Unfolings of Networks of Time Automt Frnk Cssez Thoms Chtin Clue Jr Ptrii Bouyer Serge H Pierre-Alin Reynier Rennes, Deemer 3, 2008
Unfolings [MMilln 93] First efine for Petri nets Then extene to other true onurreny moels [Esprz, Römer 99] Compt representtion of the exeutions Expliit representtion of onurreny Avoi omputtion of interlevings Moel-heking, ignosis, synhronous iruits Optimiztion: equte orers [Esprz, Römer, Vogler 02]
Time Automt [Alur, Dill 94] L, l 0, Σ, X, T, Inv Trnsitions t = ef l, g,, R, l, with ef soure: l = α(t) L trget: l = ef β(t) L ef gur: g = γ(t) ef lel: = λ(t) Σ ef resette loks: R = ρ(t) X
Time Automt: Semntis Stte l, or,θ lotion: l L urrent te: θ R te of ltest reset for every lok: x X or(x) θ The trnsition t n our t te θ θ from stte l, or,θ, if: the invrint of l is stisfie until te θ : θ or = Inv(l) l = α(t) the gur of t is stisfie t te θ : θ or = γ(t)
Exmple of Time Automton te: θ = 0 or(x) = 0
Exmple of Time Automton te: θ = 0.7 or(x) = 0 (, 0.7)
Exmple of Time Automton te: θ = 3 or(x) = 0 (, 0.7), (, 3)
Exmple of Time Automton te: θ = 3.5 or(x) = 0 (, 0.7), (, 3), (, 3.5)
Exmple of Time Automton te: θ = 4 or(x) = 4 (, 0.7), (, 3), (, 3.5), (, 4)
Exmple of Time Automton te: θ = 5 or(x) = 4 (, 0.7), (, 3), (, 3.5), (, 4), (, 5)
Networks of Time Automt synhroniztion y shre lels lol loks te: θ = 0 or(x) = 0 or(y) = 0 l 4 y 2
Networks of Time Automt synhroniztion y shre lels lol loks te: θ = 0.7 or(x) = 0 or(y) = 0 l 4 y 2 (, 0.7)
Networks of Time Automt synhroniztion y shre lels lol loks te: θ = 3 or(x) = 0 or(y) = 0 l 4 y 2 (, 0.7), (, 3)
Networks of Time Automt synhroniztion y shre lels lol loks te: θ = 4 or(x) = 0 or(y) = 4 l 4 y 2 (, 0.7), (, 3), (, 4)
Networks of Time Automt synhroniztion y shre lels lol loks te: θ = 4 or(x) = 4 or(y) = 4 l 4 y 2 (, 0.7), (, 3), (, 4), (, 4)
Networks of Time Automt synhroniztion y shre lels lol loks te: θ = 5 or(x) = 4 or(y) = 4 l 4 y 2 (, 0.7), (, 3), (, 4), (, 4), (, 5)
Networks of Time Automt synhroniztion y shre lels lol loks te: θ = 6 or(x) = 6 or(y) = 6 l 4 y 2 (, 0.7), (, 3), (, 4), (, 4), (, 5), (, 6)
Networks of Time Automt synhroniztion y shre lels lol loks te: θ = 7 or(x) = 6 or(y) = 6 l 4 y 2 (, 0.7), (, 3), (, 4), (, 4), (, 5), (, 6), (, 7)
Proesses of Networks of Untime Automt l 1 l 1 l 4
Proesses of Networks of Untime Automt l 1 l 1 l 4
Proesses of Networks of Untime Automt l 1 l 1 l 4 l 4,
Proesses of Networks of Untime Automt l 1 l 1 l 4 l 4 l 1 l 4,,
Proesses of Networks of Untime Automt l 1 l 1 l 4 l 4 l 1 l 4,,,
Proesses of Networks of Untime Automt l 1 l 1 l 4 l 4
Unfolings of Networks of Untime Automt l 1 l 1 l 4 l 4 l 1 l 4
Proesses of NTA l 1 (, 0.7), (, 4), (, 4), (, 5) (0.7) (4) l 4 (4) l 4 y 2 l 1 l 4 (5)
Proesses of NTA l 1 (, 0.7), (, 4), (, 4), (, 5) (0.7) (4) l 4 (4) l 4 y 2 l 1 l 4 (5) Other tes re possile with the sme struture
Symoli Proesses of NTA l 1 (,θ 1 ), (,θ 2 ), (,θ 3 ), (,θ 4 ) (θ 1 ) (θ 2 ) l 4 (θ 3 ) l 4 y 2 l 1 l 4 (θ 4 ) Other tes re possile with the sme struture prmeters
Symoli Proesses of NTA: Symoli onstrints inue y gurs invrints uslity onvex union of zones [Ben Slh, Bozg, Mler, 06] nlog of [Aur, Lilius, 00] for NTA (θ 1 ) l 1 (θ 3 ) (θ 2 ) l 4 θ 1 1 θ 3 θ 2 2 θ 4 θ 3 1 θ 1 θ 3 θ 2 θ 3 θ 3 θ 4 θ 4 θ 3 2 (θ 4 ) l 1 l 4
