ON VERIFICATION OF FORAL ODEL OF COPLEX YTE A A myrn EA Luyanova Lets tate Techncal Unversty Russa VI Vernasy Crmean Feeral Unversty Russa ABTRACT The aer consers the alcaton of the theory of Dohantne equatons for the analyss of formal moels of comlex systems n artcular Petr comonent moels For the comonent-lace the corresonng to ths net LIDE has been examne ts consstency has been establshe whch means that the networ always has the ablty to go from one gven state to another After analyzng the wor of the CN-net tang nto account the wor of the comonent-lace we come to the concluson that all the lste roertes of the constructe moel are reserve Keywors: verfcaton formal moel comlex system INTRODUCTION Investgaton of rocesses an systems of any comlexty s effectvely carre out by constructng an stuyng the roertes of ther moels To mlement ths aroach t s necessary to bul a qualtatve moel of the system uner stuy Petr comonent moel (CN-net) can be use as ths moel [ ] Petr comonent moel s an otmal extenson of Petr nets (PN) for constructng moels of systems wth arallelsm characterze by a large number of nteractng rocesses an ther conserable mensonalty The constructon of the CN-net allows us to go from the ntal etale moel to the smlfe escrton whch guarantees the relablty of the conclusons about the roertes of the etale moel of the system uner nvestgaton [3] ubectng the resultng rouct to formal analyss usng for examle lnear algebra methos [4] we can obtan mortant nformaton about the structure an behavor of the system beng smulate any roertes of the PN are stue by methos of lnear algebra both statc an ynamc ones In ths case when verfyng Petr comonent moels t s necessary to solve both systems of lnear homogeneous Dohantne equatons (LHDE) an systems of lnear nhomogeneous Dohantne equatons (LIDE) The ossbltes of verfcaton of CN-moels of comlex systems are consere n the wor on the moel of the ralway staton wth the use of LHDE for fnng of - an T - nvarants of a gven moel as well as LIDE for establshng the ossblty of functonng of the subnet (comonentlace) of the CN-moel PETRI ODEL ANALYI TOOL For PN N = W ) (where = ( P T F) s net W : F N functon of the multlcty of the ( arcs F P T T P the ncence relaton ntal maru of the net) wth m laces an n transtons a matrx of ncence A s constructe sze m n wth nteger coeffcents a = I ( t ) I ( t ) where s the ncence functon of ths PN an the coeffcents I reresent the number of toens that move change an are ae to the locaton a when the transton t occurs n the net N Accorng to the ncence matrx A a system of equatons of states of a gven PN s constructe ubmt Date: 98 Accetance Date: 38 DOI NO: 7456/8E/4 Coyrght The Tursh Onlne Journal of Desgn Art an Communcaton 348
Ax = () where = ( x = = u resectvely the ntal an fnal gr layouts of net N ) where u control vector consstng of n zero comonents an one comonent equal to sgnalng the oeraton at the ste of the transton t If the maru from the maru the system () always has a soluton [5] The soluton x of LHDE Ax = soluton y LHDE A T y = s calle -nvarant of the net N s reachable when n () = s calle T -nvarant of the net N The The calculaton of - и T -nvarants for a gven PN maes t ossble to verfy such roertes as reachablty bouneness reeatablty conservatsm consstency A secal role n solvng the roblems of bouneness s laye by mnmal generatng sets of - an -nvarants [6] The generatng set of -nvarants (T -nvarants) s calle the mnmal generatng set of -nvarants (T - nvarants) f there s no non-emty subset of t whch s also a generatng set It follows from the theory of Dohantne equatons [5 6 7] that the truncate set of solutons of LHDE corresonng to a gven PN s the mnmal generatng set of ) -nvarants to construct ths set one can use the T -algorthm [8] To establsh the comatblty of the LIDE the comatblty crteron of the LIDE s use [7] Theorem A system the auxlary system For LIDE of lnear nhomogeneous Dohantne equatons s comatble f an only f s comatble (T T L = a x + a x + + a x = b ; q q L = a x + a x + + a x = b ; q q = L = a x + a x + + a qxq = b auxlary system has the form L = a x + a x + + a = ; q xq L = a x + a x + + a = ; q xq = L = a x + + a = ; q xq L = a x + a x + + a x = b q q ubmt Date: 98 Accetance Date: 38 DOI NO: 7456/8E/4 Coyrght The Tursh Onlne Journal of Desgn Art an Communcaton 349
