nd edges. Eh edge hs either one endpoint: end(e) = fxg in whih se e is termed loop t vertex x, or two endpoints: end(e) = fx; yg in whih se e is terme

Theory of Regions Eri Bdouel nd Philippe Drondeu Iris, Cmpus de Beulieu, F-35042 Rennes Cedex, Frne E-mil : feri.bdouel,philippe.drondeug@iris.fr Astrt. The synthesis prolem for nets onsists in deiding whether given grph is isomorphi to the mrking grph of some net nd then onstruting it. This prolem hs een solved in the literture for vrious types of nets rnging from elementry nets to Petri nets. The generl priniple for the synthesis is to inspet regions of grphs representing extensions of ples of the likely generting nets. We follow in this survey the grdul development of the theory of regions from its foundtion y Ehrenfeuht nd Rozenerg, with prtiulr insistene on the strt mening of the theory, whih is generl produt deomposition of grphs into tomi omponents determined y prmeter lled type of nets, nd on the derivtion of eient lgorithms for net synthesis sed on liner lger. Tle of Contents: 1 Terminology of Grphs 2 Regionl Representtion of Prtil 2-Strutures 3 The Synthesis of Elementry Net Systems 4 Cutset Representtion of Finite Grphs 5 Flip-Flop Nets nd their Synthesis 6 Regions for Aritrry Types of Nets 7 Polynomil Time Algorithms for the Synthesis of Petri Nets 8 Regions in Step Trnsition Systems 9 Dul Adjuntions etween Trnsition Systems nd Nets 10 Some Applitions 1 Terminology of Grphs Sine the terminology on grph theory vries lot from one uthor to the other, we found it neessry to egin y dening the terminology used in this doument. 1.1 Grphs A grph G = (X; E) is olletion X of verties or nodes together with olletion E of edges. The grph is sid to e nite if it hs nitely mny verties

nd edges. Eh edge hs either one endpoint: end(e) = fxg in whih se e is termed loop t vertex x, or two endpoints: end(e) = fx; yg in whih se e is termed link etween verties x nd y. A grph is simple if it is loop-free: eh edge is link, nd hs no multiple edge: end(e 1 ) = end(e 2 ) ) e 1 = e 2. Therefore n edge of simple grph my e identied with the pir of its endpoints. The inidene mtrix of grph G is mtrix A with elements 0 nd 1, where eh row orresponds to vertex, eh olumn orresponds to n edge, nd A(x; e) is 1 if nd only if x is n endpoint of e. A hin of length n 1 with endpoints fx 1 ; x n+1 g is nite sequene (x 1 ; e 1 ; x 2 ; : : : ; x n ; e n ; x n+1 ) of verties nd edges suh tht end(e i ) = fx i ; x i+1 g for ll 1 i n. We sy tht the hin onnets its endpoints. For onveniene, we onsider tht every vertex is onneted to itself y n empty hin. The onneted omponent of vertex is the set of verties onneted to this vertex y some hin; the grph is onneted if it hs only one onneted omponent. A non empty hin is sid to e simple if ll edges re distint, hin is sid to e elementry if ll the verties ut possily the endpoints re pirwise distint. A yle is simple hin whose endpoints oinide: x 1 = x n+1. A tree is grph with no yle or lterntively grph in whih ny two verties re onneted y unique hin. G 0 = (X 0 ; E 0 ) is sugrph of G = (X; E) if X 0 X, E 0 E, nd the mppings tht send n edge e 2 E 0 to its endpoints in G 0 nd in G oinide. G 0 spns G if X 0 = X; spnning tree of G is sugrph whih is tree spnning G. 1.2 Direted Grphs An orienttion of n edge e is n ordered pir of verties (x; y) suh tht end(e) = fx; yg, thus loop t x hs only one possile orienttion: (x; x), while link etween x nd y hs two possile orienttions: (x; y) nd (y; x). We let e : (x; y) denote the ssignement of the orienttion (x; y) to the edge e; the verties x = @ 0 (e) nd y = @ 1 (e) re respetively lled the soure nd trget of edge e. An oriented edge is sometimes lled n r. A direted grph is grph whose edges re given n orienttion. A direted grph is simple if it is loop-free nd hs no multiple r in the sense tht two edges with the sme endpoints re neessrily given opposite orienttions: (e 1 : (x; y) ^ e 2 : (x; y)) ) e 1 = e 2. Therefore n edge of simple direted grph my e identied with the ordered pir of its endpoints, nd in tht se we write e = (x; y) when @ 0 (e) = x nd @ 1 (e) = y. Notie tht the underlying grph of simple oriented grph my not e simple s we n nd two edges with the sme endpoints ut with opposite orienttions. A sugrph of direted grph G is sugrph of the underlying grph with the orienttions of edges inherited from G. The notions of hin, yle, tree, spnning sugrph nd spnning tree do not depend on the orienttion of edges; therefore hin (yle, tree,...) of direted grph is hin (yle, tree,...) of the underlying grph. The spei notions tht tke the orienttion into ount re the following. A pth of length n 1 from x 1 to x n+1 is nite sequene (x 1 ; e 1 ; x 2 ; : : : ; x n ; e n ; x n+1 ) of verties nd edges suh tht @ 0 (e i ) = x i nd @ 1 (e i ) = x i+1 for ll 1 i n. For onveniene, we onsider tht there exists n empty pth from ny vertex to itself. A non empty pth is sid to

e simple if ll edges re distint. A pth is sid to e elementry if ll the verties ut possily the endpoints re distint. A iruit is simple pth whose endpoints oinide: x 1 = x n+1. Thus pths nd iruits re respetively hins nd yles of the underlying grph whose edges hve omptile orienttions. The inidene mtrix of direted grph is the mtrix A : X E! f 1; 0; 1g ( 1 if @ 0 (e) = x given y A(x; e) = 1 if @ 1 (e) = x. 0 otherwise 2 Regionl Representtion of Prtil 2-Strutures The theory of regions ws founded y Ehrenfeuht nd Rozenerg in [22] with the im to otin set-theoreti representtion of direted grphs (X; E), enrihed with n equivlene on edges. The resulting strutures (X; E; ) re termed prtil 2-strutures. The representtion prolem for prtil 2-strutures onsists in tthing properties p to nodes x so tht the Kripke struture so otined my e strted without loss of informtion to the dt fx j x 2 Xg nd fe j e 2 Eg, where node is enoded y the set x = fp j x j= pg of properties it stises nd n edge y the pir e = (x n y ; y n x ) where x nd y re the respetive soure nd trget of e. The min diulty is to reonstrut the equivlene reltion, nd this nnot e done unless the onsidered properties re ltered uniformly when pssing long every edge in eh equivlene lss. These spei properties, seen s sets of nodes when identied with their extensions fx j x j= pg, re lled regions in [22]. The presenttion of regions in prtil 2-strutures given elow is diretly inspired from [22], where the proofs of the results my e found. The lgorithmi spets of elementry net synthesis will e exmined in the next setion. 2.1 Prtil 2-Strutures nd their Regions Denition 2.1 A prtil 2-struture is triple G = (X; E; ) where X is nite non empty set of nodes, E E 2 (X) = f(x 1 ; x 2 ) 2 X Xj x 1 6= x 2 g is set of 2-edges over X, nd is n equivlene reltion on E. When E = E 2 (X) is the whole set of 2-edges over X, G is lled 2-struture. Prtil 2-strutures my e viewed s equivlene lsses of lelled simple direted grphs, where two grphs re equivlent if their lelling funtions hve the sme kernel. Of prtiulr interest re the prtil set 2-strutures dened s follows. Denition 2.2 A prtil set 2-struture of nite set B is prtil 2-struture G = (X; E; ) where X P(B) nd is the kernel of the funtion ((M; M 0 )) = (M nm 0 ; M 0 nm) for M; M 0 2 X. Let S2S(B) denote the (full) set 2-struture of B; i.e. when X = P(B) nd E = E 2 (X).

