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1 ISSN , ODEN RUXUEW E-mal: Journal of Sofware, Vol8, No7, July 2007, pp hp://wwwosorgcn DOI: 0360/os8553 Tel/Fax: by Journal of Sofware All rghs reserved Per,2+, 2, (, 60054) 2 (, 60039) A Reducon Technque of Per Nes Based on Logc rcu YE Jan-Hong,2+, SONG Wen 2, SUN Sh-Xn (School of ompuer Scence and Engneerng, Unversy of Elecronc Scence and Technology of hna, hengdu 60054, hna) 2 (School of Mahemacs and ompuer Engneerng, Xhua Unversy, hengdu 60039, hna) + orrespondng auhor: Phn: , E-mal: leafever@63com, hp://wwwuesceducn Ye JH, Song W, Sun SX A reducon echnque of Per nes based on logc crcu Journal of Sofware, 2007, 8(7): hp://wwwosorgcn/ /8/553hm Absrac: In radonal mehods, he local srucure of Per ne s requred o compare wh all reducon rules The process s complcae and does no f for nes wh nhbor arcs Ths paper presens a reducon mehod Frsly, Per ne s dvded no several maxmal acyclc subnes and each one s expressed wh logc form Then, logc algebra s used o reduce he logc form Fnally, he resul s reconsruced and embedded n he orgnal ne Ths paper esablshes a mehod o fnd and reduce he maxmal acyclc subnes and presens he correlave proofs Ths mehod can be appled o Per nes or subnes wh nhbor arcs and acyclc Key words: : Per ne; reducon; logc algebra; maxmal acyclc subne Per,,,,, Per, : Per ; ; ; : TP30 : A Per [,2], Per,,,,, Per Suppored by he Naonal Naural Scence Foundaon of hna under Gran No ( ); he Fundamenal Research Foundaon of Scence and Technology Bureau of Schuan Provnce of hna under Gran No ( ) Receved ; Acceped

2 554 Journal of Sofware Vol8, No7, July 2007,, [3] Per,Muraa [4,5] T-, ; [6] Muraa, T-, ;Aals [7] ;Mugarza [8] Ferscha [9], Per ; [0,] Per, Per Per ; [2] P/T, S(T) [3,4] [2],,,, [5,6], Per [7] Per Per [8,9] Per, Per, Per, RM-L (reducon mehods based on logc crcu), Per,,, Per Per [,2,20], Per : [2] 8,, Per B A & A A B A A A B B Fg Smulaon of gae crcu based on Per ne Per RM-L Per, N=(S,T;F,I) E/N,,I S T,I F= T, Power()={s (s,) F}; onrol()={s (s,) I} 2 Σ=(S,T;F,I,) E/N,,(S,T;F,I), c :c[,

3 : Per 555 s, s Power( ) c( s) = Poson ( s, ) = s, s onrol( ) c( s) = 0 2,c 0 =(,,0,0,0),Power()={s,s 2 };onrol()={s 3 };Poson(s,)=s ;Poson(s 2,)=s 2 ; Poson ( s3, ) = s3, Poson( s4, ) = s s2 s3 ; Poson( s5, 2 ) = s s2 s3, Poson(s,),=,2,3 s 4,s 5 ; s Poson(s,),=,2,3 s 4,s 5 s E/N Σ, T, c:c[ c, Poson( s, ) = Poson( s, ),,s, (s ), 2 E/N Σ,,, α T, α,c,c 2,,c α :c [, α, α c, α Poson( s, ) = Poson( s, ),, s α, = s E/N α (conac) [2] 2, α 3,c=(,,0),Poson(s, )=s ;Poson(s 2, 2 )=s 2, 2,Poson(s,)=s +s 2, + 3 E/N Σ, T, ={s}, ={s },onrol()={s},c[ c Poson ( s, ) = s,, (s ) 4,c=(0,0),c[ c,, Poson ( s, ) = s s s 4 s 2 s 5 s 2 3 Fg2 Illusrang he defnon 2 and proposon 2 2 s s 2 2 s 3 s s Fg3 Illusrang he proposon 2 Fg4 Illusrang he proposon Per, : 3 N=(S,T;F,I) E/N : () N N 0, s 0,s 2 S : s s 2 ; s Power( ); s onrol( 2 ), 2 s2 ; (2) N N 0, : s = s N I ; (3) N N 0, : s 2 ; s s = s 2, N 2 II ; (4) N N 0, : s 3 ; s s = s 2, N 3 III E/N Σ,c[σ 2 c, s ω ω 2, ( s ), s, Poson(s2, 2 )=s

