite state automaton plus some clock variables , which are used to constraint time distances between events. Clocks: x, y E1:x < 5, y := 0 x<5 A y<8 B E2: y < 8, x := 0
  • 3. Background (timed automaton network) ♦ A timed automaton network is a finite set of timed automata which interact with each other. ♦ These timed automata may interact with each other through a finite set of shared variables. ♦ For each timed automaton network, an equivalent timed automaton can be built.
  • 4. Background (timed automaton network) ♦ An example: Clocks: x Clocks: y E11:x < 5, x:=0 x<5 A v:=1 E21:y < 8, y:=0 x<8 B E12: x < 8, x := 0 v==0 v==1 1 y<8 y<3 2 E12: y < 3, y := 0 v:=0
  • 5. Background (reachability analysis 1) ♦ Many interesting properties (for example, safety) can be expressed as reachability of locations of timed automata. ♦ Because the state spaces of timed automata are infinite, model checking techniques can not be applied to timed automaton directly. – Symbolic representation of states are used in automatically reachability analysis.
  • 6. Background (Symbolic States) ♦ A symbolic state of a timed automaton network is a tuple (l,s, D) – l is the global location of the network. – s is the valuation of the set of shared variables. – D is a conjunction of formulas like x-y<c. ♦ A symbolic state (l,s, D ) represents a set of concrete states (l,s,v), where v satisfies D. ♦ Given a symbolic state S, the set of concrete states which are reachable from a concrete state in S through a given transition t can also be represented as a symbolic state. We call it as the successor of S w.r.t. t.
  • 7. Background (Basic reachability analysis algorithm 1) Wait = { S0}, Passed = {}, where S0 is the initial symbolic state while (Wait != {} ) do { S = a symbolic state in Wait; Wait = Wait – {S} for each transition t leaving S do { S’ = successor of S w.r.t. t; if (S’!= Φ and S’ is not contained by any state in Passed) Wait = Wait + {S’} if (the location of S’ is the target location) return true; } Passed = Passed + {S} }
  • 8. Background (Basic reachability analysis algorithm 2) ♦ The algorithm explores the state space by generating successors of generated states continuously. ♦ The algorithm will not generated the successors of a generated symbolic state (l,s, D1 ) only if – another symbolic state (l, s, D2 ) containing (l,s, D1 ) has already been generated. – a symbolic state S1 contains another one S2, if the set of concrete states represented by S1 contains the one represented by S2.
  • 9. Compatibility between shared variable valuations ♦ A shared variable valuations s1 is compatible with s2 on a tuple (l,D) if for each transition e leaving l, one of the following conditions holds. – s1 and s2 are identical. – The conjunction of D and g is false, where g is the time guard of e. – Neither s1 nor s2 satisfies the shared variable guards of e. – The variable guard of e is satisfied by s1, and the transition e sets s1 and s2 to two compatible variable valuations.
  • 10. An example of Compatibility Shared variables: v1, v2 Clocks: y Clocks: x e11 : x > 5; v2 = 3 x:=0, v1:=0 B A e12 : x < 3; v1 = 3 x:=0, v1:=v1+1 M e21 : y < 10; v1:=v2+1, y:= 0 N C ♦ (v1 = 3;v2 = 3) is compatible with (v1 = 2;v2 = 3) on ((A,M), (x>3 ^y<10))
  • 11. Compatibility contain ♦ Definition 3. Let (l,s1,D1) and (l,s2,D2) be two symbolic states of a timed automaton network. We say (l,s1,D1) compatibility contains (l,s2,D2) – if s1 is compatible with s2 on (l, D1) and – D1 contains D2.
  • 12. A lemma about the compatibility contain ♦ Lemma – Let S1,and S2 be two symbolic states of a timed automaton network. We have that all the locations reachable from S2 are also reachable from S1 if S1 compatibilitycontainsS2. ♦ Intuitively, (l,s1,D1) is more like to reach the target location than (l,s2,D2) is. ♦ The algorithm can avoid generating successors of a generated symbolic state (l, s, D1 ) if – another symbolic state which compatibility-contains (l, s, D) has already been generated. ♦ This condition is weaker than the basic one.
  • 13. Find the compatible valuations ♦ During the reachability analysis, if a symbolic state (l,s,D) is generated, an algorithm can be used to find valuations with which s is compatible on (l,D). ♦ This algorithm uses a backward propagation method to compute such valuations based on the definition of compatibility. ♦ All these valuations are recorded in valuation sets attached to the generated states. ♦ For each generated state (l, s’,D’), it is compatibility contained by (l,s,D) if D’ is contained by D and s is found to be compatible with s’.
  • 14. A compact data structure ♦ Let v1, v2, …, vn be a set of shared variables. We proved that the attached valuation sets can be represented as Cartesian products s1× s2 × … × sn ♦ This observation leads to a compact data structure to record the compatible shared variable valuations.
  • 15. The optimization ♦ The algorithm is optimized as follows – A shared variable valuation set is attached to each generated state. (using the compact data structure) – Avoid generating successor of (l,s,D) if there is another generated state (l, s’, D’) such that s is in the attached set of (l, s’, D’) and D’ contains D – During the reachability analysis, the attached sets are continuously expanded by backward propagation.
  • 16. The performance (1) (The bounded retransmission protocol)
  • 17. The performance (2) (the Bang&Olufsion audio protocol) ♦ The optimized algorithm uses only about 40% memories as the original one does.