Diffiulties with Time in Unfolings In untime nets, fesility of n event is lol property. In NTA, it epens on the ontext. In simple se (no invrints), it is still lol property. l 4 y 2 (0.7) (3) l 1 l 4
Conurrent Opertionl Semntis for NTA How to simulte NTA without using loks, ut with s muh onurreny s possile? Look for lol onitions to ply trnsition. Exeutions must respet the usul semntis. Notion of prtil stte L: for eh utomton, either l i, or i,θ i or. or(x) =? or(y) = 0 l 4 y 2
Conurrent Opertionl Semntis for NTA How to simulte NTA without using loks, ut with s muh onurreny s possile? Look for lol onitions to ply trnsition. Exeutions must respet the usul semntis. Notion of prtil stte L: for eh utomton, either l i, or i,θ i or. or(x) = 0 or(y) =? l 4 y 2
Conurrent Opertionl Semntis for NTA How to simulte NTA without using loks, ut with s muh onurreny s possile? Look for lol onitions to ply trnsition. Exeutions must respet the usul semntis. Notion of prtil stte L: for eh utomton, either l i, or i,θ i or. or(x) = 0 or(y) =? l 4 y 2
Conurrent Opertionl Semntis for NTA How to simulte NTA without using loks, ut with s muh onurreny s possile? Look for lol onitions to ply trnsition. Exeutions must respet the usul semntis. Notion of prtil stte L: for eh utomton, either l i, or i,θ i or. or(x) = 0 or(y) = 0 l 4 y 2
Lol Conitions to Tke Trnsitions To tke t t θ from L, we wnt: for ll ontext S of L, t n our t θ from L S. We hve: t n our t θ from L S if { the utomt onerne y t gree no invrint in L S expires efore θ l 4 y 2
Lol Conitions to Tke Trnsitions To tke t t θ from L, we wnt: for ll ontext S of L, t n our t θ from L S. We hve: t n our t θ from L S if { the utomt onerne y t gree L is stle in S until θ l 4 y 2
Lol Stility Conition Intuition LSC(L,θ) = for ll ontext S of L, L is stle in S until θ. Completeness Glol sttes re stle until the te where one of their invrints expires.
Lol Stility Conition Severl hoies to efine LSC(L,θ): trivil hoie: L is glol stte. BHR: L involves ll the utomt tht hve invrints. more generi: L ontins enough informtion to hek tht no utomton of L my e fore to synhronize erlier thn θ with nother utomton.
A proposition for LSC(L, θ) Definition: LSC(L, θ) hols iff L ontins enough informtion to hek tht no utomton of L my e fore to synhronize erlier thn θ with nother utomton: i J L θ or i = Inv i (l i ) t Syn I t J L i I I t J L = t J L l i α i (t i ) i I t J L θ or i = γ i (t i ) i I t \ J L Inv(α i (t i )) true where I t is the set of utomt involve in trnsition t; J L is the set of utomt whose stte is efine in the prtil stte L.
Symoli Unfolings of NTA In symoli unfolings: keep trk of ll the prtil stte L (not only the prt tht prtiiptes in t) use re rs. Any onfigurtion (proess) of the unfoling mps (y removing the re rs) to pre-proess (i.e. prefix of proess) of the NTA. Use only miniml sets L to inrese onurreny. l 1 l 4 l 1 l 4
Conlusion onurrent opertionl semntis for NTA prmeterize lol stility onition solve onstrints on the tes of the events stuy of the form of the onstrints finite omplete prefix of the unfoling if there is no urgeny, the unfoling is simply the superimposition of the proesses