Thus in prtiulr, ny prtil set 2-struture G of B is sustruture of S2S(B). In nottion, G S2S(B) where (X 1 ; E 1 ; 1 ) (X 2 ; E 2 ; 2 ) if X 1 X 2, E 1 E 2 nd 1 is the restrition of 2 on E 1 E 1. The representtion prolem for prtil 2-strutures my e stted s follows. Whih prtil 2-strutures re isomorphi to sustrutures of S2S(B) for some nite set B (of tokens)? The est wy to grsp this prolem is to exmine the extents R of representtion tokens 2 B in the struture S2S(B) itself, let R = fm 2 P(B)j 2 Mg. So, 2 M if nd only if M 2 R. The following my e oserved. 1. For every pir of equivlent 2-edges (M 1 ; M 0 1) nd (M 2 ; M 0 2), nd for every 2 B, 2 M 1 n M 0 1 entils 2 M 2 n M 0 2 nd symmetrilly 2 M 0 1 n M 1 entils 2 M 0 2 n M 2. This n lso e expressed s follows: { (M 1 2 R ^ M 0 1 62 R ) ) (M 2 2 R ^ M 0 2 62 R ); { (M 1 62 R ^ M 0 1 2 R ) ) (M 2 62 R ^ M 0 2 2 R ). Thus, ll the 2-edges in n equivlene lss re inident to R outwrds, or they re inident to R inwrds, or they re not inident to R. 2. 8M 1 ; M 2 2 P(B) M 1 6= M 2 ) (9 2 B M 1 2 R, M 2 62 R ). 3. For every pir of inequivlent 2-edges (M 1 ; M 0 1) nd (M 2 ; M 0 2), there exists some token 2 B suh tht one 2-edge is inident to R nd the other is not, or one 2-edge is inident to R inwrds nd the other is inident to R outwrds. These properties re lso vlid for sustrutures (X; E; ) of S2S(B), where R is the set fm 2 Xj 2 Mg. Denition 2.3 A region in prtil 2-struture G = (X; E; ) is suset of nodes R X suh tht for every pir of equivlent 2-edges (x 1 ; x 0 1) nd (x 2 ; x 0 2) in E: (x 1 2 R ^ x 0 1 62 R) ) (x 2 2 R ^ x 0 2 62 R), nd (x 1 62 R ^ x 0 1 2 R) ) (x 2 62 R ^ x 0 2 2 R). Let R G denote the set of (non trivil) regions of G, nd for x 2 X, let R G (x) = fr 2 R G j x 2 Rg. It is worth noting tht the omplement X n R of region R is region. In prtiulr X nd ; re regions (the trivil regions). Now the non trivil regions my serve s representtion tokens for sttes, tht is nodes, nd t the sme time for events, tht is lsses of equivlent 2-edges. One otins in this wy regionl versions of prtil 2-strutures dened s follows. Denition 2.4 Given prtil 2-struture G = (X; E; ), the regionl version of G is the prtil set 2-struture regv(g) = (X 0 ; E 0 ; ) with omponents X 0 = fr G (x)j x 2 Xg nd E 0 = f(r G (x); R G (x 0 ))j (x; x 0 ) 2 Eg. In this onstrution, illustrted in Fig. 1, node x is mpped to the set R G (x) of the regions whih inlude x. It ppers from Fig. 1, where equivlent edges er ommon lel, tht the mp regv is not n equivlene of prtil 2- strutures. The following theorem sttes when regv mps prtil 2-struture isomorphilly to prtil set 2-struture (the regionl representtion of the ltter).

1 2 g 3 4 R 0 = f1; 2; 3; 4g R 0 = ; R 1 = f1; 2g R 1 = f3; 4g R 2 = f1; 3g R 2 = f2; 4g (fr 1 g; fr 1 g) fr 0 ; R 1 ; R 2 g fr 0 ; R 1 ; R 2 g (fr 2 g; fr 2 g) regv(g) (fr 2 g; fr 2 g) fr 0 ; R 1 ; R 2 g fr 0 ; R 1 ; R 2 g (fr 1 g; fr 1 g) Fig. 1. prtil 2-struture nd its regionl version Theorem 2.5 A prtil 2-struture G = (X; E; ) is isomorphi to sustruture of some set 2-struture if nd only if G = regv(g) (with R G () s the isomorphism) if nd only if the following two xioms of seprtion re stised: 1. sttes seprtion: 8x 1 ; x 2 2 X x 1 6= x 2 ) 9R 2 R G (x 1 2 R, x 2 62 R). 2. events seprtion: for ll (x 1 ; x 0 1); (x 2 ; x 0 2) 2 E with (x 1 ; x 0 1) 6 (x 2 ; x 0 2) there exists some region R 2 R G suh tht either (x 1 ; x 0 1) is inident to R outwrds nd (x 2 ; x 0 2) is not or (x 2 ; x 0 2) is inident to R outwrds nd (x 1 ; x 0 1) is not. There my exist nodes x 1, x 2, x 3 nd x 4 suh tht (x 1 ; x 2 ) 2 E, (x 1 ; x 2) = (x 3; x 4), nd (x 3 ; x 4 ) 62 E. Therefore regv(g) is not hrterized y the sets fx j x 2 Xg nd fe j e 2 Eg. In order to redue the mismth, one should impose the dditionl xiom: 8(x 1 ; x 2 ) 2 E 8x 3 ; x 4 2 X (R G (x 1 ); R G (x 2 )) = (R G (x 3 ); R G (x 4 )) ) (x 3 ; x 4 ) 2 E. Further on this wy, one n even impose one or two stronger xioms: forwrd losure: 8(x 1 ; x 2 ) 2 E 8x 3 2 X (R G (x 1 ) n R G (x 2 ) R G (x 3 ) ^ R G (x 3 )\R G (x 2 )nr G (x 1 ) = ;) ) 9x 4 2 X (x 3 ; x 4 ) 2 E ^ (R G (x 1 ); R G (x 2 )) = (R G (x 3 ); R G (x 4 )). kwrd losure: 8(x 1 ; x 2 ) 2 E 8x 4 2 X (R G (x 2 ) n R G (x 1 ) R G (x 4 ) ^ R G (x 4 )\R G (x 1 )nr G (x 2 ) = ;) ) 9x 3 2 X (x 3 ; x 4 ) 2 E ^ (R G (x 1 ); R G (x 2 )) = (R G (x 3 ); R G (x 4 )). Prtil 2-strutures my e onsidered too generl from prtil point of view, nd one my prefer fousing on rehle prtil 2-strutures, suh tht ll nodes n e rehed y pths with ommon origin. A fmilir exmple of rehle prtil set 2-strutures is the lss of sequentil se grphs of elementry net systems. Denition 2.6 An elementry net is direted iprtite grph N = (P; E; F ) suh tht dom(f ) [ rn(f ) = P [ E. Elements of P, respetively E, re lled onditions (or ples), resp. events. Let x 2 y nd y 2 x e lterntive nottions of (x; y) 2 F. A se (or mrking) of N is suset of onditions M 2 P(P ). An event e hs onession in se M (noted M[e>) if nd only if ( e; e ) = (M; M 0 ) for some se M 0 (thus uniquely dened). The event e

