On interleaving in {P {P,A}-Tim ime P e Pet etri i net ets s - - PowerPoint PPT Presentation

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On interleaving in {P {P,A}-Tim ime P e Pet etri i net ets s - - PowerPoint PPT Presentation

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On interleaving in {P {P,A}-Tim ime P e Pet etri i net ets s with stron rong semantics Hanifa Boucheneb (1) , Kamel Barkaoui (2) (1) Laboratoire VeriForm , cole Polytechnique de Montral (Canada) (2) Laboratoire CEDRIC,. CNAM (France)

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SLIDE 1

On interleaving in {P {P,A}-Tim ime P e Pet etri i net ets s with stron rong semantics

Hanifa Boucheneb(1), Kamel Barkaoui(2)

(1) Laboratoire VeriForm , École Polytechnique de Montréal (Canada) (2) Laboratoire CEDRIC,. CNAM (France)

1 21-24 September, Infinity’10, Singapore

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SLIDE 2

Outline

  • Reachability analysis of timed models
  • Interleaving in {P,A,T}-TPN  SCG and CSCG
  • Conclusion

2 21-24 September, Infinity’10, Singapore

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SLIDE 3

Reachability analysis of timed models

Reachabilty properties

Counter-example

Property not satisfied

Model

checki king

Property satisfied

A finite/ infinite set of states = an abstract state I nfinite transition system

 Abstraction  a finite representation which preserves properties of interest.  Challenge:

  • More coarser abstraction preserving properties of interest.
  • Computed with minor resources (time and space).

Abstraction

3 21-24 September, Infinity’10, Singapore

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SLIDE 4

Reachability analysis of timed models

Linear properties

State space abstractions in the literature preserving linear properties:

  • State Class Graph (SCG),
  • Contracted State Class Graph (CSCG),
  • Zone graphs.

 may differ in:

  • Characterization of states (interval states or clock states),
  • Agglomeration criteria of states,
  • Size

 Three levels of abstraction

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SLIDE 5

Three levels of abstraction:

1 . Tim e abstraction OR 2 . States reachable by the sam e firing sequence independently of their firing tim es are grouped in the sam e node. 3 . The grouped states are then considered m odulo som e relation

  • f equivalence  Abstract states ( state classes, state zones)

Reachability analysis of timed models

θ t s1 s2 t s1 s2 s’1 θ t s1 s2 s’1 s1 s2 s’1 t t1 t1 t2 t2 t2 t2 t3 t1 t1 t1 t1 t1 t2 t2 t3 t1 t1 t2 t2 t3

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SLIDE 6

Reachability analysis of timed models

 Finite reachability graphs for bounded {P,A,T}-TPN and timed automata  Reachability problem is decidable.  State explosion problem: Abstract states reached by different interleavings of the same set of transitions are in general not equal.  Abstraction by inclusion 6

t1

1 4

t1 t2 t2

2 3 5

t3

5 ⊆ 6

6

t1

1 4

t1 t2 t2 t3

2 3

6 21-24 September, Infinity’10, Singapore

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SLIDE 7

Reachability analysis of timed models

Abstraction by convex-union Convex-union abstractions are much more compact than inclusion abstractions Test of convexity  very expensive operation: Smallest-enclosing-DBM (5,6) – 5 ⊆ 6 6

t1

1 4

t1 t2 t2

2 3 5

t3

6

t1

1 4

t1 t2 t2 t3

2 3 5

5 ∪ 6 is convex?

7 21-24 September, Infinity’10, Singapore

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SLIDE 8

Reachability analysis of timed models

 Approach of Maler et al. (2006): CCS-like composition of timed automata

  • compute abstract states in breadth-first manner,
  • group abstract states reached by different interleavings of the same set of

transitions.  The union of abstract states reached by different interleavings of the same set of transitions is convex  Test of convexity is not needed 16 abstract states 12 abstract states

1 2 3 4 6 7 8 9 10 12 11 5 13 14 15 16 t1 t2 t3 t3 t2 t1 t3 t2 t1 t3 t3 t1 t1 t2 t2 1 2 3 4 6 7 8 9 12 13 11 16 14 15 t1 t2 t3 t3 t2 t1 t3 t2 t1 t3 t1 t2 5 10

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SLIDE 9

Interleaving in {P,A,T}-TPN

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P-TPN A-TPN T-TPN

 Availability intervals of tokens  Implicit / explicit firing intervals  Firing intervals of transitions  Strong or weak time semantics  A transition

  • cannot fire outside its firing interval
  • Strong time semantics  cannot loose its firability by time progression
  • Weak time semantics  may loose its firability by time progression
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SLIDE 10

Interleaving in {P,A,T}-TPN

What about expressiveness?

