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. Sketch-de-preuve, Nous procédons, pour chacune des règles (TR1) et (TR2), en montrant

. @bullet-la-chaîne, V al@N " si V al = unset et V ar est de la forme undef ined(E, N )

. @bullet-la-chaîne, V al@V ar " sinon

. Enfin, les trois règles suivantes correspondent à toutes les étapes d'après la première concaténation : uoiLabel(E, L, N + 1) :? not undef (E, V ar1); V al1 = unset

. Théorème and P. Soit, P ) sa traduction en réseau Booléen. Soit I 0 une interprétation de P , et I 0 , I 1 , I 2 , . . . l'orbite de I 0 par rapport à P . Alors S(I 0 ) ? sy S(I 1 ) ? sy S(I 2 ) ? sy, est l'unique trace de la dynamique synchrone de B(P )

. Preuve and P. Soit, ) le RB obtenu à partir de P par la traduction introduite dans [Ino11] et donnée dans la section 5.3. Soient I une interprétation de P et s l'état global de B(P ) tel que S(I) ? sy s . Soit v i ? var(P ) une variable propositionnelle de P, Par construction de B(P ), v i ? V . Nous montrons d'abord que s = S(T P (I))

S. Soit, S. Carte, S. De, and O. =. , Nous dénotons par A = {a 1 , o m } l'ensemble des opérateur logiques de S, par O AN D (resp. O OR , O N OT ) l'ensemble des opérateurs logiques AND de S (resp. OR, NOT) Pour une activité a i ? A de S, nous dénotons par req(a i ) (resp. stim(a i ), inh(a i )) les sources de l'ensemble des stimulations nécessaires (resp. stimulations, inhibitions) ciblant a i . Pour un opérateur logique o i ? O de S, nous dénotons par in(o i ) l'ensemble des noeuds (activités ou opérateurs logiques) à la source d'un arc logique ciblant o i, Nous montrons maintenant la proposition zsh :1 : command not found : :w Soient ? T RAD (S) la traduction de S en SBGNLog-AF, ? ON T O l'ensemble des axiomes ontologiques de SBGNLog-AF limités à ceux traduisant les relations is_a Soit ?(S) le programme défini par : ?(S), pp.1-18

. Finalement, At (S) le programme obtenu de ?(S) en supprimant la notion de temps de ce programme, Nous appliquons les trois étapes de transformation à ? At (S) pour obtenir le programme ? At (S) f

+. (. Comme-body, M n?1 = I n?1 , par définition de M n , body + (R ) ? M Comme body ? (R) ? I n?1 = ?, M n?1 = I n?1 et que pour tout variable propositionnelle v l , v l ? M n?1 ssi present

I. Pour-une-interprétation-de-herbrand and ?. , nous dénotons par S t (I) l'état global de B(S) correspondant à I au temps t : S t (I) = (b 1 , . . . , b n ) où b i = 1 si present(a i , t) ? I

S. Étant-donné-un-graphe-d-'influences and R. Le, S) modélisant S d'après les principes généraux (B1?7) comporte une fonction Booléenne par activité de S. Choisir une fonction Booléenne à appliquer à chaque pas de temps revient donc à choisir exactement une activité de S. Ce choix peut être encodé par la règle ASP suivante, ) : activity(A)}1 :? time(T ), t < tmax. (A19), p.1

L. Ensuite, A16?18) ne peut se faire que pour l'activité qui a été choisie au temps T . Ceci peut être encodé en ajoutant à chacun de ces axiomes la condition apply(A, T ), pour former les nouveaux axiomes suivants : present(A, T ) :? time(T ); time(T ), A, vol.next activity

. Apply-(-a, not hasM odulator(A); present(A, T )

. Apply-(-a, hasM odulator(A); not hasStimulator(A)

. Not-hasp-resentinhibitor, A21) present(A, T ) :? time(T ); time(T

. Apply-(-a, hasM odulator(A); hasP resentStimulator

. Enfin, une activité qui n'a pas été choisie au temps T conserve sa valeur au temps T , qui est le successeur de T : present(A, T ) :? time(T ); time(T )

@. Rad, S) est la traduction de S en SBGNLog-AF écrite en ASP

@. On, ensemble des axiomes ontologiques de SBGNLog-AF écrits en ASP

. Considérons-Également-un-entier, qui est la version ASP du programme logique ? T (T max ), et le programme

D. Alors and . La, même façon que les traces finies de la dynamique synchrone peuvent être calculées à partir de ? T r (S, T max ), les traces de sa dynamique synchrone peuvent être calculées à partir de ? T r (S, T max ) Nous avons la propriété suivante

C. Propriété, . 1s, ·. Sy, S. Sy, and . Tmax, T max un entier positif non nul, et {M 1 Alors l'ensemble {s ? sy S 1 (M i )

. Notons-qu-'à-deux-modèles-stables-de-?-t-r-(-s and . Peut-correspondre-la-même-trajectoire, En effet, chaque modèle stable M i contient un ensemble d'atomes {apply(a i,t , t) | a i,t ? V, 0 ? t < T max } tel que, pour deux modèles stables M i et M j donnés, il existe un entier t pour lequel a i,t = a j,t . Or choisir, à un instant t donné, c'est-à-dire à partir d'un état global donné, d'appliquer telle fonction d'un RB plutôt que telle autre n'influe pas sur la trace partant de cet état

. @bullet-p-=-{p-1, p m } l'ensemble des processus de la carte

@. O-=-{o-1, o q } l'ensemble des opérateurs logiques de la carte

@. Pour-chaque-processus-p and ?. , nous dénotons par reac(p) (resp. prod(p)) l'ensemble des réactifs (resp. produits) de p qui ne sont pas de type source ou puits ; par req(p) (resp. stim, inh(p)) l'ensemble des stimulateurs nécessaires (resp. stimulateurs, inhibiteurs) qui modulent p

@. Pour-chaque-opérateur-logique-o, nous dénotons par in(o) l'ensemble des noeuds (EPNs ou opérateurs logiques) sources d'un arc logique entrant sur o

S. Nous-dénotons-par and . {s-1, S r } l'ensemble valide d'histoires considéré, et par ?S = S?S S l'ensemble des EPNs qui sont dans une histoire

R. Nous-encodons-la-carte-considérée-en-un, S. Défini-par-le-triplet-(-?, and T. , où ? est l'ensemble des automates de ce RA, S l'ensemble de ses états globaux, et T l'ensemble de ses transitions locales, suivant l'encodage défini dans la section 6