4 556 Journal of Sofware Vol8, No7, July 2007 () I, Poson ( s, 2) = s, 2, Poson( s ω, ) = s s s ω s s s s = s, 5(a) ; (2) II, Poson( s, = s + s, 2 Poson( s ω, ) s + s s 2 ) (ADeMorgen), s + s s = s s ) s, I, s s s = ohers ( ohers ohers = ohers 5(b) ; (3) III, Poson( s, 2 ) = s sohers, 2 Poson( s ω, ) = s sohers s, s sohers s = ( s s + sohers s), s ω, 2 s, 2 s s ohers, s ω sohers, s s s s + s, s s, s s = 0,, s = s s ohers ohers ohers 4 Σ=(S,T;F,I,) E/N, N=(S,T;F,I) I II, s s = s ; III, s s = 0 s I srucure s ω s sω s 2 s ω s s 0 srucure s II srucure s s oher s 2 III srucure 2 s s s s oher (a) (b) (c) Fg5 Illusrang he defnon 3, defnon RM-L Per, : ( ),, ( ) [20],, E/N : 2 N Per,w=(s 0,,s n ) n, s = s, =,, n, n, s s s s w s 0 ; w, s s n s 0 = n 22 N E/N,T T,N T- =(S,T ;F,I ),S = (T ) (T ),F =F ((S T ) (T S )),I =I (S T ) 23 E/N N=(S,T;F,I),T T,N T T-, N, TT, T T-, T =T, N N, 3, : N, :S I S, (S I ) T =, s S I, / x S, s x < s ;S O S,(S O ) T =, s SO, / x S,s s < x 3 : S I T I ={ =(Power() onrol()) S I } ((S I T I )

5 : Per 557 (T I (T I ) )) F,(S I T I ) I ; S O ; S = S S S ) T T I (( S T ) ( T S )) F,( S T ) I medal medal medal 6,S I ={s 0,s,s 2 },S O ={s 5,s 6,s 7,s 8 } medal ( I O s 5 s 0 s 9 4 s 7 s s 8 s 5 s 2 s 6 s 0 Fg6 The maxmal acyclc subne T I,, ( ),, T I, σ,σ ~ 3, 3 c c σ c 0 [σ [ O,,σ (T T I ) c O c O (s) Poson(s,),s S O, (s),poson(s,) S I T, c R(c 0 ): c[, T dead ={ T, };S -dead ={s s S, T dead,s }; S = { s s S c ( s) 0} { Poson( s, ) s Sdead, s } ;SO-Logc={Poson(s,) s S O, s } dead dead O ; S Tdead Logc 4 3 S O-Logc S, s s, s s s T dead Logc s + s ohers s, 4, s s s Per 5 4 ( (Karnaugh map),q-m(qune Mcluskey) ) 5, Per S O-Logc, S S O-, S T dead Logc T dead 3 2, Per, 6( S O- ) () S O- S I, S I s, sum, ~ 3 S I S O (2) Poson(s,) S O- Poson(s,)=0, σ σ s, s, s= ohers =

