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Mahdi Dolati
Mahdi Dolati

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?.???????? ??????????????????? ?????????????? ???????? ???? ?4??????? ????? ????? ???????????? ?????? ???????? ???? ?????? ???? ???? ?????? ???? ???? ??????? ????? ???? ??????.?????? ???????? ????? ????? ???? ??????? ???????????? ?????? ??????????? ?????? ???? ????? ???????????????5????????? ????? ???????6??????? ???? ????:????? ??????? ????? ?????? ???? 1.?????? ?????? ???????? ???? ???? ???????????? ????? ????? ?????? ?????? ??????? ???? ?????? ???????? ????? ????? ??????? ???? ???????? .???? 2.?????? ?????? ?????????? ?????? ???? ???????? ???????? ???? ???? ?????? ??????? ??????? ?????? ???????? ????? ????? ??????? ???? ?????? .???? ????? ???? ?????????2???????? ?????t1????? ???? ???? ??? ????? ?????? ?????? ??????? ???? ?????? ???? ?????????????????? ????3??????? .????? ?????? ??????? ?????? ?????? ???????????? ???? ????? ??????? ??????? ?????? ?????? ????:?????? ??(?) = { ?(?, ?) , ??? (?, ?) ¡Ê ? 0 , ??? (?, ?) ? ? ?+(?) = { ?(?, ?) , ??? (?, ?) ¡Ê ? 0 , ??? (?, ?) ? ? ???? ????????????? ????? ??????t:????? ????? ?????? ??: ? ? ??(?) ¡Ü ?(?) ???????? ????? ???t??????? ????? ???????????????? ?????? ????? ???:????? ??: ? ? ?¡ä(?) = ?(?) ? ??(?) + ?+(?) ??????????????¡ä ????m??????? ????? ???? ????????? ????? ?????? ? ¡ú ? ?¡ä ???? ???m???????????? ????? ??????? ¡ú ? ? ?¡ä ????¡ú ? ? ??????? ????????? ???????¡ú ?????? ????? ???? ?????? ?????? ??????? ??????? ????.????? ?????? ?????? ???? ???????????? ?????? ????????? ????? ????? ????????????? ??????? ??????? ???? ?????? ???? ??????? ?????? ???? ??????? ??????? ????????? 0m.??????????????????? ????????? ?????????????????? ??????????? ????????????(?) ? {?¡ä |?0 ¡ú(?,?,?) ? ?¡ä }.????? ??????? ?????????????7N??????????????? ???¡ú ????? ???? ????? ???????????????????? ????????????(?)????? ????? ??????????? ??? ?????? ?????? ?????? ????8?????????? ?????? ???? ??? ???????????? ???? ?????? ????????????????? ??????? ???? ?????? ??????? ?????????? .????? ?.?????? ?????????? ???????????? ????? ?????? ???????? ???? ?????????????? ??????? ??????? ???????? ?????? ????? ??????? ?????? ???? ?????? ???????? ???? ???????? ???? ????????????? P1 3 2 21 1 P1 P2 P3 t1 t2 P4 ?????3?????? ???????????? ???2?????? ????? ???? ???? ????????t1 3 2 21 1 P1 P2 P3 t1 P1 t2 P4 ?????2?????? ?????????? ???
