Kleene Theorems for Free Choice Nets Lbelled with Distributed Alphbets Rmchndr Phwde Indin Institute of Technology Dhrwd, Dhrwd 580011, Indi Emil: prb@iitdh.c.in Abstrct. We provided [15] expressions for free choice nets hving distributed choice property which mkes the nets direct product representble. In recent work [14], we gve equivlent syntx for lrger clss of free choice nets obtined by dropping distributed choice property. In both these works, the clsses of free choice nets were restricted by product condition on the set of finl mrkings. In this pper we do wy with this restriction nd give expressions for the resultnt clsses of nets which correspond to free choice synchronous products nd Zielonk utomt. For free choice nets with distributed choice property, we give n lterntive chrcteriztion which uses properties checkble in polynomil time. Free choice nets we consider re 1-bounded, S-coverble, nd re lbelled with distributed lphbets, where S-components of the ssocited S-cover respect the given lphbet distribution. 1 Introduction There re severl different notions of cceptnce to define lnguges for lbelled Petri nets, depending on restrictions on lbelling nd finl mrkings [12]. The lnguge of net with n initil mrking nd finite set of finl mrkings, is clled L-type lnguge [7]. One gol of this work is to give syntx of expressions for L-type lnguges for vrious subclsses of 1-bounded, free choice nets lbelled with distributed lphbets. One dvntge of using distributed lphbet is tht we cn see free choice nets s products of utomt [11], enbling us to write expressions for the nets using components. This lso enbles us to compre expressiveness of nets nd products of utomt. We construct expressions for L-type lnguges of free choice nets vi free choice Zielonk utomt. Consider the net N of Figure 1 with G = {{r 1, s 1 }, {r 2, s 2 }} s its set of finl mrkings, with its decomposition into finite stte mchines in Figure 2. Becuse this net is decomposble, its mrkings cn be written in tuple form, where ech s-component hs plce in the tuple: for exmple G = {(r 1, s 1 ), (r 2, s 2 )}. For the finl mrking {(r 1, s 1 )}, its lnguge is expressed s fsync((b+c), (d+e) ). Similrly, for the finl mrkings {(r 2, s 2 )} equivlent expression is fsync((b + c), (d + e) ). In generl, if the plces involved in the finl mrkings form product, then its lnguge is specified by tking product of component expressions, using free choice Zielonk utomt with product-cceptnce [15]
78 PNSE 18 Petri Nets nd Softwre Engineering r 1 s 1 r 1 s 1 r 2 r 3 s 2 s 3 b c d e Fig. 1. Free Choice Net without distributed choice r 2 r 3 s 2 s 3 b c d e Fig. 2. S-cover of the net in Fig. 1 s intermediry. Even though r 1 nd s 2 prticipte in finl mrkings, mrking {r 1, s 2 } does not belong to G, hence set G do not form product. The lnguge L of net system (N, G) cn be described by, fsync((b+c), (d+e) )+fsync((b+ c), (d+e) ). The key ide is bility to express the lnguge of net s union of lnguge of nets hving product condition on finl set of mrkings. This closure under union my not be lwys possible for restricted clsses of lnguges defined over distributed lphbet. For exmple, union of direct product lnguges L 1 = {c, cb} nd L 2 = {c, cbb} defined over Σ 1 = {c, } nd Σ 2 = {c, b} respectively, is not expressible s direct product lnguge. But this lnguge is ccepted by synchronous products : the direct products extended with subsetcceptnce. For the restricted clss of direct product representble free choice nets, with its set of finl mrkings hving product condition-we gve expressions vi product systems with mtchings nd product-cceptnce [15, 16]. As second gol, we develop syntx for free choice nets with distributed choice, now extended with subset-cceptnce. For net in this clss lso, its lnguge cn be expressed s union of lnguges ccepted by product system with mtchings nd product-cceptnce. This union is ccepted by product systems with mtching nd extended with subset-cceptnce ( free choice synchronous products ). As third contribution, we develop n lternte chrcteriztion of this clss of nets, vi free choice Zielonk utomt with product-moves. Lnguge equivlent expressions for 1-bounded nets hve been given by Grbowski [5], Grg nd Rgunth [4] nd other uthors [8], where renming opertor hs been used in the syntx to dismbigute synchroniztions. We hve chosen to not use this opertor nd to lso exploit the S-decompositions of nets. The syntx for smller sublclsses of nets such mrked grphs nd free choice nets with initil mrkings s feedbck vertex set hs been given erlier [9, 13]. Rest of the pper is orgnized s follows. In the next section, we begin with preliminries on distributed lphbets nd nets. In Section 3 we define product systems with globls nd subset-cceptnce, nd show tht their lnguges cn be expressed s union of lnguges ccepted by product systems with globls nd product-cceptnce. The following section reltes these product systems to nets. In Section 5 we develop syntx of expressions for product systems with subset