my then re t M, resulting in the step M[e>M 0. Thus, M[e> if nd only if e M ^ M \ e = ;, nd then M[e>M 0 where M 0 = (M n e) [ e. A net is pure if 8x 2 P [ E x \ x = ;; it is simple if 8x; y 2 P [ E (x = y ^ x = y) ) x = y. The elementry nets onsidered from now on re ssumed to e pure nd simple. Denition 2.7 An elementry net system is struture N = (P; E; F; M 0 ) where N = (P; E; F ) is the underlying net nd M 0 (in P(P )) is the initil se. The sequentil se grph of N is the prtil set 2-struture sg(n ) = (X 0 ; E 0 ; ) where X 0 P(P ) is the smllest set of ses rehle from M 0 y sequenes of steps M[e>M 0 nd E 0 is the set of orresponding pirs (M; M 0 ). Lemm 2.8 A prtil set 2-struture G = (X; E; ) is the sequentil se grph of n elementry net system if nd only if it is rehle nd the following property is stised: 8(x 1 ; x 2 ) 2 E 8x 3 2 X (x 1 n x 2 x 3 ^ x 3 \ x 2 n x 1 = ;) ) 9x 4 2 X ((x 3 ; x 4 ) 2 E ^ (x 1 ; x 2 ) = (x 3 ; x 4 )). From Theo. 2.5 nd Lem. 2.8, one otins the following. Corollry 2.9 A prtil 2-struture G = (X; E; ) is isomorphi to the sequentil se grph of n elementry net system if nd only if it is rehle nd stises the xioms of sttes seprtion, events seprtion, nd forwrd losure. The elementry net system in the ove orollry is essentilly the set of the ordered symmetri dierenes (R G (x); R G (y)) for 2-edges (x; y) 2 E. The representtion prolem for prtil 2-strutures set t the eginning of the setion hs in ft een given the solution x = R G (x). The ples of the net re the regions r 2 R(G), the events re the equivlene lsses of edges, nd the ow reltion is suh tht: F ([e] ; r), r 2 R G (y) n R G (x) for some (x; y) 2 E; nd F (r; [e] ), r 2 R G (x) n R G (y) for some (x; y) 2 E. The initil se of the net system is dened s R G (x 0 ) for some x 0 2 X suh tht every node of G is rehle from x 0. 2.2 Elementry Automt The seond prt of the setion pves the wy for the lgorithmi nlysis of the region sed orrespondene etween rehle grphs nd elementry net systems. With this ojetive in mind, we rest the results otined so fr into the frmework of trnsition systems, nd illustrte the modied orrespondene on omplete exmple. Denition 2.10 A (lelled) trnsition system is triple A = (S; E; T ) with set of sttes S, set of events E, nd set of trnsitions T S E S. Let s e! s 0 e n equivlent nottion for (s; e; s 0 ) 2 T. An event e is enled t stte s (noted s e!) if s e! s 0 for some s 0. An event e is o-enled t s 0 (noted e! s 0 ) if s e! s 0 for some s. An utomton is struture A = (S; E; T; s 0 ) onsisting of n underlying trnsition system A = (S; E; T ) nd n initil stte s 0 2 S.

A prtil 2-struture G = (X; E; ) my e identied with the trnsition system (X; E= ; T ) where x [e]! x 0 if nd only if (x; x 0 ) e. This trnsition system is loopfree: s! e s 0 ) s 6= s 0, hs no multiple r: s e 1! s 0 ^ s e 2! s 0 ) e 1 = e 2, nd it is redued: 8e 2 E 9s; s 0 2 S s! e s 0. The sequentil se grphs of the redued net systems dened herefter fll in this sulss of trnsition systems. Denition 2.11 An elementry net system N = (P; E; F; M 0 ) is redued if every event e 2 E hs onession t some se M rehle from M 0, nd for every two distint onditions p; p 0 2 P there exists some se M rehle from M 0 suh tht p 2 M, p 0 62 M. The dul of redued elementry net system N is the utomton N = (S; E; T; M 0 ) where S is the set of ses rehle from M 0 y sequenes of steps M[e>M 0 nd T is the set of the orresponding trnsitions (M; e; M 0 ). Thus N is essentilly the imge of sg(n ) through the mp whih sends the equivlene lss of 2-edges f(m; M 0 )j (M; M 0 ) = ( e; e )g to the event e. Sine N is simple nd redued, this mp is one to one nd onto. By onstrution, N is rehle from M 0, deterministi: M e! M 0 ^ M e! M 00 ) M 0 = M 00, nd o-deterministi: M 0 e! M ^ M 00 e! M ) M 0 = M 00. The denition of regions my e rried to utomt in the following form. Denition 2.12 A region in n utomton A = (S; E; T; s 0 ), or in the underlying trnsition system (S; E; T ), is suset of sttes R S suh tht e e s1 2 R ^ s 8e 2 E 8s 1; s 2; s 3; s 4 2 S s 1! s2 ^ s 3! s4 ) 2 62 R ) s 3 2 R ^ s 4 62 R s 1 62 R ^ s 2 2 R ) s 3 62 R ^ s 4 2 R Let R A denote the set of (non trivil) regions of A, nd for s 2 S let R A (s) = fr 2 R A j s 2 Rg. Thus, R is region if nd only if the lel e of trnsition sues to determine whether the trnsition is inident to R inwrds (R is then termed n output region for e, noted e R), or it is inident to R outwrds (R is then termed n input region for e, noted R e), or it is not inident to R (it is internl to R or externl to R). In prtiulr, if A is rehle nd redued, the non trivil regions of A my e represented s mps R : E! f 1; 0; 1g suh tht R (e) = 1 if e R, R (e) = 1 if R e, nd R (e) = 0 otherwise; the hrteristi funtion of R, let R : S! f0; 1g where R (s) = 1, s 2 R, is then the unique mp suh tht s e! s 0 ) R (e) = R (s 0 ) R (s). It is esily seen tht for every ondition p of net system N, the set of the rehle ses M tht ontin p is region of N. This region, denoted y p nd lled the extension of p, is suh tht e p, e 2 p nd p e, e 2 p. Reversing the proess whih leds from net systems to sequentil se grphs, let us rest the denition of regionl versions in terms of nets nd net systems. Denition 2.13 Given n utomton A = (S; E; T; s 0 ), the dul of A is the (redued) elementry net system A = (R A ; (E= ) n f"g; F; s 0) where: is the equivlene on E indued y regions, let e 1 e 2, (8R 2 R A e 1 R, e 2 R ^ R e 1, R e 2 );

" is the equivlene lss of the events whih re inputless nd outputless i.e. whih re internl or externl to ll regions, if suh events exist; F is the ow reltion suh tht F ([e] ; R), e R nd F (R; [e] ), R e; nd s 0 = fr 2 R A j s 0 2 Rg. The net system A is lso lled the sturted net version of A (for resons explined in the sequel). The ounterprt of Cor. 2.9 for utomt is the following. Theorem 2.14 An utomton A = (S; E; T; s 0 ) is isomorphi to the dul N of n elementry net system if nd only if A = A if nd only if A is simple (it hs neither loop nor multiple r), redued, rehle nd it stises the following properties of seprtion: ssp (Sttes Seprtion Property): 8s; s 0 2 S s 6= s 0 ) 9R 2 R A (s 2 R, s 0 62 R) esp (Events Seprtion Property): 8e; e 0 2 E e 6= e 0 ) 9R 2 R A (R e ^ not(r e 0 )) _ (e R ^ not(e 0 R)) essp (Events-Sttes Seprtion Property): 8e 2 E 8s 2 S not(s e!) ) 9R 2 R A (R e ^ s 62 R) _ (e R ^ s 2 R) An utomton stisfying these onditions is termed n elementry utomton. Oserve tht every event in n elementry utomton hs input regions nd output regions (from ssp), hene the mp sending e to [e] is ijetion etween E nd (E= ) n f"g (from esp). The isomorphism from A to A (the sequentil se grph of the sturted net version of A) mps e to [e] nd s to s = fr 2 R A j s 2 Rg. This isomorphism pplies in prtiulr to sequentil se grphs, whene N = N for every elementry net system. However, N = (P; E; F; M 0 ) is generlly not isomorphi to its doule dul N. In ft, every ondition p of N indues orresponding region p of N whih inludes the rehle ses in whih ondition p holds, nd N is isomorphi to the full sunet system of N with set of events E= (= E) nd set of ples fp j p 2 P g. Thus, whenever N 0 = N, N 0 is isomorphi to sunet system of N whih is for tht reson termed the sturted version of N. Now, for n elementry utomton A, A = A entils tht A = A, hene A is lwys sturted net system. The im of the next setion is to optimize the synthesis proess y looking t dmissile sunets N of A suh tht A = N. Before tkling the synthesis prolem, we proeed to simplifying the presenttion of elementry utomt, nd retrieve the usul presenttion given in [11, 19, 34]. Proposition 2.15 Let utomton A e simple, redued nd rehle, then A is elementry if nd only if the seprtion properties ssp nd essp re stised.