where b an the frst equatons are obtane as lnear combnatons: for all = до f b then L = b L b L f b then L + = b L b L Theorem A system u + u + + u tb = s comatble f an only f equaton has at least one soluton ) n a set N such that the coornates of the vector e e ( v = e + + + are a multle of the number Here are the values L (x) on the vectors system e t e e Let s conser a system equaton (x) L that are vectors of the subsystem T of the frst equatons of the If for the frst t N = до n whch the equaton wth a free term equatons of the system by Theorem t s necessary to fn at least one soluton of equaton u + u + + u t = to establsh the comatblty of the system Let any two coeffcents then assgnng to all an u r an b = can be chosen as the t s ossble to construct T then satsfyng the hyothess of the theorem of the last equaton be mutually rme numbers of ooste sgns ( r = r ) an a an the extene Euclean algorthm we fn last equaton wll be constructe t value an usng the Bezout relaton for The followng conton s obtane for establshng the comatblty of the LIDE u an u Thus the esre soluton of the Theorem 3 If there s an equaton L (x) wth a free term equal to n the LIDE an f at least two values L (x) on the vectors T of the frst numbers wth ooste sgns then such a system s comatble equatons of the system are mutually rme ANALYI OF THE PETRY ODEL Let s conser a Petr net from [9] shown n Fg a) Ths net smulates the movement of trans through some ea-en ralway staton wth one nut trac an three nternal tracs In Fg b) the moel s resente as a CN-net In ths CN-net the lace P s a comonent-lace whch s searately shown n Fg c) Havng wrtten the ncence matrx for the CN-net an establshe that for ths net = = () we bul the LHDE Ax = Its ostve nteger solutons are T -nvarants of CN-net The roblem of the comatblty of the resultant LHDE s solve by constructng a truncate set of solutons by an T -algorthm The result s a vector e = ( ) T -nvarant To ubmt Date: 98 Accetance Date: 38 DOI NO: 7456/8E/4 Coyrght The Tursh Onlne Journal of Desgn Art an Communcaton 35
search for the - nvarants we construct a LHDE A T y = the comatblty of whch s also establshe by the T -algorthm The result are vectors e ( ) e ( ) = = a b c Fg oels: a) a ea-en ralway staton n the form of PN; b) a ea-en ralway staton n the form of a CN-net; c) comonent-lace For the net that slays the comonent-lace maru at the corresonng auxlary system P P the ntal maru ( ) an the - ste = ( ) o not conce so we bul the LIDE = и an ts x x x3 = ; x 4 = ; x = ; x3 = ; = x x4 = ; x = x3 x6 = ; x4 5 = x x x3 x 4 = ; x = ; x3 = ; = x x4 = ; x = x3 x6 = ; x4 5 = 4 5 6 = ; Vectors e `= ( ) e ` = ( ) are T subsystems comrse of the frst 7 equatons Values L 8 ( x ) on vectors T : L 8 ( e` ) = L 8 ( e` ) = Theorem 3 s not alcable Accorng to Theorem we obtan the equaton u + u t = Let s tae one of ts roots for examle ( ) Then the vector-soluton of the system has the form e ` + e` = ( ) s consstent By Theorem the system s comatble to the corresonng system CONCLUION ubmt Date: 98 Accetance Date: 38 DOI NO: 7456/8E/4 Coyrght The Tursh Onlne Journal of Desgn Art an Communcaton 35
The analyss of the constructe Petr nets allows us to establsh the roertes of the formal moel Thus from the set of -nvarants e = ( ) e = ( ) CN-networ of the ea-en ralway staton t s clear that all ts laces are covere by non-zero coornates ths means that the networ s lmte You can get an uer score for toens that ft anywhere for examle for a lace T P : e ( P ) = The same score for the number of toens can be obtane for the rest of the CNnet n queston Ths means that the consere net s e ( P ) safe The resence of T -nvarants ncates the ossesson of the net by the roerty of reeatablty whch s nterrete as the constant movement of trans through the staton The net has the consstency roerty because there s a sequence of transtons trggere on the net whch starts at least once whch can be nterrete for the staton as the earture of trans For the comonent-lace the corresonng to ths net LIDE has been examne ts consstency has been establshe whch means that the networ always has the ablty to go from one gven state to another After analyzng the wor of the CN-net tang nto account the wor of the comonent-lace we come to the concluson that all the lste roertes of the constructe moel are reserve ACKNOWLEDGEENT The reorte stuy was fune by RFBR accorng to the research roect 6-7-854 а REFERENCE E Luyanova Comonent moelng: on connectons of etale Petr moel an comonent moel of arallel strbute system //ITHEA 3 P 5- EA Luyanova The structural elements of a Petr net comonent //Problemy rogramuvannya No -3 P 5-3 EA Luyanova AV Dereza The stuy Of the structural elements of the sngle-tye CN-networ urng the comonent moelng an analyss of comlex systems wth concurrency //Cybernetcs an systems analyss No 6 P-9 T urata Petr Nets: Proertes Analyss an Alcatons //Proceengs of the IEEE 989 Vol 77 No 4 P 54-58 L Kryvy Algorthms for comutaton of systems of lnear Dohantne equatons n nteger omans //Cybernetcs an systems analyss 6 No P3-7 L Kryvy On some methos of comutaton an comatblty crtera for systems of lnear Dohantne equatons n the regon of natural numbers //Cybernetcs an systems analyss 999 No 4 P-36 L Kryvy A comatblty crteron for systems of lnear Dohantne equatons over the set of natural numbers // Cybernetcs an systems analyss 999 No 5 P7- L Kryvy On the calculaton of the mnmal set of nvarants of Petr nets //Artfcal Intellgence No 3 P99-6 A hmyrn IA eyh EA Luyanova On the reresentaton of the Petr comonent net by a neghborhoo system //Vestn of LTU 6 No 4 (3) P35-4 ubmt Date: 98 Accetance Date: 38 DOI NO: 7456/8E/4 Coyrght The Tursh Onlne Journal of Desgn Art an Communcaton 35