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) 21-24 September, Infinity’10, Singapore 1
Outline Reachability analysis of timed models Interleaving in {P,A,T}-TPN SCG and CSCG Conclusion 21-24 September, Infinity’10, Singapore 2
Reachability analysis of timed models Abstraction A finite/ infinite Model set of states = an abstract state checki king Reachabilty properties I nfinite transition system Counter-example Property Property not satisfied satisfied Abstraction a finite representation which preserves properties of interest. Challenge: More coarser abstraction preserving properties of interest. Computed with minor resources (time and space). 21-24 September, Infinity’10, Singapore 3
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
Reachability analysis of timed models Three levels of abstraction: 1 . Tim e abstraction θ t t s’ 1 s 1 s 2 s 1 s 2 OR θ t t s’ 1 s’ 1 s 1 s 2 s 1 s 2 2 . States reachable by the sam e firing sequence independently of their firing tim es are grouped in the sam e node. t3 t3 t2 t2 t2 t2 t1 t1 t1 t1 t3 t1 t1 t1 t2 t2 t1 t2 t2 t1 3 . The grouped states are then considered m odulo som e relation of equivalence Abstract states ( state classes, state zones) 5 21-24 September, Infinity’10, Singapore
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 5 ⊆ 6 4 t3 4 t3 2 t2 t2 5 2 t1 1 t1 1 t1 6 t1 3 t2 6 3 t2 6 21-24 September, Infinity’10, Singapore
Reachability analysis of timed models Abstraction by convex-union 5 ∪ 6 is convex? 4 4 t3 t3 t2 2 2 t2 5 t1 t1 1 5 1 3 3 t1 t1 6 6 t2 t2 Convex-union abstractions are much more compact than inclusion abstractions Test of convexity very expensive operation: Smallest-enclosing-DBM (5,6) – 5 ⊆ 6 7 21-24 September, Infinity’10, Singapore
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 t1 1 1 t1 t1 t3 t3 t2 t2 2 3 4 2 3 4 t2 t1 t3 t2 t3 t2 t1 t3 t2 t1 t3 5 6 7 8 9 10 5 10 6 7 8 9 t1 t2 t2 t3 t3 t1 t3 t2 t1 12 11 14 11 12 13 14 15 16 13 16 15 21-24 September, Infinity’10, Singapore 8 16 abstract states 12 abstract states
Interleaving in {P,A,T}-TPN 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 9 21-24 September, Infinity’10, Singapore
Interleaving in {P,A,T}-TPN What about expressiveness? More expressive model Some models are incomparable 10 21-24 September, Infinity’10, Singapore
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 [ State = (M, Deadp, Ip) 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. 21-24 September, Infinity’10, Singapore 11