?????????? ?????? ???? ????? ?????? ?????????????? ????? ????????? .??????? ?????? ?????? ?????? ?????? ?.??????????????? ?????? ???? ???? ??????? ????????????? ???N?????? ???? ?????????? m????? ?m ? R(N)??????? ????? ???? ?????? .???? ???? ????? ???????? ??????? ?????? ????? ???? ????????????????? ????????? ???? ?????? ???? ???????????? ?????? ???? ???????????? ?????????????? ??????? ???????????? .?????? ?????? ???? ???? ??????? ?????????????? ???????????? ??????? ?????? ?????? ???????????????? ????????? ??????? ???? ????????.????? ?.???????????0 ?????? ???? )?(??????? ??????? ???? ??????? ?????????????????? ????? ??? ???? ????? ??????? ?????????. ?.??????????????? ?????? ???? ???? ?????? ?????????? ???k-??????????? ??????????? ???? ????? ???????????????????? ?????????? ?(????? ????????????)??????? ??????? ???? ???????k???? ?????? .?????? ???????? ??????????? ????? ????? ???? ????? 1-?????????? ?????? ??? ?????? ??????????? ????????? ?????? ????? ????? ????? k??????k-????????.?????? ????? ?????? ?????????? ???k-???????????? ?????????? ????? ???? ??? ??????????? ????? ??? ??????????? ???????? ??????.????????? ?????????? ??????????????? ?????????12??????? ?????? ?????? ???? ????? ??????????????????? ????? ?????? ??????.?????? ??????????? ???? ???? ????? ?????? ?????????? ??????? ????????????? ?????? ????? ????? ?????.?????? ???????? ???? ??????? ???????????? ?????? ?????? ???? ???????????? ?????? ?????? ???? ????? .????? ?????? ???????? ??????? ??????? ?.???????????? ?????????????????? ??????????? ??????????? ??????? ?????? ??????????? ???????? ????? ???? ???????????? ???? ???????? ???????? ????????? ???????? ??? ?????? ???? ?????? ???????????? ??????? ?????? ?????? ??????? ?????? ????? ????????? ?????? ?????? ????? ???? ?????? ??? ????????????????? ???? ??????? ???? ?????? ?????? ??? ?????? ??????? ???????????????? ??????????????? ???? ????? ???? ????? ???? ??????? ???? ????.????????? ?.??????? ???????????? ??????? ??????? ???? ?????? ????? ????? ??????? ???? ???????? ???? ??????? ?????????? ????? ??????? ????? ?????????? ?????? ?????.????????? ?.?.??????? ??????? ?????? ????????????? ?????? ???????? ???? ?????? ?????????????? ????? ????????? ?????? ?????? ???? ?????????? ???? .?????????? ????? ???? ???? ???? ?????????.?????????? ?????? ????? ????? ???? ???? ????? ?????.?????????? ???????? ?????? ????? ???? ?????? ?????t2?????? ??????????? (1, 1, 0, 5)????(1, 0, 1, 0).????? ?????????????? ????? ????? ?????????? ???????? ???? ?????? ???.????? ?.?.????????? ??????? ?????4?????? ????? ???? ?????5???????? ????? ????
??? ????? ?????????? ????? ????? ????? ???? ????????? ????????? ??????? ????? ???? ?????? ??????????????? ???? ????????? .?????t2????? ?????? ????? ?????? ???? ???????? ??????? ???? ?????P1???? ????? ????? ?????? ??????? ?????? ?????? ??????? ?????? ?????????? ?????P1????????.???? ??????? ?????? ???? ?????????????(1, 1, 0, 1)????(1, 0, 1, 1)???????? ?????. ?.?.???????????? ??????5 ????????? ?????? ???????? ????? ?????? ??????? ???????? ???? ????? ?????? ???? ?????? ????????????? ????????? .?????? ?????? ????? ???? ?????? ????? ???????????? ???? ??????? ??????? ????????? ????????? ????? ???? ?????? ???? ????????? ??????? ?????? ????? ???? ??????? ?????? ???????? ???? ?????? ????????? ?????? ??????? ???? ????? ??????????? ????????????? ???? .?????t1???? ?????? ??????? ???? ???????????P4???P3???? ???????1?????????? ????? ????? ????? ??? ????? ?????.??????????? ?????? ????? ????????? ?????? ??? ???? ????????.????? ?.?????????? ????????????? ????????? ??????? ???? ????? ????????? ?????????? ???? ????? ?????? ?????? ??????? ????????? ?????? ?????? ???? ????????? ???? ??????? ????????? ????? .????? ?????? ?????? ?????? ???? ?????? ??????? ???????? ?????? ????????????? ?????? ?????? ????? ?????? ???????????? ???? ??????? ???????? ??????????? ??????? ?????? ????? ???? ?????? ????? ?????? ????????? ?????????????? ???? ????? ????? ?????? ???? ???? ??? ?????????? ?????? ????????? ?????? ???????? ??????? ??? ????? ????? ??????? ???? ???????? ???? ????? ????? ?????? ???? ????????? ???? ?????? ????? ??????? ???? ????? ????????? ?????? ???? ???? ?????? ????? ?????.?????? ?.?.????????????? ?????? ?????????6 ?????? ?????????? ?????? ???????? ???? ?????? ?????? ???? ?????????? ???? ???????:????? ????????, I)0Z=(N?????? ????:????? ?????? ??? ????? ????? ?????? ??? 