Proof: Let A = (S; E; T; s 0 ), nd ssume for ontrdition e 6= e 0 nd 8R 2 R A (R e, R e 0 ) ^ (e R, e 0 R). We show tht s e! s 0 entils s e0! s 0 ontrditing the ssumption tht A is simple. Assume s! e s 0 nd not s!, e0 then y essp: 9R 2 R A (R e 0 ^ s 62 R) _ (e 0 R ^ s 2 R) nd the ontrdition of s! e s 0 follows from the denition of regions. Let s 00 2 S suh tht s! e0 s 00, then R A (s 00 ) = R A (s) n e 0 [ e 0 = R A (s) n e [ e = R A (s 0 ) nd s 0 = s 00 follows from esp. For omplete proofs of the results whih hve een stted in this susetion, the reder is referred to [19] where prtil 2-strutures re y-pssed. As n illustrtion, let us onsider the elementry net system nd the se grph given in Fig. 2. In Fig. 3 re displyed some of the non trivil regions of x 3 x2 z x 1 y 1 ' y 2 ' ' y 3 s 0 = fx 1; z; y 1g s 1 = fx 2; y 1g s 2 = fx 1; y 2g s 3 = fx 3; z; y 1g s 4 = fx 1; z; y 3g s 5 = fx 3; y 2g s 6 = fx 2; y 3g s 7 = fx 3; z; y 3g ' s 1 s 2 ' ' s 6 s 0 s 5 ' ' s 4 s 3 ' s 7 ' Fig. 2. n elementry net system nd its se grph this utomton. The missing items n e otined y symmetry. Eh drwing ' ' ' ' ' X ' 1 ' ' ' ' X 3 ' ' :X 1 ' ' :X 3 ' ' ' X 2 ' ' ' Z ' ' ' ' ' :X 2 ' ' ' ' :Z ' Fig. 3. some regions of the se grph of the elementry net system of Fig. 2 nd their ssoited tomi net systems represents region R onsisting of lk sttes. The ow reltions for the region R nd for its omplement :R = S n R re lso represented pitorilly; nlly one token indites whih of these omplementry regions ontins the initil stte.

We end up with the elementry net system of Fig. 4, whih is the originl net of Fig. 2 enrihed with dditionl ples (indited y dshed lines) ut with unhnged ehviour. The originl net system is emedded into its sturted :X 1 :Y 1 X 1 Z Y 1 ' ' X 3 X 2 Y 2 Y 3 :Z :X 3 :Y 3 ' :X 2 :Y 2 Fig. 4. the emedding of the elementry net system of Fig. 2 into its doule dul version y the mp tht sends ple x to its extension in the stte grph i.e. the set of mrkings fm 2 Sj x 2 Mg. 3 The Synthesis of Elementry Net Systems All utomt onsidered in this setion re ssumed to e pre-elementry, i.e. simple, rehle nd redued. The synthesis prolem of elementry net systems [19] is s follows: Given nite utomton A = (S; E; T; s 0 ), deide whether A = N for some elementry net system N with the sme set of events E, nd if so, onstrut N. Sine the set R A of ll the regions of A is nite, we lredy know from Prop. 2.15 tht this prolem n e deided in exponentil time y simultneously exploring R A, for heking stisftion of the seprtion properties esp nd essp, nd onstruting N = A. The im of this setion is to improve on this rute fore solution. We review rst Desel nd Reisig's study of dmissile sets of regions nd their tehniques for eliminting redundnt regions. Next we ount for Bernrdinello's results on the synthesis of stte mhine deomposle net systems, sed on the ruil remrk tht the miniml regions of n utomton form n dmissile set, nd for susequent work y Cortdell et l. on the reliztion of utomt y elementry nets up to some quotient of utomt. We nlly report the results otined on the omplexity of the synthesis prolem in [25, 3].

3.1 Admissile sets of regions In n elementry net system N = (P; E; F; M 0 ), eh ondition p 2 P determines n tomi sunet system of N, let N p = (fpg; E; F p ; M 0;p ) where F p is the restrition of F nd M 0;p (p) = M 0 (p). If we do not re out the isolted events P in N p, these tomi sunet systems re elementry nd N is just their sum p2p N p, where nets re glued together on events e 2 E. This deomposition my e used to isolte the ontriution of eh ondition p 2 P to the glol struture of the sequentil se grph N. This utomton my e seen s deterministi reognizer of nite sequenes, in whih every stte (i.e. se) is epting. An utomton of this type is hrterized up to isomorphism y the lnguge L it epts plus the equivlene on L whih identies these sequenes tht led to ommon (epting) stte. Now in the se of N, L nd re the intersetions for p rnging over P of the respetive lnguges nd equivlenes hrteristi of Np : L = \ p2p L p nd = \ p2p p. Thus the role of eh ondition p is twofold: on the one hnd, p uts o sequenes u e suh tht u 2 L ut u e 62 L p, nd on the other hnd p seprtes pirs of words u; v 2 L suh tht u 6 p v. Returning to the synthesis prolem, let us now lrify the reltionship etween utomt nd tomi net systems. Let A = (S; E; T; s 0 ) e nite deterministi utomton, with lnguge L nd equivlene, nd let N p = (fpg; E; F p ; M 0;p ) e n tomi net system, induing dul utomton Np with lnguge L p nd equivlene p. The utomton Np hs two sttes, ; nd fpg, one of whih is M 0;p, nd it hs trnsitions ;! e fpg if F p (e; p), fpg! e ; if F p (p; e), nd otherwise ;! e ; nd fpg! e fpg. Suppose L L p nd p. Let R p e the u u suset of sttes s 2 S suh tht s 0! s in A nd M0;p! fpg in N p for some sequene of events u 2 E. Then R p is region of A, s 0 2 R p, M 0;p = fpg, nd for every e 2 E: R p e, F p (p; e) nd e R p, F p (e; p). Conversely, for ny region R p of A, the elementry net system N p dened y the ove reltions indues dul utomton Np suh tht L L p nd p. Moreover, R p seprtes two distint sttes s 0 nd s 00 u v suh tht s 0! s0 nd s 0! s 00 in A if u nd only if u 6 p v, nd R p seprtes stte s suh tht s 0! s from n event e suh tht not(s!) e if nd only if u e 62 L p. Therefore, given net system N = (P; E; F; M 0 ) = P p2p N p, the dul utomton N is isomorphi to the utomton A if nd only if L = \ p2p L p nd = \ p2p p, if nd only if for ll p 2 P, N p is n tomi net system dened from some orresponding region R p in A nd the following properties re stised: ssp': 8u; v 2 L u 6 v ) 9p 2 P u 6 p v, essp': 8u 2 L 8e 2 E u e 62 L ) 9p 2 P u e 62 L p, if nd only if the fmily of regions fr p j p 2 P g is dmissile ording to the following denition. Denition 3.1 Given n utomton A = (S; E; T; s 0 ), suset of regions fr p j p 2 P g R A is dmissile if nd only if it inludes witnesses for the stisftion of every instne of the following seprtion prolems where e 2 E nd s; s 0 ; s 00 2 S re suh tht s 0 6= s 00 nd not(s e!):

