th Worksho on Agorthms and Toos for Petr Nets, Setember - October, 4, Unversty of Paderborn, Germany, 75-8 Sovng the fundamenta equaton of Petr net usng the decomoston nto functona subnets Dmtry A Zatsev Odessa Natona Teecommuncaton Academy Kuznechnaya,, Odessa, 59 Ukrane htt://wwwgeoctescom/zsoftua Abstract The technque of souton of the fundamenta equaton of Petr net based on decomoston nto functona subnets s roosed Soutons of the fundamenta equaton of the entre Petr net are cacuated out of soutons of the fundamenta equatons of functona subnets for dua Petr net Acceeraton of comutatons obtaned s exonenta wth the resect to dmenson of Petr net Keywords: Petr net, Fundamenta equaton, Decomoston, Functona subnet INTRODUCTION Matrx methods [,,4] are the most rosectve for arge-scae rea-fe systems Petr net modes anayss The fundamenta equaton of Petr net consttutes a system of near Dohantne equatons [5] Soutons of ths system are nterreted as the frng count vectors for the aowed sequences of transtons and so have to be nonnegatve nteger numbers that stuates the secfcs of the task Methods for these systems souton are reresented n [,4,,9,] Unfortunatey, a the known methods have an asymtotcay exonenta comexty that makes ts acaton for rea-fe systems anayss dffcut The goa of the resent work s the constructon of the comostona methods for souton of fundamenta equaton of Petr net aowng the consderabe acceeraton of comutatons Reay, modes of comex systems are assembed out of modes of ts comonents usuay Moreover, n the cases the comoston of mode out of subnets s not gven, we suggest to ay the methods of Petr net decomoston reresented n [8] to artton of a gven Petr net nto set of ts functona subnets [7] Earer the anaogous technque was aed successfuy for nvarants cacuaton [,5] The acceeraton of comutatons obtaned s estmated wth an exonenta functon Snce the dmenson of subnets as a rue s essentay tte than the dmenson of entre net, the actua acceeraton of comutatons may be extremey consderabe that was confrmed by the resuts of ths technque acaton to communcaton rotocos anayss [,,] BASIC CONCEPTS Petr net s a quadrue N = ( P, T, F, W ), where P = {} s a fnte set of nodes named aces, T = {t} s a fnte set of nodes named transtons; fow reaton F P T U T P defnes a set of arcs connectng aces and transtons, mang W : F Ν defnes a mutcty of arcs; Ν denotes a set of natura numbers Markng of net s a mang µ : P Ν, defnng a dstrbuton of dynamc eements named tokens over aces; Ν s a set of nonnegatve nteger numbers Marked Petr net s a coue M = ( N, µ ) or a quntue M = ( P, T, F, W, µ ), where µ s nta markng
A fundamenta equaton of Petr net [5] may be reresented as foows x A = µ (), where µ = µ µ, x s a frng count vector, A s a transosed ncdence matrx or ncdence matrx of dua Petr net [5] Notce that each equaton of ths system corresonds to a transton of dua Petr net It s known [,,5] that the sovabty of fundamenta equaton n nonnegatve nteger numbers s a necessary condton of the reachabty of a gven markng Soutons of system () are used for the constructon of the requred frng sequences 4 t t4 t t 5 t5 t t 5 t t 4 t5 t4 Fg Petr net N ~ Fg Dua Petr net N Accordng to [,4] we sha reresent a genera souton of homogeneous system as the near combnaton of bass soutons wth nonnegatve nteger coeffcents Notce that a bass conssts of mnma n nteger nonnegatve attce soutons of system As dstnct from cassc theory of near systems for reresentaton of genera nonnegatve nteger souton of nonhomogeneous system t s necessary to nvove not one arbtrary but a set of mnma artcuar soutons FUNCTIONAL SUBNETS Net wth nut and outut aces s Petr net the seca subsets of aces namey nut and outut are ndcated n whch Functona net s a tre Z = ( N, X, Y ), where N s Petr net, X P s a set of nut aces, Y P s a set of outut aces, at that sets of nut and outut aces do not ntersect: X I Y =, and, moreover, nut aces do not have nut arcs and outut aces do not have outut arcs: X: =, Y : = Paces of set Q = P \ ( X UY ) w be named an nterna and aces C = X UY a contact Petr net N = ( P, T, F ) s a subnet of net N f P P, T T, F F Functona net Z = ( N, X, Y ) w be named a functona subnet of net N and denoted as Z f N f N s subnet of N and, moreover, Z s connected wth resduary art of net ony by arcs ncdent to ether nut or outut aces, at that nut aces may have ony nut arcs and outut aces ony outut arcs Thus we have: X :{(, t) =, Y :{( t, ) =, Q :{(, t) = {( t, ) = Functona subnet Z f N s a mnma f t does not contan any other functona subnet of the source Petr net N