1.0S(Z) = N?????? ???????? ???? ???? ????? ?????? ??? ???????17Z???? ??????????? 2.? ¡Ã ? ¡ú ?0 + ¡Á (?0 + ¡È {¡Þ})???(t)2(t)¡ÜI1I ???? ??????t??? ????? ????????t))(2(t), I1I(t) = (I. ???? ??????? ????? ????I?????? ??????18?)(t1I????? ?????? ????????? ( ]????? ?????? ???? ?[??? ??????1?eft??? )(t)2I?????? ????????? ( ]????? ?????? ???? ?[??? ?????? ?????20lft)???? ?????????? .??????????? ???? ?????? ?????? ????? ???? ?????? ???? ????? ???????????? ??????I???? ???? .????? ??????? ??????? ??????? ????? ???? ?????? ??????? ???????? ????? ?????? ?????? ???? ??????? ?????? ???????????M??????? ?????? ?????? ???? ???? ? H?????? ????? ???????? ???? ?????? ???? ??????? ?????????? ????? ?????? ????????? ???? ??????? ???? ??????? ????? ???? ????? ?????? ???? ?????? ??????? ??????? ?????? ????? ??????? ?????? ???????????????? ?????? .????? ??????? ?????? ??????:????? ??????? ????? ????? ???? ????? ??????? ????? ?????, I)0W, m, F,Z=(P, T?????? ?????????? ?????? ???????? ??? ??????? ¡Ã ? ¡ú ?0 + ¡È {?}???????z = (m, h)?????? ???? ????? ?????? ??? ????? ????? ???????m?????? ??????? ??????? ???????h ?????? ?????????? ??????? ???????Z.??????? ??? ?????? ?????? ????? ???? ???? ??????? ????? ?????? ??????? ????? ???? ???? ????? ???????? ???? ?????? ??????????????? ???? ???? ?????? ???? ????? .??? ???? ??????? ????? ????? ???? ??????:?????? ?????6??????????? ????? ????
?????? ?????????????? ???Z?????? ????z=(m, h)?????? ?????? ???? ??????? ????z¡¯=(m¡¯, h¡¯)???? ????z ? ¡ú ?¡ä???? ?????????? ??? ?????? ????:????? ?????? ??? ????? ????? 1.m' = m 2.? ?: ???( ?) ¡Ù?¡ú ?( ?) + ? ¡Ü ???(?). ???????? ?????? ?????? ????????? ??????? ?????? ?????? ????? ???? 3.? ?: ? ? ?¡ä ( ?) = { ?( ?) + ? ??? ?? ¡Ü ? ? ??? ?? ? ? ???????? ?????? ????? ?????? ???? ?????? ????????????? ????z ?? ¡ú ?¡ä?????? ???????? ???? ??? ??????:????? ??????? ??? ????? ????? 1.??? ¡Ü ??????? ?????? ???????? ?????? ? 2.???(??) ¡Ü ?(??)???????? ???? ?????? ??????? ???? ?????? ??? ?????????????? ??????? 3.?¡ä = ? + ¦¤?? 4.??????? ???? ??????t:??????? ?¡ä(?) = { ? ??? ?? ? ?¡ä ?(?) ??? ?? ¡Ü ? ¡Ä ? ¡Ù ?? ¡Ä ?? ¡É ?? = ¦µ? 0 ?. ?. ?????? ????4-?????? ?????? ???? ??????? ????? ????? ????:????? ?? = {?: ? |?(?, ?) > 0} ?????? ???????? ?????? ????? ??????? ??????????? ?????????????? ????? ???????? ????? ????????? ???? ?????? ?????? ?????? ?????? ???? ??????? ????0???? ??? ???? ????????? ?????????? ?????????????????? ??????? ???????? ????? ? ???? ???? ???? ?????? ?????? ????? ??????0???? ?????????????? ??? ????? ??????? ?????? ?????? ?????? ????? ???? ???? ???????H(t)????? ???? ?????.????? ?????? ??????? ?????????? ???? ???????? ???? ?????? ??????? ??????? ?????? ??? ???? ??????? ?????? ???? ?????? ????.?????? ???????.?????7?????? ????????????? ????????????? ?????? ???? ????? .????? ?????? ??????????? ???7???? ????:?????? ?????? ????? ????? ???? ?????? ( ?0, ( 0 ? ? 0 ) ) ??????0???????? ????? ?????? ???? ??????? ???????????1???4??????? ???????? ??? ??????????2???3????? ???? ????? ?????? ?????? ????????? ?????? ???????? ???? ???????????1???4???????? ?????? ????? ????????????eft?????? ???? ????????0????? ????? .?????1.3???? ??????? ??????? ??????? ?????? ???? ????? ??????:????? ( ?1, ( 1.3 ? ? 1.3 ) ) ???????? ??????? ???? ?????? ????? ????1????????? ????? ????? ??????? )¡Ü1.31eft(t??????? ???? ?????? ??????? ?????? ???? ????? .????? ???? ????? ?????? ???? ???????:????? ( ?2, ( 2.3 ? ? 2.3 ) ) ???????? ?????? ????? ????4t???? ?????????? ????? ????? .????? ????? ??????? 4t???? ????? ?????? ???? ??????? ????? ?????:?????? ( ?3, ( 2.3 ? 0 ? ) ) ?????7?????? ?????????? ?????? ?????????? ?????? ???? ?????