ssp(s 0 ; s 00 ) : 9p 2 P s 0 2 R p, s 00 62 R p, essp(s; e) : 9p 2 P (R p e ^ s 62 R p ) _ (e R p ^ s 2 R p ). It is esily seen tht prolem ssp(s 0 ; s 00 ) nnot e solved positively in nondeterministi utomton A where s e! s 0 nd s e! s 00 for s 0 6= s 00. One redisovers in this wy si result estlished in [19]. Theorem 3.2 An utomton A = (S; E; T; s 0 ) is isomorphi to N for N = (P; E; F; M 0 ) if nd only if for every p 2 P, the tomi sunet system N p of N my e dened from some orresponding region R p of A, nd the set of regions fr p j p 2 P g is dmissile. In view of Def. 3.1 nd Theo. 3.2, the synthesis prolem for A = (S; E; T; s 0 ) my e solved y onsidering t most jsj (jsj + jej) regions of A. Nevertheless, this does not indite how to selet these regions from R A. The purpose is to onstrut suset of regions R R A s smll s possile suh tht R is dmissile if nd only if the whole set of regions R A is dmissile. Some struturl rules re proposed in [19] for the stepwise elimintion of redundnt regions, strting from R A. Denition 3.3 Let R R A e set of regions. A region R 2 R is redundnt in R if the following ssertions re equivlent: (i) R is dmissile (ii) R n frg is dmissile. Proposition 3.4 Let A = (S; E; T; s 0 ) nd R 2 R R A. In eh of the following ses R is redundnt in R. 1. S n R 2 R, 2. 9R 1 ; R 2 ; R 3 ; R 4 2 R R = R 1 \ R 2 ^ S n R = R 3 \ R 4, 3. 9R 1 ; R 2 ; R 3 ; R 4 2 R R = R 1 [ R 2 ^ S n R = R 3 [ R 4, 4. 9R 1 ; R 2 2 R R = R 1 \ R 2 ^ 8s 2 R 8e 2 E 8s 0 2 S n R s! e s 0 ) s 0 62 R 1 [ R 2. One redued set of regions R hs een otined from R A, one n hek diretly from Def. 3.1 whether it is dmissile, proving tht A is elementry, nd then extrt from R miniml suset fr p j p 2 P g suh tht A = ( P p2p N p). It is worth noting tht there exists in generl no lest dmissile set of regions. This ft is illustrted in Fig. 5 y the so-lled \four sesons" exmple reprodued from [19]. The \four sesons" utomton my e relized y two miniml sunet systems of the dul sturted net system: one hs four onditions nd is onttfree while the other one hs three onditions ut is not ontt-free. Denition 3.5 An elementry net system N = (P; E; F; M 0 ) is ontt-free if e M ) M \ e = ; for every event e nd for every rehle se M. Thus, the sulss of elementry net systems whih re ontt-free nd redued oinides with the sulss of the redued nd one-sfe Petri nets. Now, every sturted net system N = (P; E; F; M 0 ) is ontt-free: every ondition p 2 P indues two omplementry regions R p nd R p in N, nd sine N = N there

f1g f2g d 1 2 f4g f3g d 4 3 d f1; 3g d f2; 3g f3; 4g Fig. 5. the four sesons exmple: the utomton (on the left), the sturted net system (on the middle) nd two elementry net systems orresponding to miniml sets of regions (on the right) should exist some ondition p 2 P suh tht R p = R p. Therefore, every elementry utomton my e relized y one-sfe Petri net. The following dpttion of Theo. 3.2, sed on the use of omplementry regions, is estlished in [19] Proposition 3.6 An utomton A = (S; E; T; s 0 ) is isomorphi to N for ontt-free net system N = (P; E; F; M 0 ) = P p2p N p if nd only if every tomi sunet system N p of N my e dened from orresponding region R p 2 R A nd the following properties of seprtion re stised: ssp : 8s; s 0 2 S s 6= s 0 ) 9p 2 P s 2 R p, s 0 62 R p essp ] : 8e 2 E 8s 2 S not(s!) e ) 9p 2 P R p e ^ s 62 R p. 3.2 Miniml Regions Among the dmissile sets of regions of n elementry utomton, the set of miniml regions plys distinguished role euse it leds nturlly, s shown in [11], to stte mhine deomposle (nd hene ontt-free) net system relizing the utomton. Denition 3.7 An elementry net system N = (P; E; F; M 0 ) is stte mhine if its initil se is singleton nd every event hs one preondition nd one postondition. A stte mhine omponent of N = (P; E; F; M 0 ) is stte mhine N 0 = (P 0 ; E 0 ; F 0 ; M 0 0) suh tht P 0 P, E 0 = fe 2 Ej( e [ e ) \ P 0 6= ;g, F 0 = F \(E 0 P 0 [P 0 E 0 ), nd M 0 0 = M 0 \P 0. A stte mhine deomposition of N = (P; E; F; M 0 ) is fmily of stte mhines, let N i = (P i ; E i ; F i ; M 0;i ), suh tht P = [ i P i, E = [ i E i, F = [ i F i, nd M 0 = [ i M 0;i. A stte mhine is nothing else thn rehle utomton, s n e seen from Fig. 6 where the elementry net system given in Fig. 2 is deomposed into three stte mhine omponents. The respetive stte mhine omponents model sequentil proesses whih re synhronized on their ommon events. In this exmple, the synhroniztion prevents the leftmost nd rightmost proesses from

x 1 z y 1 ' ' ' x 3 x 2 x 2 y 2 y 2 y 3 ' ' Fig. 6. three stte mhine omponents of the net system of Fig. 2 entering simultneously the ritil setion gured y the mutully exlusive onditions x 2 nd y 2. Eh stte mhine omponent N i of net system N = P i N i my e seen s sequentil oserver of N, projeting ses of N on oservle onditions p 2 P i. By denition of stte mhine omponents, eh se of N projets to one nd extly one ondition p 2 P i, hene eh se of N elongs to extly one region R p of N suh tht p 2 P i. Proposition 3.8 Every stte mhine omponent N i = (P i ; E i ; F i ; M 0;i ) of n elementry net system N = P i N i determines regionl prtition fr p j p 2 P i g of the sequentil se grph N. Conversely, every regionl prtition fr p j p 2 P g of N determines stte mhine omponent of the sturted net system N. Returning to the exmple, the regionl prtitions of N (Fig. 3) whih determine the three stte mhine omponents shown in Fig. 6 re respetively fx 1 ; X 2 ; X 3 g, fx 2 ; Z; Y 2 g, nd fy 1 ; Y 2 ; Y 3 g where: Z = fs 0; s 3; s 4; s 7g X 1 = fs 0; s 2; s 4g X 2 = fs 1; s 6g X 3 = fs 3; s 5; s 7g Y 1 = fs 0; s 1; s 3g Y 2 = fs 2; s 5g = fs 4; s 6; s 7g Y 3 It my e oserved tht ll these regions re miniml w.r.t. set inlusion in R N. The prtiulr interest of miniml regions for the net system reliztion of elementry utomt is shown y the following proposition nd orollries. Proposition 3.9 Given n utomton A = (S; E; T; s 0 ), the following properties re stised y the set R A of regions of A: 1. If R 1 nd R 2 re disjoint regions then R 1 [ R 2 is region with (R 1 [ R 2 ) = ( R 1 [ R 2 ) n (( R 1 \ R 2 ) [ ( R 2 \ R 1 )) (R 1 [ R 2 ) = (R 1 [ R 2 ) n (( R 1 \ R 2 ) [ ( R 2 \ R 1 )):