10 21-24 September, Infinity’10, Singapore

Some models are incomparable More expressive model

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SLIDE 11

Interleaving in P-TPN

P-TPN model

p [a,b]  A token created in p, at date θ, is (unless it is consumed):

unavailable in [θ, θ+a [

available in [ a+θ, b+θ ]

dead token in ] b+θ, ∞ [

A transition t is firable if all its required tokens are available.

Its firing takes no time. State = (M, Deadp, Ip)

11 21-24 September, Infinity’10, Singapore

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SLIDE 12

Interleaving in P-TPN

s0 s2 s1 s5 s3

(p1+ p2, ∅, I(p1) = [0,2], I(p2)=[1,3]) (p2+ p3, ∅, I(p2) = [1,3], I(p3)=[1,1]) (p1+ p2, ∅, I(p1)=[0,1], I(p2) = [0,2]) (p2+ p3, ∅, I(p2) = [0,2], I(p3)=[1,1])

1 2 t1 t2

s4

t1

State s = (M, Deadp, Ip)

(M,Deadp,Ip) --- d  (M,Deadp,Ip`) iff ∀p∈ M-Deadp, d ≤ ↑Ip(p) and Ip’(p)=[Max(0, ↓Ip(p)-d), ↑Ip(p) – d]

(M,Deadp,Ip) --- t  (M’,Deadp,Ip’) iff Pre(t) ⊆ M - Deadp, ∀p∈ Pre(t), ↓Ip(p) =0 M’= (M – Pre(t)) ∪ Post(t), ∀p’∈ M’-Deadp, Ip’(p’)= Ip(p’) if p’ ∉ Post(t), and Ip’(p’) = Isp(p’) otherwise.

(M,Deadp,Ip) --- Err  (M, Deadp’,Ip`) iff No friable transition and no time progression from (M,Deadp,Ip) Deadp’ = Deadp ∪ {p’∈M-Deadp | ↑Ip(p’)=0 }, ∀p’∈ M-Deadp’, Ip’(p’)= Ip(p’).

cannot over-pass intervals of non dead tokens

All tokens of t have reached their intervals timelock state

…

(p1+ p4, ∅, I(p1) = [0,1], I(p4)=[2,2])

21-24 September, Infinity’10, Singapore

(p1+ p2, ∅, I(p1) = [1,3], I(p2)=[2,4])

Semantics

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SLIDE 13

Interleaving in P-TPN

…

s0 s2 s1 s5 s3

(p1+ p2, ∅, I(p1) = [0,2], I(p2)=[1,3]) (p2+ p3, ∅, I(p2) = [1,3], I(p3)=[1,1]) (p1+ p2, ∅, I(p1)=[0,1], I(p2) = [0,2]) (p2+ p3, ∅, I(p2) = [0,2], I(p3)=[1,1])

1 2 t1 t2

1 ≤ 3 ∧ 1 ≤ 4 2 ≤ 3 ∧ 2 ≤ 4

s4

t1

(p1+ p4, ∅, I(p1) = [0,1], I(p4)=[2,2])

State class { states reached by the same firing sequence } = (M, Deadp, φ)

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(p1+ p2, ∅, I(p1) = [1,3], I(p2)=[2,4])

SCG

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SLIDE 14

Interleaving in P-TPN

State class = (M, Deadp, φ) = { states reached by the same firing sequence }

(M, Deadp, φ) –t-> (M’,Deadp’, φ’) iff

  • φ ∧ /\pf ∈Pre(t), pi ∈ M-Deadp

pf – pi ≤ 0 is consistent

  • M’ = (M – Pre(t)) ∪ Post(t), Deadp’= Deadp,
  • φ’ ?
  • φ’ =φ ∧ /\pf ∈Pre(t), pi ∈ M-Deadp

pf – pi ≤ 0

  • Rename each pf s.t. pf ∈ Pre(t) in t
  • Add /\pn ∈Post(t), ↓Isp(pn) ≤ pn – t ≤ ↑ Isp(pn)
  • Replace each pi by pi + t and eliminate pi.