? 5.????????????? ????????????? ?????? ?????????? ?????? ????????????????? ?????????? ???? ??????? ??????????????? ???? ??????? ????? ??????? ?????????? ??? ???????????? ????? .????? ????? ?????? ?????? ?????? ?????? ?????? ???????????? ???? ??? ?????? ???????? ??????? ???????? ????????? ???????????? ???? ????????????? ??????? ???? ???? ??????? ??????? ?????? ?????? ??????? ????? ???? .????? ???????? ???????? ?????? ????? ???????????? ??? ???????? ????? ??????? ????? ?????? ?????? ????? ??? ??????? ?????? ????????????????? ??? ???????? ??????????? ?????? ?????????? ?????? ???? .?????? ??????? ??????????? ?????? ????????? ???????? ????? ??????? ???????????? ??? ????????? ??? ??????????? ??????? ?????? ?????? ??????? ??????? ?????? ?????? ????????????? ????? ???? ??? ????? ?????? ?????? ???? .???? ?????? ??????? ?????? ?????? ???? ???????? ???? 6.??????? [1] Prof. Louchka Popova-Zeugmann, ¡°Time Petri Nets: Theory, Tools and Applications Part I¡±, Humboldt-Universit?t zu Berlin, Course Lecture [2] Prof. Morgan Magnin, ¡°Computational Approaches to Analyze Complex Dynamic Systems : Model-Checking and its Applica- tions¡±, NII International advanced lectures series on ICT: series of 4 lectures of 2 hours in Spring 2013 at National Institute of Informatics, , T¨­ky¨­, Japan [3] Louchka Popova-Zeugmann, ¡°Time and Petri Nets¡±, Springer, Chapter 2, 2013 [4] http://en.wikipedia.org/wiki/Petri_net 1 Transition 2 Place 3 Carl Adam Petri 4 Enabled 5 Nondeterministic 6 Fire 7 Reachability graph 8 State Space ? Operational semantic 10 Liveness 11 Boundedness 12 Structurally bounded 13 Reset arc 14 Read arc 15 Inhibitor arc 16 Time Petri net 17 Skeleton 18 Interval function 1? Earliest firing time 20 Latest firing time

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  • 1. ???????? ????? ?????? ????? ???? ????????????????? ?????????? ????? ??????? ?????? ?????????????? ?????????????? ????? mehdidolati@hotmail.com ?.??????? ?????????????? ???? ?????? ???????????????? ?????????? ????????????? ?????????????? ?????????? ??????????? .?????? ???? ?????? ?????????????? ????????? ????? ????????????? ???? ???? ???? ???????????? ?????????????????????1???????????????2????????? .???????????????? ?????? ???? ??????? ???????????? ?????? ???? ???? ?????? ??????? ?????? ????? ???? ??? ?????? ?????????? .??????????? ?????? ???? ???????? ?????? ??????? ??????? ????? ???? ??? ??????? ??????.?????????????????? ????????? ?????? ??????????? ?????? ???? ??????????/?? ????????? ???.??????? ???????? ???? ????????? ???????????? ?????? ?????? ???? ?????3??????????? ????? ???? ???? ??????1?3????? ????13???????? ??????? ?????? ???????.????? ?????? ??????? ????????? ????????????????? ????????????? ????????????? ?????????????????? ???????? ??????? ??? ??????? ????????? ?????? ?????????? ????????? ???????? ????????????.?????????? ???????????????? ??????????? ??????? ??????? ??????? ????.????? ???????? ??????? ???? ???? ?????? ?????? ????? ?.????? ?.???????? ????????????? ??? ?????? ????????? ??????????N=(P, T, F):???? ???? ???? ????? 