2. If R nd R 0 re regions nd R 0 R then R n R 0 is region. If moreover R 0 is miniml then e (R n R 0 ) for every event e 2 R 0 whih is not inident to R (i.e. suh tht e 62 R [ R ). 3. If R is region nd s 2 R, then s 2 R 0 for some miniml region R 0 R. 4. If R is region nd e n event suh tht R e, then R 0 e for some miniml region R 0 R; symmetrilly if e is n event suh tht e R, then e R 0 for some miniml region R 0 R. 5. Every region is disjoint union of miniml regions. Corollry 3.10 A pre-elementry utomton is elementry if nd only if its set of miniml regions is dmissile. It my e further oserved tht the set of miniml regions of pre-elementry utomton A is dmissile w.r.t. the seprtion properties ssp nd essp if nd only if it is dmissile w.r.t. the seprtion properties ssp nd essp ]. In ft, let fr 1 ; : : : ; R n g e ny prtition of the set of sttes of A into miniml regions, then eh instne of the prolem essp(s; e) solved y region R i suh tht e R i nd s 2 R i n lso e solved y region R j suh tht R j e nd s 62 R j. Sine the set of ll prtitions of the set of sttes of A into miniml regions indues stte mhine deomposition of the net system P p N p dened from the set of ll miniml regions R p of A, one dedues lso the following. Corollry 3.11 Every elementry utomton my e relized y stte mhine deomposle (nd hene ontt-free) elementry net system. An lgorithm sed on miniml regions hs een proposed in [14] for vrint prolem of reliztion of utomt y net systems whih my e stted s follows. Given pre-elementry utomton A, deide whether exists nd onstrut (miniml) elementry net system N suh tht N = A 0 for some quotient A 0 of A. We rell tht A 0 = (S 0 ; E; T 0 ; s 0 e 0) is quotient of A = (S; E; T; s 0 ) if s 1! s2 in A if nd only if (s 1)! e (s 2) in A 0 for some surjetive mp : S! S 0 suh tht s 0 0 = (s 0 ). This prolem is similr to the originl synthesis prolem, up to the ft tht the sttes seprtion property ssp is ignored. Now the events-sttes seprtion property essp ] is vlid in A if nd only if for every event e the set of sttes fs 2 Sj s!g e oinides with the intersetion of the miniml regions R suh tht R e. The lgorithm strts from the sets fs 2 Sj s!g e nd inreses them into miniml regions, whih re generted until the vlidity of essp ] n e deided upon. The net N is then onstruted from miniml set of miniml regions dmissile with respet to essp ]. A vrint form of this lgorithm hs een integrted to softwre tool for the synthesis of synhronous iruits [15]. It should e noted tht the prolem of relizing utomt y nets up to quotient diers signintly from the prolem of relizing utomt y nets up to ehviourl equivlene (equlity of the epted lnguges). In order to mke the dierene visile, let us fous on nite nd deterministi utomt.

In this ontext, ehviourl equivlene oinides with isimilrity. Given - nite deterministi utomton A, with lnguge L nd hrteristi equivlene on L, the prolem of relizing A up to ehviourl equivlene onsists in onstruting n elementry net system N suh tht N reognizes L. For the prolem of relizing A up to quotient, it is set s further requirement tht ny two equivlent sequenes in L led to the sme se when they re red from the initil se of N. In orther words, it is sked tht N. The reson why this onstrint mkes notle dierene is tht the elementry utomt re not losed under quotient. This ounterft is illustrted in Fig. 7: the utomton shown on the middle is isomorphi to the se grph of the net displyed on the left, ut its minimized version shown on the right is not elementry (ny region R suh tht R must inlude stte 3, hene the prolem essp(3; ) nnot e solved). 1 1 2 3 d 2 3 d d 4 5 d 6 7 4 d 6 5 Fig. 7. elementry utomt re not losed under quotient 3.3 Complexity Results Hirishi proved in [25] tht the seprtion prolems ssp(s; s 0 ) nd essp ] (s; e) re NP-omplete in the respetive dt (A; s; s 0 ) nd (A; s; e). Sine regions in A re losed under omplementtion, the prolem essp(s; e) is lso NP-omplete. It does not follow therefrom tht the synthesis prolem for elementry net systems is NP-omplete; however this is the se. The synthesis prolem is oviously in NP sine the totl numer of instnes of seprtion prolems in n utomton A is qudrti in the size of A, nd it n e heked in polynomil time whether non-deterministilly hosen suset of sttes is region solving xed seprtion prolem. Now polynomil redution of 3-SAT to the synthesis prolem of elementry net systems ws estlished in [3], showing NP-hrdness sine 3- SAT is NP-omplete (see e.g. [23]). Rell tht 3-SAT is the prolem whether, given nite set of oolen luses over V, with three litterls per luse, there exists some truth ssignment for V vlidting eh luse. Eh lusl system of this form is ssoited in [3] with n utomton suh tht the lusl system is stisle if nd only if the utomton is elementry if nd only if the seprtion property essp ] is vlid. Therefore, the synthesis prolem for elementry net systems is NP-omplete, nd so is the prolem of relizing utomt y nets

up to quotient. The prolems of relizing utomt y nets up to ehviourl equivlene, or up to n unfolding (given A nd N suh tht A is isomorphi to quotient of N ) hve unknown omplexity. 4 Cutset Representtion of Finite Grphs We hve seen tht the region sed synthesis of elementry net systems from initilized prtil 2-strutures (X; E; ; x 0 ) is NP-omplete prolem. Nevertheless, this prolem is trivil when the lelling equivlene is disrete: in tht se, the prtil 2-struture is essentilly stte mhine with set of ples X; even etter, this stte mhine is equivlent to net system with jxj 1 ples, whose se grph is prtil set 2-struture isomorphi to the given prtil 2-struture. There exists lrge vriety of set-theoreti representtions for n unlelled grph (X; E), ll of whih using t most jxj 1 tokens. These representtions, sed on uts nd utsets, my e omputed y liner lgeri methods whih re quite stndrd in pplied grph theory. The purpose of this setion is to review these methods, nd therey shed light on regions in two respets. First, we exmine the lose reltionship etween regions nd uts (this nlogy ws rst pointed out to us y T. Murt). Seond, we indite the ostles to using liner lgeri methods for the region sed representtion of lelled grphs. On ount of this nlysis, vrint denition of regions is proposed in the next setion. 4.1 Cuts nd Cutsets Let G = (X; E) e nite, onneted nd simple direted grph with set of nodes X = fx 1 ; : : : ; x n g nd set of 2-edges E = fe 1 ; : : : ; e m g. So, G is free of loops multiple rs, lthough 2-edge e = (x l ; x k ) my hve n inverse e 1 = (x k ; x l ) in E. A utset of G is miniml set of 2-edges whose removl inreses the numer of onneted omponents y one. A ut of G is utset or n edge disjoint union of utsets. Sine G is onneted, every ut or utset C E determines two omplementry susets of nodes p nd X n p, oth non empty, suh tht for every 2-edge e = (x k ; x l ), e 2 C if nd only if x k 2 p, x l 62 p. Conversely, every non trivil suset p X determines ut etween p nd X n p, whih is utset when oth p nd X n p re onneted. An orienttion of the ut C results from the hoie of one of the two omplementry susets of nodes determined y the ut, let p. An oriented ut C my e oded y vetor C 2 IR m suh tht for every 2-edge e i = (x k ; x l ), C(i) = 1 if x k 62 p nd x l 2 p, C(i) = 1 if x k 2 p nd x l 62 p, nd C(i) = 0 if x k 2 p, x l 2 p. Let X = fx 1 ; : : : ; x n g nd E = fe 1 ; : : : ; e m g. We will ddress the prolem of onstruting vriety of sets of properties fp 1 ; : : : ; p n 1 g where p i X suh tht the prtil 2-struture (fx j x 2 Xg; fe j e 2 Eg; ) where x = fp i j x 2 p i g, (x k ; x l ) = (x k ; x l ), nd (x k ; x l ) = (x k n x l ; x l n x k ) is isomorphi to G viewed s prtil 2-struture: G = (X; E; id E ). Eh fmily of tokens fp 1 ; : : : ; p n 1 g