SCG is finite for all bounded P-TPNs and preserves linear properties

14 21-24 September, Infinity’10, Singapore

SCG

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SLIDE 15

Interleaving in P-TPN

In the P-TPN SCG, the union of state classes reached by different interleavings of the same set of transitions is not necessarily convex.

15 21-24 September, Infinity’10, Singapore

c0

(p1+ p2, ∅, 1 ≤ p1 ≤ 3 ∧ 2 ≤ p2 ≤ 4)

c2 c1 c4 c3

(p1+ p4, ∅, 0 ≤ p1 ≤ 1 ∧ p4 = 2) (p3+ p4, ∅, p3 = 1 ∧ 1 ≤ p4 ≤ 2) (p2+ p3, ∅, 0 ≤ p2 ≤ 3 ∧ p3 = 1) (p3+ p4, ∅, 0≤ p3 ≤ 1 ∧ p4 = 2)

t2 t1 t1 t2

1 ≤ p1 ≤ 3 ∧ 2 ≤ p2 ≤ 4 ∧ p2 ≤ p1 1 ≤ p1 ≤ 3 ∧ 2 ≤ p2 ≤ 4 ∧ p2 ≤ p1

c3 ≠ c4 c3 ⊄ c4 c4 ⊄ c3 c3 ∪ c4 is not convex

SCG

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SLIDE 16

Interleaving in P-TPN

c0

(p1+ p2, ∅,

  • 3 ≤ p1 - p2 ≤ 1)

c2 c1 C4 c3

(p1+ p4, ∅,

  • 2 ≤ p1 - p4 ≤ -1)

(p3+ p4, ∅,

  • 1 ≤ p3 -p4 ≤ 0)

(p2+ p3, ∅,

  • 1 ≤ p2 - p3 ≤ 2)

(p3+ p4, ∅,

  • 2≤ p3 - p4 ≤ -1)

t2 t1 t1 t2

  • 3 ≤ p1 - p2 ≤ 1

∧ p2 - p1 ≤ 0

  • 3 ≤ p1 - p2 ≤ 1

∧ p2 - p1 ≤ 0

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CSCG

 CSCG is the quotient graph of the SCG w.r.t. ≈: (M, Deadp, φ) ≈ (M’, Deadp’, φ’) M= M’, Deadp = Deadp’ and φ’ and φ’ have the same triangular constraints  ≈ is a bisimulation over the SCG

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SLIDE 17

Interleaving in P-TPN

c0

(p1+ p2, ∅,

  • 3 ≤ p1 - p2 ≤ 1)

c2 c1 C4 c3

(p1+ p4, ∅,

  • 2 ≤ p1 - p4 ≤ -1)

(p3+ p4, ∅,

  • 1 ≤ p3 -p4 ≤ 0)

(p2+ p3, ∅,

  • 1 ≤ p2 - p3 ≤ 2)

(p3+ p4, ∅,

  • 2≤ p3 - p4 ≤ -1)

t2 t1 t1 t2

  • 3 ≤ p1 - p2 ≤ 1

∧ p2 - p1 ≤ 0

  • 3 ≤ p1 - p2 ≤ 1

∧ p2 - p1 ≤ 0

c3 ≠ c4 c3 ⊄ c4 c4 ⊄ c3 c3 ∪ c4 is convex Theorem In the P-TPN CSCG, the union of state classes reached by different interleavings of the same set of transitions is convex.

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CSCG

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SLIDE 18

Interleaving in A-TPN

(p,t) [a,b]  A token created in p, at date θ, is (unless it is consumed):

unavailable in [θ, θ+a [ for t

available in [ a+θ, b+θ ] for t

dead token in ] b+θ, ∞ [ for t

A transition t is firable if all its input arcs are available.

Its firing takes no time. State = (M, Deada, Ia)

18 21-24 September, Infinity’10, Singapore

A-TPN model

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SLIDE 19

Interleaving in A-TPN

State s = (M, Deada, Ia)

(M,Deada,Ia) --- d  (M,Deada,Ia`) iff ∀(p,t)∈ EE(M)-Deada, d ≤ ↑Ia(p,t) and Ia’(p,t)=[Max(0, ↓Ia(p,t)-d), ↑Ia(p,t) – d]

(M,Deada,Ia) --- t  (M’,Deada,Ia’) iff Pre(t)x{ t)⊆ EE(M) - Deada, ∀p∈ Pre(t), ↓Ia(p,t) =0 M’= (M – Pre(t)) ∪ Post(t), ∀(p’,t`)∈ EE(M’)-Deada, Ia’(p’,t’)= Ia(p’,t’) if p’ ∉ Post(t), and Ia’(p’,t’) = Isa(p’,t’) otherwise.