1.P???T???????? ??????? ?????????? ?????? ???? ???? ????????? ???????? ???.??????? ???? 2.? ? (? ¡Á ?) ¡È (? ¡Á ?)?????????????????????????????????????? ???.???? ?????????1????????????????? ???:???? ???? ???? ????? ???????? ?????? 1.P = {P1, P2, P3, P4} T = {t1, t2} 2.F = {(P1, t1), (P2, t1), (P2, t2), (P3, t3), (t1, P4), (t2, P4)} ?????? ?????????? ??????? ???????? ???? ????? ????PN=(N, M, W) :???? 1.N = (P, T, F).????? ?????? ???? 2.?: ? ¡ú ????? ?????????? ????????????? ?????????? ???? ???????? ???????????? ??????? ??? ??????????? ???? ????? ???? ?????? ????.?????0m??????????????????? ?????????? .????? ??????? 3.?: ? ¡ú ???? ????????????? ???? ????? ???? ????? ???? ???? ????.????? ?????2????? ?????????????? ??????? ?????? ???? ?????? ??????? ???? ???? ????? ?0 = (1, 1, 0, 0).????? P1 P2 P3 t1 P1 t2 P4 ?????1?????? ????
  • 2. ?.???????? ??????????????????? ?????????????? ???????? ???? ?4??????? ????? ????? ???????????? ?????? ???????? ???? ?????? ???? ???? ?????? ???? ???? ??????? ????? ???? ??????.?????? ???????? ????? ????? ???? ??????? ???????????? ?????? ??????????? ?????? ???? ????? ???????????????5????????? ????? ???????6??????? ???? ????:????? ??????? ????? ?????? ???? 1.?????? ?????? ???????? ???? ???? ???????????? ????? ????? ?????? ?????? ??????? ???? ?????? ???????? ????? ????? ??????? ???? ???????? .???? 2.?????? ?????? ?????????? ?????? ???? ???????? ???????? ???? ???? ?????? ??????? ??????? ?????? ???????? ????? ????? ??????? ???? ?????? .???? ????? ???? ?????????2???????? ?????t1????? ???? ???? ??? ????? ?????? ?????? ??????? ???? ?????? ???? ?????????????????? ????3??????? .????? ?????? ??????? ?????? ?????? ???????????? ???? ????? ??????? ??????? ?????? ?????? ????:?????? ??(?) = { ?(?, ?) , ??? (?, ?) ¡Ê ? 0 , ??? (?, ?) ? ? ?+(?) = { ?(?, ?) , ??? (?, ?) ¡Ê ? 0 , ??? (?, ?) ? ? ???? ????????????? ????? ??????t:????? ????? ?????? ??: ? ? ??(?) ¡Ü ?(?) ???????? ????? ???t??????? ????? ???????????????? ?????? ????? ???:????? ??: ? ? ?¡ä(?) = ?(?) ? ??(?) + ?+(?) ??????????????¡ä ????m??????? ????? ???? ????????? ????? ?????? ? ¡ú ? ?¡ä ???? ???m???????????? ????? ??????? ¡ú ? ? ?¡ä ????¡ú ? ? ??????? ????????? ???????¡ú ?????? ????? ???? ?????? ?????? ??????? ??????? ????.????? ?????? ?????? ???? ???????????? ?????? ????????? ????? ????? ????????????? ??????? ??????? ???? ?????? ???? ??????? ?????? ???? ??????? ??????? ????????? 0m.??????????????????? ????????? ?????????????????? ??????????? ????????????(?) ? {?¡ä |?0 ¡ú(?,?,?) ? ?¡ä }.????? ??????? ?????????????7N??????????????? ???¡ú ????? ???? ????? ???????????????????? ????????????(?)????? ????? ??????????? ??? ?????? ?????? ?????? ????8?????????? ?????? ???? ??? ???????????? ???? ?????? ????????????????? ??????? ???? ?????? ??????? ?????????? .????? ?.?????? ?????????? ???????????? ????? ?????? ???????? ???? ?????????????? ??????? ??????? ???????? ?????? ????? ??????? ?????? ???? ?????? ???????? ???? ???????? ???? ????????????? P1 3 2 21 1 P1 P2 P3 t1 t2 P4 ?????3?????? ???????????? ???2?????? ????? ???? ???? ????????t1 3 2 21 1 P1 P2 P3 t1 P1 t2 P4 ?????2?????? ?????????? ???