6 558 Journal of Sofware Vol8, No7, July 2007 (3) Poson(s,) S O- Poson(s,)=,,,s, 7( S ) T dead () S SI, S I s, 6() T dead (2) T dead, T dead, S -dead, s S dead, () s (3) T dead,s s S O, (,s), Poson(s, ) s 8 () s S I,s, S O- S T dead 7(a) s,poson(s,)=s,, ( s, ) = s, 7(a) 7(b), (2) s S I,s S O-Logc S S S O- T dead s, s = 8 s 0 T dead Logc,, s s s s s s s s s Orgnal srucure Reconsrucon Orgnal srucure Reconsrucon (a) (b) Fg7 Illusrang he rule 8() 7 8() 5 s 0 s 4 s s 9 s 5 s 6 s 2 s 7 s 0 s8 Fg8 The reducon resul of 8 9 6~ 8, 4 N 0 3 N, S I S O, s S I, s, (,s); s S O, s, (s, ), N 2 : 3, S I S O 6~ 8 S I

7 : Per 559 S I l,,s I S O N S I S O, 24 N,N 2 N,N 3 N,N 2, N N 3,N 2 N 3 22,N N,, N, N + + :() N S, s S s,,s O S N O S I N + s, T T,,T,T,N T-, T 23 + (2) ( N N N T, T T T, S O,,, ) S,,S S-,,, S N, N N N S T m n, :() (s, ) F, (s, )=s ;(2) (s, ) I, 3 ( s, ) = s ;(3) (, s ) F, ( s, ) = ; (4) (s, )=0 3 N : N=(S,T;F); : N () T, T ={ }, T-, (2) (2), ) ( ) T, :, ; ( S O, I, T, T T- (3) (3) (2), TT T N = S, T, ) N S I O ( F (4) (4) T, (), (2) (3),, N N = S, T, ) (5) 2 ( 2 2 F2 (5) (4), N, 2,, T, N, 2,, (6) (6) N, 2,, 32 : N ; : S I, S O S SO-Logc S S O-Logc, S, SO- S T dead Logc T dead T dead N, N T dead Logc, N N 3 32, 22, 3 P, N,

8 560 Journal of Sofware Vol8, No7, July 2007 T-, 32 NP,, NP,RM-L, 4 RM-L : :, ;, ;, ( 9 ) s 2 2 s 5 6 s 7 s s 3 4 s 6 7 s 8 8 s 0 o o s 4 3 s 9 5 Fg9 The example 9 : ; ; 2 ; 3 ; 4 ; 5 ; 6 ; 7 ; 8 ; o N,S I ={, },S O ={o} : Marx NeReducon(Marx ) {Se acyclcsubnes =DvdeNe( ); } DO { } =Ge(Se acyclcsubnes ); Se acyclcsubnes =Se acyclcsubnes ; // //, Se acyclcsubnes,, // Se acyclcsubnes DvdeSubNe(,SI,S O ); // TransMarx(,SO-Logc, S ); //, T dead Logc S O-Logc, S T dead Logc S O-New =Q-M(S O-Logc ); // Q-M S O-Logc, S O-New S =Q-M( ); // Q-M, S T dead ST dead Logc ST dead Logc T dead ReconsrucExpresson(,SO-New, S ); //SO-New, S T dead T dead RecoveryMarx(, ); // Whle (Se acyclcsubnes!=null); Oupu ( ); // 0,