will determine orresponding set of (oriented) uts fc 1 ; : : : C n 1 g whih re linerly independent s vetors C i 2 IR m. The interesting ft here is tht one n esily onstrut liner ses of uts, given s sets of fundmentl utsets of G with respet to ritrry spnning trees. Rell tht spnning tree is set of edges U E, free of yles nd onneting X. The fundmentl utsets w.r.t. U re the uts whih inlude extly one rnh of U. Eh rnh of U determines two onneted omponents of U (nd thus of G), with set of nodes p nd X n p, suh tht every other rnh of U is internl either to p or to X n p. The fundmentl utsets w.r.t. U my e omputed y lssil methods of liner lger. These methods re relled elow, following the nottions of [16]. 4.2 Computing Cutsets The grph G = (X; E) is hrterized up to isomorphism y its inidene mtrix. We rell tht this mtrix A = [ i;j ] is n n m mtrix with entries in f 1; 0; 1g, with i;j = 0 if edge e j is not inident to node x i, i;j = 1 if x i is the soure of e j, nd i;j = 1 if x i is the trget of e j. Sine every olumn ontins extly two non zero entries (1 nd 1) every row n e omputed from the other rows, nd the mtrix A hs the sme rnk s the mtrix A 1 otined y ersing its lst row. Let A = A1 A 2 where A 1 is n (n 1) m mtrix nd A 2 is n 1 m mtrix. Atully A 1 nd A hve rnk n 1. Assume w.l.o.g. tht the (n 1) rnhes of the spnning tree U re the edges e 0 j = e j+(m n+1) for j 2 f1; : : : ; n 1g. Then A 1 = A 11 A 12 where A12 is the (n 1) (n 1) mtrix orresponding to the edges of the tree (the rnhes) nd A 11 is the (n 1) (m n + 1) mtrix orresponding to the other edges (the hords). The fundmentl utset C i of G determined y the edge e 0 i of the spnning tree is given y the i th row of the fundmentl utset mtrix Q f = A 1 12 A 1. This (n 1) m mtrix hs the form Q f 11 I n 1 where Ik is the identity mtrix of rnk k. The i th row of Q f ssoited with the fundmentl utset C i is n m vetor with entries in f 1; 0; 1g. Let p i nd X n p i e the two onneted omponents of G seprted y C i, suh tht e 0 i hs its soure in X n p i nd its trget in p i. Then for every j 2 f1; : : : ; mg, C i (j) = 0 if e j is not in C i, C i (j) = 1 if e j is oriented from X n p i to p i nd C i (j) = 1 if e j is oriented from p i to X n p i. A omplete exmple is shown in Fig. 8. It is worth noting tht the mtrix A 1 12 n e omputed diretly from G without inverting mtrix A 12, for it oinides with the pth mtrix P = [p i;j ] dened s follows. For eh j 2 f1; : : : ; n 1g, let j e the unique hin (in the tree U) onneting x j nd the referene node x n ; then for 1 i; j n 1, let p i;j = 0 if e 0 i does not elong to j, p i;j = 1 if e 0 i elongs to j nd is oriented towrds the referene node x n, nd p i;j = 1 if e 0 i elongs to j nd is oriented towrds node x j.

4.3 Cutset Representtion of Grphs The nodes of G my e oded injetively y f0; 1g vetors ording to their memership to the properties p j determined y the uts C j, resulting in n n (n 1) mtrix S = [s i;j ], lled the stte mtrix, suh tht s i;j = 1 if x i 2 p j, nd s i;j = 0 if x i 62 p j. Let S = [X 1 X n] t, where the X i re olumn vetors. The set fxi tj i ng of rows of S, representing nodes x i, together with the set fc i j i < ng of rows of Q f, representing fundmentl utsets, provide representtion of G. These dt re lso suient for retrieving the spnning tree. Atully, there is extly one wy to ssemle the row vetors C i into mtrix of the form Q f = Q f 11 I n 1 ; nd n ordered pir of vetors (Xk ; X l ) represents n edge e j = (x k ; x l ) if nd only if X l X k = Q f (; j). 4.4 Vrint Representtions A vrint representtion of G is given y the pir of mtries P nd Q f11. As mtter of ft, the redued inidene mtrix A 1 = A 11 A 12 my e omputed y A 12 = P 1 nd A 11 = P 1 Q f11. The pth mtrix P n in turn e reonstruted from X n nd the redued stte mtrix S = [X 1; : : : ; X n 1] t. Atully, for every j < n, X n = X j +P j where P j is the j th olumn of P (oding the hin j onneting x j nd x n ), hene the pth mtrix P nd the redued stte mtrix (n 1) times z } { S re onneted y the identity S t = [X n; : : : ; X n] - P. In prtiulr, S = P t if ll edges e 0 j of U re oriented wy from the referene node x n. 4.5 Fundmentl Cyles It hs some importne for the sequel to note tht the informtion provided y the fundmentl mtrix Q f is extly the sme s the informtion provided y the fundmentl yle mtrix 1 B f, dened s follows from the spnning tree U. Eh hord (i.e. edge in E n U) determines yle in G, onsisting of this edge nd the unique hin in U tht onnets its endpoints. This yle my e represented y n m vetor B i with entries in f 1; 0; 1g s follows: B i (j) = 0 if e j is not ontined in the yle, else B i (j) = 1 or 1 depending on whether the orienttion of e j grees with, or is opposite to the orienttion of e i within this yle. The fundmentl yle mtrix B f is the (m n + 1) m mtrix dened y B f (i; j) = B i (j). This mtrix is of the form B f = I m n+1 B f 12, where B f12 = Q t f11 (in prtiulr, rnh elongs to the fundmentl yle dened y hord if nd only if the hord elongs to the fundmentl utset dened y the rnh). Therefore, B f Q t f = 0, nd the vetor spes V B nd V Q respetively generted over IR y the fundmentl yles (rows of B f ) nd y the fundmentl utsets (rows of Q f ) re orthogonl. These two vetor spes, whih do not depend on the hoie of the spnning tree, re indeed orthogonl omplements of IR m. 1 lled fundmentl iruit mtrix in [16]

e s 5 1 s 2 e 2 e1 e 4 s 4 s 3 e 3 Inidene Mtrix: A = e 1 e 2 e 3 e 4 e 5 s 1 1 0 0 0 1 s 2 0 1 0 1 1 s 3 0 0 1 1 0 s 4 1 1 1 0 0 = 2 4 A11 A12 A 2 3 5 n = 4 verties m = 5 edges r = n 1 = 3 rnk C 5 Pth Mtrix: s 1 s 2 s 3 P = A 1 12 = e 3 1 0 0 e 4 1 1 0 e 5 1 1 1 e s 5 1 s 2 C 4 Fundmentl Cutset Mtrix: e1 e 2 s 4 e 3 s 3 e 4 Q f = e 1 e 2 e 3 e 4 e 5 C 3 1 1 1 0 0 C 4 1 1 0 1 0 C 5 1 0 0 0 1 = Q f11 I r C 3 Stte Mtrix: Q f = P A 1 S = C 3 C 4 C 5 s 1 0 0 0 s 2 1 0 0 s 3 1 1 0 s 4 1 1 1 e s 5 1 s 2 1 e 2 e1 e 4 2 s 4 s 3 e 3 Fundmentl Cyle Mtrix: B f = e 1 e 2 e 3 e 4 e 5 1 1 0 1 1 1 2 0 1 1 1 0 = I m r Bf12 B f 12 = Q t f 11 Fig. 8. fundmentl utsets nd yles