  • 3. ?????????? ?????? ???? ????? ?????? ?????????????? ????? ????????? .??????? ?????? ?????? ?????? ?????? ?.??????????????? ?????? ???? ???? ??????? ????????????? ???N?????? ???? ?????????? m????? ?m ? R(N)??????? ????? ???? ?????? .???? ???? ????? ???????? ??????? ?????? ????? ???? ????????????????? ????????? ???? ?????? ???? ???????????? ?????? ???? ???????????? ?????????????? ??????? ???????????? .?????? ?????? ???? ???? ??????? ?????????????? ???????????? ??????? ?????? ?????? ???????????????? ????????? ??????? ???? ????????.????? ?.???????????0 ?????? ???? )?(??????? ??????? ???? ??????? ?????????????????? ????? ??? ???? ????? ??????? ?????????. ?.??????????????? ?????? ???? ???? ?????? ?????????? ???k-??????????? ??????????? ???? ????? ???????????????????? ?????????? ?(????? ????????????)??????? ??????? ???? ???????k???? ?????? .?????? ???????? ??????????? ????? ????? ???? ????? 1-?????????? ?????? ??? ?????? ??????????? ????????? ?????? ????? ????? ????? k??????k-????????.?????? ????? ?????? ?????????? ???k-???????????? ?????????? ????? ???? ??? ??????????? ????? ??? ??????????? ???????? ??????.????????? ?????????? ??????????????? ?????????12??????? ?????? ?????? ???? ????? ??????????????????? ????? ?????? ??????.?????? ??????????? ???? ???? ????? ?????? ?????????? ??????? ????????????? ?????? ????? ????? ?????.?????? ???????? ???? ??????? ???????????? ?????? ?????? ???? ???????????? ?????? ?????? ???? ????? .????? ?????? ???????? ??????? ??????? ?.???????????? ?????????????????? ??????????? ??????????? ??????? ?????? ??????????? ???????? ????? ???? ???????????? ???? ???????? ???????? ????????? ???????? ??? ?????? ???? ?????? ???????????? ??????? ?????? ?????? ??????? ?????? ????? ????????? ?????? ?????? ????? ???? ?????? ??? ????????????????? ???? ??????? ???? ?????? ?????? ??? ?????? ??????? ???????????????? ??????????????? ???? ????? ???? ????? ???? ??????? ???? ????.????????? ?.??????? ???????????? ??????? ??????? ???? ?????? ????? ????? ??????? ???? ???????? ???? ??????? ?????????? ????? ??????? ????? ?????????? ?????? ?????.????????? ?.?.??????? ??????? ?????? ????????????? ?????? ???????? ???? ?????? ?????????????? ????? ????????? ?????? ?????? ???? ?????????? ???? .?????????? ????? ???? ???? ???? ?????????.?????????? ?????? ????? ????? ???? ???? ????? ?????.?????????? ???????? ?????? ????? ???? ?????? ?????t2?????? ??????????? (1, 1, 0, 5)????(1, 0, 1, 0).????? ?????????????? ????? ????? ?????????? ???????? ???? ?????? ???.????? ?.?.????????? ??????? ?????4?????? ????? ???? ?????5???????? ????? ????