9 : Per 56, o Fg0 The ne afer reducon 0 5 (), ;,,, 3 32 (2) Muraa, (3) Muraa RM-L Type of ne Ordnary ne Ordnary ne whou conflc Table omparson beween RM-L and classcal reducon mehods orrespondng reducon mehods Presen sx rules basc operaons: Fuson or elmnaon of places and ransons Presen four seps for maxmal acyclc subne Basc operaons: Logc algebra RM-L Dfference Sepwse reducon Area reducon Effcency Low, dffcul o generae algorhm Fnsh he reducon maybe only ones Easy o generae algorhm Ne wh nhbor arcs No Yes Reduce for cycle Yes No,,, References: [] Resg W Per Nes: An Inroducon Berln, Hedelberg: Sprnger-Verlag, [2] Yuan Y The Theory and Applcaon of Per Nes Beng: Publshng House of Elecroncs Indusry, (n hnese) [3] Jang J The PN Theory of Dscree Even Dynamc Sysem Beng: Scence Press, (n hnese) [4] Muraa T, Koh JY Reducon and expanson of lve and safe marked graphs IEEE Trans on rcu Sysems, 980,AS-27(): 6870 [5] Muraa T Per nes: Properes, analyss, and applcaons Proc of he IEEE, 989,77(4):54580 [6] Jang J Some reducon operaons for a weghed T-graph Journal of hna Insue of ommuncaons, 994,5(2):9702 (n hnese wh Englsh absrac) [7] van der Aals WMP, Basen T Inherance of workflows: An approach o acklng problems relaed o change Theorecal ompuer Scence, 2002,270():25203 [8] Mugarza J, amus H, Genna J, Teruel E, Slva M Reducng he compuaonal complexy of schedulng problems n Per nes by means of ransformaon rules In: IEEE In l onf on Sysems, Man, and ybernecs [9] Ferscha A oncurren execuon of med Per nes In: Tew JD, Manvannan S, Sadowsk DA, Sela AF, eds Proc of he Wner Smulaon onf Orlando: Socey for ompuer Smulaon Inernaonal Press, [0] Ln On refnemen of model srucure for sochasc Per nes Journal of Sofware, 2000,():0409 (n hnese wh Englsh absrac)

10 562 Journal of Sofware Vol8, No7, July 2007 [] Ln, Qu Y, Zheng B, Tan LQ An approach o performance equvalen smplfcaon and analyss of sochasc Per nes ATA ELETRONIA SINIA, 2002,30(): (n hnese wh Englsh absrac) [2] Xu AG, Jang J The reducon operaons and her properes for P/T nes Journal of Sofware,997,8(7): (n hnese wh Englsh absrac) [3] L JQ, Fan YS Research of Per nes based workflow model reducon mehods Informaon and onrol, 2002,30(6): (n hnese wh Englsh absrac) [4] Zhou JT, Sh ML, Ye XM A mehod for semanc verfcaon of workflow processes based on Per nes reducon echnque Journal of Sofware, 2005,6(7):2425 (n hnese wh Englsh absrac) hp://wwwosorgcn/ /6/24hm [5] Palmer J, Perlman D, Wroe; hen WK, Xu PP, Trans Schaum s Oulnes Inroducon o Dgal Sysems Beng: Scence Press, (n hnese) [6] Lu BQ Dgal rcu and Sysem Beng: Tsnghua Unversy Press, (n hnese) [7] Schaefer DH Per ne represenaons of compuaonal and communcaon operaors In: Yakovlev A, ed Hardware Desgn and Per Nes Boson: Kluwer Academc Publshers, [8] Yakovlev AV, Koelmans AM, Semenov A, Knnmen DJ Modellng, analyss and synhess of asynchronous conrol crcus usng Per nes Inegraon, he VLSI Journal, 996,2(3):4370 [9] Zhao BH, Jng L, Yan YG Hardware mplemenaon of Per nes Journal of Sofware, 2002,3(8): (n hnese wh Englsh absrac) hp://wwwosorg cn/ /3/652pdf [20] Wu ZH An Inroducon of Per Nes Beng: hna Machne Press, (n hnese) [2] van der Aalsa WMP, er Hofsede AHM Verfcaon of workflow ask srucures: A Per-ne-based approach Informaon Sysems, 2000,25():4369 : [2] Per :, [3] PN :, [6] T-,994,5(2):9702 [0] Per,2000,():0409 [],,, Per,2002,30(): [2], P/T 997,8(7): [3], Per,2002,30(6): [4],, Per,2005,6(7):2425 hp://wwwosorg cn/ /6/24hm [5] Palmer J,Perlman D, ;,, :, [6] :, [9],, Per,2002,3(8): hp://wwwosorgcn/ /3/652pdf [20] Per :, (976 ),,,,,Per (940 ),,,,, (956 ),,,F, Per

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