(M,Deada,Ia) --- Err  (M, Deada’,Ia`) iff No friable transition and no time progression from (M,Deada,Ia) Deada’ = Deada ∪ {(p’,t’)∈EE(M)-Deada | ↑Ia(p’,t’)=0 }, ∀p’∈ EE(M)-Deada’, Ia’(p’,t’)= Ia(p’,t’).

cannot over-pass intervals of non dead arcs All input arcs of t have reached their intervals timelock state

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Semantics

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SLIDE 20

Interleaving in A-TPN

c3 ≠ c4 c3 ⊄ c4 c4 ⊄ c3 c3 ∪ c4 is not convex In the A-TPN SCG, the union of state classes reached by different interleavings of the same set of transitions is not necessarily convex.

20 21-24 September, Infinity’10, Singapore

c0

(p1+ p2, ∅, 1 ≤ pt1 ≤ 3 ∧ 2 ≤ pt2 ≤ 4)

c2 c1 c4 c3

(p1+ p4, ∅, 0 ≤ pt1 ≤ 1 ∧ pt4 = 2) (p3+ p4, ∅, pt3 = 1 ∧ 1 ≤ pt4 ≤ 2) (p2+ p3, ∅, 0 ≤ pt2 ≤ 3 ∧ pt3 = 1) (p3+ p4, ∅, 0≤ pt3 ≤ 1 ∧ pt4 = 2)

t2 t1 t1 t2

1 ≤ pt1 ≤ 3 ∧ 2 ≤ pt2 ≤ 4 ∧ pt2 ≤ pt1 1 ≤ pt1 ≤ 3 ∧ 2 ≤ pt2 ≤ 4 ∧ pt2 ≤ pt1

SCG

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SLIDE 21

Interleaving in A-TPN

c0

(p1+ p2, ∅,

  • 3 ≤ pt1 - pt2 ≤ 1)

c2 c1 C4 c3

(p1+ p4, ∅,

  • 2 ≤ pt1 - pt4 ≤ -1)

(p3+ p4, ∅,

  • 1 ≤ pt3 -pt4 ≤ 0)

(p2+ p3, ∅,

  • 1 ≤ pt2 - pt3 ≤ 2)

(p3+ p4, ∅,

  • 2≤ pt3 - pt4 ≤ -1)

t2 t1 t1 t2

  • 3 ≤ pt1 - pt2 ≤ 1

∧ pt2 - pt1 ≤ 0

  • 3 ≤ pt1 - pt2 ≤ 1

∧ pt2 - pt1 ≤ 0

c3 ≠ c4 c3 ⊄ c4 c4 ⊄ c3 c3 ∪ c4 is convex

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Theorem : In the A-TPN CSCG, the union of state classes reached by different interleavings of the same set of transitions is convex.

CSCG

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SLIDE 22

Interleaving in T-TPN

 The union of state classes reached by different interleavings of the same set of transitions is not necessarily convex in the SCG and in the CSCG [Boucheneb et al. 2008].

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SLIDE 23

Conclusion

 The union of state classes reached by different interleavings

  • f the same set of transitions is:

not necessarily convex in the SCG of the { P,A} -TPN

convex in the CSCG of the { P,A} -TPN

is not necessarily convex in the SCG and CSCG of the T-TPN [ Boucheneb & al. 2008]

 A-TPN is the more powerful model [ Boyer & Roux 2008] and

then more suitable, than the T-TPN, for abstractions by convex-union.

 The translation of T-TPN into A-TPN needs to add several

places and transitions [ Boyer 2001] , which may offset the benefits of abstractions by convex-union.

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SLIDE 24

Conclusion

A-TPN

(11 places, 21 transitions, 42 arcs)

T-TPN

(3 places, 2 transitions, 4 arcs) [2,5] [1,3] β α

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 Too large model  Bad impact on analysis (complexity)

[ Boyer 2001]

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SLIDE 25

Thank you!

Questions?

25 21-24 September, Infinity’10, Singapore