Net generated by the ndcated set of transtons R T w be denoted as B (R) The decomoston nto functona subnets [7] has been nvestgated n [8] Invarants of functona subnets were studed n [,5] Let us enumerate the most sgnfcant roertes of functona subnets: ) Functona subnet s generated by the set of ts own transtons ) Set of mnma functona subnets I = { Z }, Z f N defnes the artton of set T nto nonntersectng subsets T, such that T = UT, T I T =, k k ) Each functona subnet Z of an arbtrary Petr net N s the sum (unon) of fnte number of mnma functona subnets Unon of subnets may be defned wth the ad of oeraton of contact aces fuson 4) Each contact ace of decomosed Petr net has no more than one nut mnma functona subnet and no more than one outut mnma functona subnet 5) Petr net N s nvarant ff a ts mnma functona subnets Z, Z f N are nvarant and there s a common nonzero nvarant of contact aces 4 FUNDAMENTAL EQUATIONS OF FUNCTIONAL SUBNETS Let us consder the structure of system (): A x A = µ Each equaton L : x = µ, where A denotes -th coumn of matrx A, corresonds to transton t (of dua net) Equaton contans the terms for a the ncdent aces At that the coeffcents are equas to weghts of arcs and the terms for nut aces have sgn mnus and for outut aces us Therefore the system () may be reresented as L = L L L n () Theorem Souton x of fundamenta equaton for Petr net N s the souton of fundamenta equaton for each of ts functona subnets Proof As x s the souton of fundamenta equaton for Petr net N, so x s a nonnegatve nteger souton of system () and consequenty x s a nonnegatve nteger souton for each of equatons L Thus x s a souton for an arbtrary subset { L } Accordng to roerty ), a functona subnet Z, Z f N s generated by the set of ts own transtons T Thus, an equaton corresondng to a transton of subnet has the same form L as for the entre net, so subnet contans a the ncdent aces of source net Therefore the system reresentng the fundamenta equaton for functona subnet Z, Z f N s a subset of set { L } and vector x s ts souton Consequenty x s the souton of fundamenta equaton for functona subnet Z Arbtrary choce of subnet Z f N n above reasonng roves the theorem Theorem Fundamenta equaton of Petr net s sovabe f and ony f t s sovabe for each mnma functona subnet and a common souton for contact aces exsts Proof We sha use equvaent transformatons of systems of equatons to not rove searatey necessary and suffcent condtons Accordng to roerty ), a set of mnma functona subnets I = { Z }, Z f N of an arbtrary Petr net N defnes a artton of set T nto nonntersectng subsets T Let number of mnma functona subnets equas k As t was mentoned n the roof of theorem, equatons contan the terms for a the ncdent aces Therefore,
k L L L L, () where L s a subsystem for a mnma functona subnet Z, Z f N Notce that f L has not soutons, than L has not soutons aso Let us a genera souton for each functona subnet has the form x + u G, (4) where u G s the genera souton of homogeneous system, x X, where X s the set of mnma artcuar soutons of nonhomogeneous system of equatons Accordng to (): k k k L x + u G + u G = + u G Therefore system k k k x + u G + u G = + u G (5) s equvaent to source system of equatons () We sha demonstrate further that the souton of system (5) requres essentay smaer quantty of equatons Let us consder a set of aces of Petr net N wth the set of mnma functona subnets { Z Z f N} : P = Q U Q U U Q k U C, where Q s a set of nterna aces of subnet Z and C s a set of contact aces Accordng to defnton each nterna ace Q s ncdent ony to transtons from set T Thus x corresondng to ths ace s contaned ony n system L Consequenty, we have to sove ony equa- tons for contact aces from set C Now we construct equatons for contact aces of net C, so ony they are ncdent more than one subnet Accordng to roerty 4), each contact ace C s ncdent not more than two functona subnets Therefore, we have equatons x + u G + u G, () where, s the numbers of mnma functona subnets ncdent to