  • 4. ??? ????? ?????????? ????? ????? ????? ???? ????????? ????????? ??????? ????? ???? ?????? ??????????????? ???? ????????? .?????t2????? ?????? ????? ?????? ???? ???????? ??????? ???? ?????P1???? ????? ????? ?????? ??????? ?????? ?????? ??????? ?????? ?????????? ?????P1????????.???? ??????? ?????? ???? ?????????????(1, 1, 0, 1)????(1, 0, 1, 1)???????? ?????. ?.?.???????????? ??????5 ????????? ?????? ???????? ????? ?????? ??????? ???????? ???? ????? ?????? ???? ?????? ????????????? ????????? .?????? ?????? ????? ???? ?????? ????? ???????????? ???? ??????? ??????? ????????? ????????? ????? ???? ?????? ???? ????????? ??????? ?????? ????? ???? ??????? ?????? ???????? ???? ?????? ????????? ?????? ??????? ???? ????? ??????????? ????????????? ???? .?????t1???? ?????? ??????? ???? ???????????P4???P3???? ???????1?????????? ????? ????? ????? ??? ????? ?????.??????????? ?????? ????? ????????? ?????? ??? ???? ????????.????? ?.?????????? ????????????? ????????? ??????? ???? ????? ????????? ?????????? ???? ????? ?????? ?????? ??????? ????????? ?????? ?????? ???? ????????? ???? ??????? ????????? ????? .????? ?????? ?????? ?????? ???? ?????? ??????? ???????? ?????? ????????????? ?????? ?????? ????? ?????? ???????????? ???? ??????? ???????? ??????????? ??????? ?????? ????? ???? ?????? ????? ?????? ????????? ?????????????? ???? ????? ????? ?????? ???? ???? ??? ?????????? ?????? ????????? ?????? ???????? ??????? ??? ????? ????? ??????? ???? ???????? ???? ????? ????? ?????? ???? ????????? ???? ?????? ????? ??????? ???? ????? ????????? ?????? ???? ???? ?????? ????? ?????.?????? ?.?.????????????? ?????? ?????????6 ?????? ?????????? ?????? ???????? ???? ?????? ?????? ???? ?????????? ???? ???????:????? ????????, I)0Z=(N?????? ????:????? ?????? ??? ????? ????? ?????? ??? 1.0S(Z) = N?????? ???????? ???? ???? ????? ?????? ??? ???????17Z???? ??????????? 2.? ¡Ã ? ¡ú ?0 + ¡Á (?0 + ¡È {¡Þ})???(t)2(t)¡ÜI1I ???? ??????t??? ????? ????????t))(2(t), I1I(t) = (I. ???? ??????? ????? ????I?????? ??????18?)(t1I????? ?????? ????????? ( ]????? ?????? ???? ?[??? ??????1?eft??? )(t)2I?????? ????????? ( ]????? ?????? ???? ?[??? ?????? ?????20lft)???? ?????????? .??????????? ???? ?????? ?????? ????? ???? ?????? ???? ????? ???????????? ??????I???? ???? .????? ??????? ??????? ??????? ????? ???? ?????? ??????? ???????? ????? ?????? ?????? ???? ??????? ?????? ???????????M??????? ?????? ?????? ???? ???? ? H?????? ????? ???????? ???? ?????? ???? ??????? ?????????? ????? ?????? ????????? ???? ??????? ???? ??????? ????? ???? ????? ?????? ???? ?????? ??????? ??????? ?????? ????? ??????? ?????? ???????????????? ?????? .????? ??????? ?????? ??????:????? ??????? ????? ????? ???? ????? ??????? ????? ?????, I)0W, m, F,Z=(P, T?????? ?????????? ?????? ???????? ??? ??????? ¡Ã ? ¡ú ?0 + ¡È {?}???????z = (m, h)?????? ???? ????? ?????? ??? ????? ????? ???????m?????? ??????? ??????? ???????h ?????? ?????????? ??????? ???????Z.??????? ??? ?????? ?????? ????? ???? ???? ??????? ????? ?????? ??????? ????? ???? ???? ????? ???????? ???? ?????? ??????????????? ???? ???? ?????? ???? ????? .??? ???? ??????? ????? ????? ???? ??????:?????? ?????6??????????? ????? ????