Interleaving in P-TPN Semantics cannot over-pass State s = (M, Deadp, Ip) intervals of non dead (M,Deadp,Ip) --- d (M,Deadp,Ip`) iff tokens ∀ p ∈ M-Deadp, d ≤ ↑ Ip(p) and Ip’(p)=[Max(0, ↓ Ip(p)-d), ↑ Ip(p) – d] All tokens of t have (M,Deadp,Ip) --- t (M’,Deadp,Ip’) iff (p1+ p2, ∅ , reached their I(p1) = [1,3], Pre(t) ⊆ M - Deadp, ∀ p ∈ Pre(t), ↓ Ip(p) =0 s0 intervals I(p2)=[2,4]) M’= (M – Pre(t)) ∪ Post(t), 2 1 ∀ p’ ∈ M’-Deadp, Ip’(p’)= Ip(p’) if p’ ∉ Post(t), (p1+ p2, ∅ , and Ip’(p’) = Isp(p’) otherwise. I(p1)=[0,1], (p1+ p2, ∅ , I(p2) = [0,2]) s1 … s2 I(p1) = [0,2], (M,Deadp,Ip) --- Err (M, Deadp’,Ip`) iff I(p2)=[1,3]) t1 No friable transition and no time progression t2 t1 from (M,Deadp,Ip) s3 Deadp’ = Deadp ∪ {p’ ∈ M-Deadp | ↑ Ip(p’)=0 }, s4 s5 (p2+ p3, ∅ , ∀ p’ ∈ M-Deadp’, (p1+ p4, ∅ , timelock state (p2+ p3, ∅ , I(p2) = [1,3], Ip’(p’)= Ip(p’). I(p1) = [0,1], I(p3)=[1,1]) I(p2) = [0,2], I(p4)=[2,2]) I(p3)=[1,1]) 21-24 September, Infinity’10, Singapore
Interleaving in P-TPN SCG (p1+ p2, ∅ , I(p1) = [1,3], I(p2)=[2,4]) s0 … 1 ≤ 3 ∧ 1 ≤ 4 2 ≤ 3 ∧ 2 ≤ 4 2 1 (p1+ p2, ∅ , (p1+ p2, ∅ , s1 s2 I(p1)=[0,1], I(p2) = [0,2]) I(p1) = [0,2], I(p2)=[1,3]) t1 t2 t1 s3 (p2+ p3, ∅ , (p1+ p4, ∅ , s5 I(p2) = [1,3], I(p3)=[1,1]) s4 I(p1) = [0,1], I(p4)=[2,2]) (p2+ p3, ∅ , I(p2) = [0,2], I(p3)=[1,1]) State class { states reached by the same firing sequence } = (M, Deadp, φ ) 21-24 September, Infinity’10, Singapore 13
Interleaving in P-TPN SCG 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
Interleaving in P-TPN SCG (p1+ p2, ∅ , c0 1 ≤ p1 ≤ 3 ∧ 2 ≤ p2 ≤ 4) t2 1 ≤ p1 ≤ 3 ∧ 2 ≤ p2 ≤ 4 1 ≤ p1 ≤ 3 ∧ 2 ≤ p2 ≤ 4 t1 ∧ p2 ≤ p1 ∧ p2 ≤ p1 (p2+ p3, ∅ , (p1+ p4, ∅ , 0 ≤ p2 ≤ 3 ∧ p3 = 1) c1 c2 0 ≤ p1 ≤ 1 ∧ p4 = 2) t1 t2 (p3+ p4, ∅ , c3 ≠ c4 (p3+ p4, ∅ , 0 ≤ p3 ≤ 1 ∧ p4 = 2) c3 ⊄ c4 c3 c4 p3 = 1 ∧ 1 ≤ p4 ≤ 2) c4 ⊄ c3 c3 ∪ c4 is not convex In the P-TPN SCG, the union of state classes reached by different interleavings of the same set of transitions is not necessarily convex. 21-24 September, Infinity’10, Singapore 15
Interleaving in P-TPN 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 (p1+ p2, ∅ , ≈ is a bisimulation over the SCG -3 ≤ p1 - p2 ≤ 1) c0 -3 ≤ p1 - p2 ≤ 1 t2 -3 ≤ p1 - p2 ≤ 1 ∧ p2 - p1 ≤ 0 t1 ∧ p2 - p1 ≤ 0 (p2+ p3, ∅ , (p1+ p4, ∅ , -1 ≤ p2 - p3 ≤ 2) c1 c2 -2 ≤ p1 - p4 ≤ -1) t1 t2 (p3+ p4, ∅ , (p3+ p4, ∅ , -2 ≤ p3 - p4 ≤ -1) c3 C4 -1 ≤ p3 -p4 ≤ 0) 21-24 September, Infinity’10, Singapore 16
Interleaving in P-TPN CSCG (p1+ p2, ∅ , -3 ≤ p1 - p2 ≤ 1) c0 -3 ≤ p1 - p2 ≤ 1 t2 -3 ≤ p1 - p2 ≤ 1 ∧ p2 - p1 ≤ 0 t1 ∧ p2 - p1 ≤ 0 (p2+ p3, ∅ , (p1+ p4, ∅ , -1 ≤ p2 - p3 ≤ 2) c1 c2 -2 ≤ p1 - p4 ≤ -1) t1 t2 (p3+ p4, ∅ , (p3+ p4, ∅ , -2 ≤ p3 - p4 ≤ -1) c3 C4 -1 ≤ p3 -p4 ≤ 0) c3 ≠ c4 c3 ⊄ c4 c4 ⊄ c3 Theorem c3 ∪ c4 is convex In the P-TPN CSCG, the union of state classes reached by different interleavings of the same set of transitions is convex. 21-24 September, Infinity’10, Singapore 17
Interleaving in A-TPN A-TPN model (p,t) [a,b] A token created in p, at date θ , is (unless it is consumed): unavailable in [ θ , θ +a [ for t State = (M, Deada, Ia) 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. 21-24 September, Infinity’10, Singapore 18
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