contact ace C and G s a coumn of matrx G corresondng to ace Equaton () may be reresented n form u G u G = x x Thus, system x = x + u G, Q C, (7) u G u G = x x, C s equvaent to source system () Ths fact cometes the roof of theorem Notce that n both cases descrbed n roof accordng to (7), we have to sove a near homogeneous system of equatons Coroary To sove the fundamenta equaton of Petr net we may sove the fundamenta equatons of ts mnma functona subnets and then to fnd a common soutons for contact aces Coroary Theorem s vad aso for an arbtrary set of functona subnets defnng a artton of the set of transton of Petr net 5 COMPOSITION OF FUNDAMENTAL EQUATIONS Takng nto consderaton the resuts obtaned n the revous secton we may formuate a comostona method for souton of fundamenta equaton of Petr net: Stage Construct a dua Petr net Stage Decomose dua Petr net nto functona subnets
Stage Cacuate soutons for each of functona subnets fnd genera soutons of nonhomogeneous systems of equatons (4) Stage Comose subnets fnd the common souton () for the set of contact aces Note that stages, consst n souton of systems of near nonhomogeneous Dohantne equatons n nonnegatve nteger numbers For ths urose the methods descrbed n [,4,,9] may be aed Let us extract out of system (7) equatons for contact aces u G u G x Or n the matrx form G u u = b, b x G Let us enumerate a the varabes and to assembe the matrxes G, u n such a way to obtan a unted vector u = u u k u G n a unted matrx K Then we obtan system u K = b System obtaned has the form (), consequenty, ts genera souton has the form (4): u = u + v J (8) Let us construct a unted matrx G of soutons (4) of system () for a the functona subnets n such a manner that x + u G (9) We substtute (8) n (9): x + ( u + v J ) G + u G + v J G Thus x + v H, x + u G, H = J G () Snce ony equvaent transformatons were nvoved, the reasonng reresented above roves the foowng theorem Theorem Exressons () reresent a genera souton of fundamenta equaton () Now we estmate the tota acceeraton of cacuatons under the obtanng of nvarants va decomoston Let r be a maxma number ether contact or nterna aces of subnets Notce that r r n Then the comexty of fundamenta equaton souton for subnet may be estmated as ~, snce the comexty of decomoston accordng to [8] s oynoma Thus, the acceeraton of comutatons s estmated as n n r r = () Therefore, acceeraton of comutatons obtaned s exonenta Notce that the exonenta acceeraton of comutatons reresented wth exresson () s vad aso n the case the genera soutons for the functona subnets have more than one mnma artcuar souton Reay, et each of mnma functona subnets has not more than n mnma soutons Then durng cacuaton of common soutons for contact aces we ought to sove n sys- tems and oynoma muter may be omtted n the comarson estmatons of exonenta functons
AN EXAMPLE OF FUNDAMENTAL EQUATION SOLUTION Let us check the reachabty of markng µ = (,,,,4 ) n Petr net N (Fg ) Thus µ = (,,,,4) ~ Stages, Dua Petr net N (Fg ) s decomosed nto four mnma functona subnets 4 Z, Z, Z, Z cometey defned by the subsets of ts transtons: T = t }, T = t, t }, T = { t 5 }, T = t } 4 { 4 Stage x x x x = ( ) + ( u, u ) Z : { + 4 + = ; { { Z : x x x5 =, x x x = ; x = ( ) + ( u, u ) x + x x x = ( 4 ) + ( u, u ) Z : { 5 = 4; x x x = ( ) + ( u 4 ) ( ) 4 Z : { 4 = ; Stage u + u u u =, u u =, 4 u = ( u 4 u u u u u u ) = u u =, 4 u u =, = ( ) + ( v, v) u u =, u u u = ; x = ( 4 ) + ( v, v) Notce that the genera souton of homogeneous equaton consttutes t-nvarant of Petr net On the mnma souton x = (,,,,4, ) we may construct the frabe sequence σ = tt5t 5t5tt4tt5 Therefore, markng µ = (,,,,4 ) s reachabe n net N In ths tny exame a the aces are contact, so we have not obtaned an acceeraton of comutatons For rea-fe exames the acceeratons may become rather consderabe [,,] 7 CONCLUSION The comexty of Petr net fundamenta equaton souton s exonenta n genera case Ths fact makes the anayss of rea-fe obects dffcut The technque roosed and studed n resent aer aows the acceeraton of the souton of fundamenta equaton Ths technque s based on the decomoston of Petr net nto functona subnets The acceeraton obtaned s exonenta wth resect to the number of nodes of source Petr net
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