  • 5. ?????? ?????????????? ???Z?????? ????z=(m, h)?????? ?????? ???? ??????? ????z¡¯=(m¡¯, h¡¯)???? ????z ? ¡ú ?¡ä???? ?????????? ??? ?????? ????:????? ?????? ??? ????? ????? 1.m' = m 2.? ?: ???( ?) ¡Ù?¡ú ?( ?) + ? ¡Ü ???(?). ???????? ?????? ?????? ????????? ??????? ?????? ?????? ????? ???? 3.? ?: ? ? ?¡ä ( ?) = { ?( ?) + ? ??? ?? ¡Ü ? ? ??? ?? ? ? ???????? ?????? ????? ?????? ???? ?????? ????????????? ????z ?? ¡ú ?¡ä?????? ???????? ???? ??? ??????:????? ??????? ??? ????? ????? 1.??? ¡Ü ??????? ?????? ???????? ?????? ? 2.???(??) ¡Ü ?(??)???????? ???? ?????? ??????? ???? ?????? ??? ?????????????? ??????? 3.?¡ä = ? + ¦¤?? 4.??????? ???? ??????t:??????? ?¡ä(?) = { ? ??? ?? ? ?¡ä ?(?) ??? ?? ¡Ü ? ¡Ä ? ¡Ù ?? ¡Ä ?? ¡É ?? = ¦µ? 0 ?. ?. ?????? ????4-?????? ?????? ???? ??????? ????? ????? ????:????? ?? = {?: ? |?(?, ?) > 0} ?????? ???????? ?????? ????? ??????? ??????????? ?????????????? ????? ???????? ????? ????????? ???? ?????? ?????? ?????? ?????? ???? ??????? ????0???? ??? ???? ????????? ?????????? ?????????????????? ??????? ???????? ????? ? ???? ???? ???? ?????? ?????? ????? ??????0???? ?????????????? ??? ????? ??????? ?????? ?????? ?????? ????? ???? ???? ???????H(t)????? ???? ?????.????? ?????? ??????? ?????????? ???? ???????? ???? ?????? ??????? ??????? ?????? ??? ???? ??????? ?????? ???? ?????? ????.?????? ???????.?????7?????? ????????????? ????????????? ?????? ???? ????? .????? ?????? ??????????? ???7???? ????:?????? ?????? ????? ????? ???? ?????? ( ?0, ( 0 ? ? 0 ) ) ??????0???????? ????? ?????? ???? ??????? ???????????1???4??????? ???????? ??? ??????????2???3????? ???? ????? ?????? ?????? ????????? ?????? ???????? ???? ???????????1???4???????? ?????? ????? ????????????eft?????? ???? ????????0????? ????? .?????1.3???? ??????? ??????? ??????? ?????? ???? ????? ??????:????? ( ?1, ( 1.3 ? ? 1.3 ) ) ???????? ??????? ???? ?????? ????? ????1????????? ????? ????? ??????? )¡Ü1.31eft(t??????? ???? ?????? ??????? ?????? ???? ????? .????? ???? ????? ?????? ???? ???????:????? ( ?2, ( 2.3 ? ? 2.3 ) ) ???????? ?????? ????? ????4t???? ?????????? ????? ????? .????? ????? ??????? 4t???? ????? ?????? ???? ??????? ????? ?????:?????? ( ?3, ( 2.3 ? 0 ? ) ) ?????7?????? ?????????? ?????? ?????????? ?????? ???? ?????
  • 6. ? 5.????????????? ????????????? ?????? ?????????? ?????? ????????????????? ?????????? ???? ??????? ??????????????? ???? ??????? ????? ??????? ?????????? ??? ???????????? ????? .????? ????? ?????? ?????? ?????? ?????? ?????? ???????????? ???? ??? ?????? ???????? ??????? ???????? ????????? ???????????? ???? ????????????? ??????? ???? ???? ??????? ??????? ?????? ?????? ??????? ????? ???? .????? ???????? ???????? ?????? ????? ???????????? ??? ???????? ????? ??????? ????? ?????? ?????? ????? ??? ??????? ?????? ????????????????? ??? ???????? ??????????? ?????? ?????????? ?????? ???? .?????? ??????? ??????????? ?????? ????????? ???????? ????? ??????? ???????????? ??? ????????? ??? ??????????? ??????? ?????? ?????? ??????? ??????? ?????? ?????? ????????????? ????? ???? ??? ????? ?????? ?????? ???? .???? ?????? ??????? ?????? ?????? ???? ???????? ???? 6.??????? [1] Prof. Louchka Popova-Zeugmann, ¡°Time Petri Nets: Theory, Tools and Applications Part I¡±, Humboldt-Universit?t zu Berlin, Course Lecture [2] Prof. Morgan Magnin, ¡°Computational Approaches to Analyze Complex Dynamic Systems : Model-Checking and its Applica- tions¡±, NII International advanced lectures series on ICT: series of 4 lectures of 2 hours in Spring 2013 at National Institute of Informatics, , T¨­ky¨­, Japan [3] Louchka Popova-Zeugmann, ¡°Time and Petri Nets¡±, Springer, Chapter 2, 2013 [4] http://en.wikipedia.org/wiki/Petri_net 1 Transition 2 Place 3 Carl Adam Petri 4 Enabled 5 Nondeterministic 6 Fire 7 Reachability graph 8 State Space ? Operational semantic 10 Liveness 11 Boundedness 12 Structurally bounded 13 Reset arc 14 Read arc 15 Inhibitor arc 16 Time Petri net 17 Skeleton 18 Interval function 1? Earliest firing time 20 Latest firing time