Every non null vetor in V B with entries 1, 0, nd 1 is sum of fundmentl yles nd/or inverses of fundmentl yles, hene it is either yle or n edge disjoint union of yles in vetor form. Similrly, every non null vetor C in V Q with entries 1, 0, nd 1 denes ut fe j j C(j) 6= 0g, ut C my dier y the sign of its omponents from the vetor whih represents this ut (nd lso from the opposite of this vetor). For ounterexmple, let C = ( 1; 1) where e 1 nd e 2 hve the sme trget nd distint soures. 4.6 Bk to set 2-Strutures We sw tht G my e represented y set of f0; 1g vetors expressing the set of properties of its nodes (x j 2 p i, X j (i) = 1), plus the set of the fundmentl utsets whih dene these properties (the utset C i dening p i is given y the i th row of Q f ). A node x j is then identied with the set of tokens x j = fij X j(i) = 1g; similrly, n edge e j = (x k ; x l ) is identied with the ordered pir e j = (x k ; x l ). We show tht the resulting prtil 2-struture G = (fx j x 2 Xg; fe j e 2 Eg; ) is tully isomorphi to the given grph G = (X; E; id E ). It is esily seen tht the ove representtion is injetive on nodes, sine two dierent nodes of the spnning tree re lwys seprted y fundmentl utset. In order to prove tht G = G, it sues therefore to show tht (e j ) = (e l ) entils e j = e l. We estlish stronger property, nmely: Lemm 4.1 Let e j = (x k ; x l ) e n edge of G, then for every pir of nodes x p nd x q, (x k ; x l ) = (x p ; x q) entils tht x p = x k nd x l = x q. Proof: Assuming the premises, let e hin onneting x p nd x q in the spnning tree U, represented y vetor 2 f 1; 0; 1g m y \orienting" the hin from x p to x q. Suppose (j) = 1, thus the edge e j is oriented wy from x q nd towrds x p in tht hin. Let p j e the property dened y the fundmentl utset whih inludes e j, then neessrily x p ; x l 2 p j nd x q ; x k 62 p j, hene x k n x l 6= x p n x q, ontrditing our ssumptions. Therefore, if we let 1 j denote the vetor with 1 t position j nd 0 elsewhere, the vetor 1 j hs ll entries in f 1; 0; 1g. Sine Q f mesures vritions of properties long, the ssumption (x k ; x l ) = (x p; x q) reds s Q f = Q f 1 j. Thus the vetor 1 j lies in V B, nd it is either yle or disjoint union of yles in vetor form. Sine there is no yle in U, it follows tht 1 j is yle, hene x p = x k nd x q = x l s ws to show. Now, ny set fp 0 1 ; : : : ; p0 n 1g of non trivil susets of X determines orresponding 2-struture G = (fx j x 2 Xg; fe j e 2 Eg; ), dened s ove y setting Xj = fij x j 2 p 0 i g nd (x k; x l ) = (x k ; x l ). For 1 i n 1, let C0 i denote the ut seprting the omplementry susets X np 0 i nd p0 i. We will show tht G = G whenever the orresponding vetors C1; 0 : : : ; Cn 1 0 re linerly independent. This is for instne the se when p 0 i = fx ig. Bewre of the ft tht G my e isomorphi to G even though C1; 0 : : : ; Cn 1 0 re not linerly independent. For n illustrtion, let p 0 1 = fx 2 ; x 3 g, p 0 2 = fx 2 ; x 4 g nd p 0 3 = fx 1 ; x 3 g in

G = (X; E) where X = fx 1 ; x 2 ; x 3 ; x 4 g nd E = fe 1 ; e 2 ; e 3 g with e i = (x 1 ; x i+1 ), then G = G ut C2 0 +C3 0 = 0. Notie tht in this representtion of G the vetors e 1, e 2, nd e 3 re not linerly independent: e 1 + 2 e 2 + e 3 = 0 even though there is no yle in G. Assuming tht C1; 0 : : : ; Cn 1 0 re linerly independent, let us prove tht G = (X; E; id E ) nd G = (fx j x 2 Xg; fe j e 2 Eg; ) re isomorphi prtil 2-strutures. Let x k 6= x l nd ssume for ontrdition x k = x l. Let e the hin in U onneting the verties x k nd x l. By onstrution of the uts Ci, 0 Ci 0 = 0 for every i n 1. Sine C0 1 ; : : : ; C0 n 1 re linerly independent, they spn the vetor spe V Q nd is yle, thus x k = x l. It remins to show tht (e j ) = (e i ) entils e j = e i. Lemm 4.2 Let e j = (x k ; x l ) e n edge of G, then for every pir of nodes x p nd x q, (x k ; x l ) = (x p; x q) entils tht x p = x k nd x l = x q. Proof: Let e hin onneting x p nd x q in the spnning tree, represented y vetor 2 f 1; 0; 1g m y \orienting" the hin from x p to x q. Suppose (j) = 1, thus the edge e j is in nd it is oriented wy from x q nd towrds x p in tht hin. From the ssumption (x k ; x l ) = (x p ; x q) nd y onstrution of the uts C 0 i, it follows tht C 0 i = 1 j C 0 i for ll i n 1, where 1 j denotes the vetor with 1 t position j nd 0 elsewhere. Thus ( 1 j ) C 0 i = 0 for ll i, nd sine C 0 1 ; : : : ; C0 n 1 form sis of the vetor spe V Q, it follows tht ( 1 j ) C k = 0 for ll k n 1 nd in prtiulr for h = j (m n + 1). Now the rst m n + 1 entries of the vetor 1 j re zeros nd the lst n 1 entries of C h re zeros ut C h (j) whih is 1. Therefore, (j) = 1 nd we hve rehed ontrdition. Thus the vetor 1 j hs ll entries in f 1; 0; 1g. Sine ( 1 j ) C 0 i = 0 for ll i, the vetor 1 j lies in V B, nd it is either yle or disjoint union of yles in vetor form. Sine there is no yle in U, it follows tht 1 j is yle, hene x p = x k nd x q = x l s ws to show. We now give n exmple (see Fig. 9) showing tht the omputtion of uts nd utsets nnot led diretly to net representtion of G = (X; E). Let fp 1 ; p 2 g e 3 e 1 x 1 x 1 x 2 x 3 e 1 e 2 p 1 1 1 0 p 2 p 1 fp 1 g e p 3 2 1 0 1 x 2 e x 3 3 e 1 e 2 e 2 e 2 e 1 ; fp 2 g Fig. 9. elementry net system ssoited with sis of uts X = fx 1 ; x 2 ; x 3 g nd E = fe 1 ; e 2 ; e 3 g with e 1 = (x 1 ; x 2 ), e 2 = (x 1 ; x 3 ) nd e 3 = (x 2 ; x 3 ). A sis of uts for G is given y the vetors C 1 = (0; 1; 1)