Phwde et.l.: Kleene Theorems for Lbelled Free Choice Nets 79 cceptnce, nd next section estblishes the correspondence between vrious clsses of product systems nd expressions. In lst section we conclude, with overview of estblished correspondences between ll three formlisms. 2 Preliminries Let Σ be finite lphbet nd Σ be the set of ll words over lphbet Σ, including the empty word ε. A lnguge over n lphbet Σ is subset L Σ. The projection of word{ w Σ to set Σ, denoted s w, is defined by: (σ ε = ε nd (σ) = ) if, σ if /. Given lnguges L 1, L 2,..., L m, their synchronized shuffle L = L 1... L m is defined s: w L iff for ll i {1,..., m}, w Σi L i. Definition 1 (Distributed Alphbet). Let Loc denote the set {1, 2,..., k}. A distribution of Σ over Loc is tuple of nonempty sets (Σ 1, Σ 2,..., Σ k ) with Σ = 1 i k Σ i. For ech ction Σ, its loctions re the set loc() = {i Σ i }. Actions Σ such tht loc() = 1 re clled locl, otherwise they re clled globl. A globl ction is globl in the loctions in which it occurs. For set S let (S) denote the set of ll its susbets. For singleton sets like {p}, sometimes we my write it s p. 2.1 Nets Definition 2. A lbelled net N is tuple (S, T, F, λ), where S is set of plces, T is set of trnsitions lbelled by the function λ : T Σ nd F (T S) (S T ) is the flow reltion. Elements of S T re clled nodes of N. Given node z of net N, set z = {x (x, z) F } is clled pre-set of z nd z = {x (z, x) F } is clled post-set of z. Given set Z of nodes of N, let Z = z Z z nd Z = z Z z. We only consider nets in which every trnsition hs nonempty pre- nd postset. For ll ctions in Σ let T = {t t T nd λ(t) = }. We re only interested in 1-bounded nets, where plce is either mrked or not mrked. Hence we define mrking of net s subset of its plces. Definition 3. A lbelled net system is tuple (N, M 0, G) where N = (S, T, F, λ) is lbelled net with M 0 S s its initil mrking nd set of mrkings G (S). A trnsition t is enbled in mrking M if ll plces in its pre-set re mrked by M. In such cse, t cn be fired to produce the new mrking M = (M \ t) t nd, we write this s M t M or M λ(t) M.
80 PNSE 18 Petri Nets nd Softwre Engineering A firing sequence λ(t 1 )λ(t 2 )... is defined by composition of lbels of trn- t sition sequence t 1 t 2... fired from M 0 i.e., M 1 t 0 2 M1.... In such firing sequence, we sy tht M j is rechble from M i, for every i j. We sy net system (N, M 0 ) is live if, for every rechble mrking M nd every trnsition t, there exists mrking M rechble from M which enbles t. Definition 4. For lbelled net system (N, M 0, G), its lnguge is defined s Lng(N, M 0, G) = {λ(σ) Σ σ T σ nd M 0 M, for some M G}. Net Systems nd its components Here we define components of net. Our notion of component does not require strong connectedness nd so it is different from notion of S-component in [3], nd therefore our notion of S-cover lso differs from theirs. Definition 5. Let N = (S X, T X, F (X X)) be subnet of net N = (S, T, F ), generted by nonempty set X of nodes of N. Subnet N is clled component of N if, For ech plce s of X, s, s X (the pre- nd post-sets re tken in N), For ll trnsitions t T, we hve t = 1 = t (N is n S-net [3]), Under the flow reltion, N is connected. A set C of components of net N is clled S-cover for N, if every plce of the net belongs to some component of C. A net is covered by components if it hs n S-cover. Fix distribution (Σ 1, Σ 2,..., Σ k ) of Σ. We define s-decomposition [6] of net into sequentil components. Definition 6. A lbelled net N = (S, T, F, λ) is clled S-decomposble if, there exists n S-cover C for N, such tht for ech T i = {λ 1 () Σ i }, there exists S i such tht the induced component (S i, T i, F i ) is in C. Now from S-decomposbility we get n S-cover for net N, since there exist subsets S 1, S 2,..., S k of plces S, such tht S = S 1 S 2... S k nd S i S i = T i, such tht the subnet (S i, T i, F i ) generted by S i nd T i is n S-net, where F i is the induced flow reltion from S i nd T i. If net (S, T, F, λ) is 1-bounded nd S-decomposble then mrking cn be written s k-tuple from its component plces S 1 S 2... S k. We give below the product condition on the set of finl mrkings of net system which is known [17, 11] to restrict clsses of lnguges. Definition 7. An S-decomposble lbelled net system with product-cceptnce (N, M 0, G) is n S-decomposble lbelled net N = (S, T, F, λ) with n initil mrking M 0 nd set of mrkings G (S), which is product: if q 1, q 2,... q k G nd q 1, q 2,... q k G then {q 1, q 1} {q 2, q 2}... {q k, q k } G. Let t be trnsition in T. Then by S-decomposbility pre-plce nd post-plce of t belongs to ech S i for ll i in loc(). Let t[i] denote the tuple p,, p such tht (p, t), (t, p ) F i, for p, p P i for ll i in loc().
Phwde et.l.: Kleene Theorems for Lbelled Free Choice Nets 81 2.2 Free choice nets nd their properties Let x be node of net N. The cluster of x, denoted by [x], is the miniml set of nodes contining x such tht if plce s [x] then s is included in [x], nd if trnsition t [x] then t is included in [x]. The set of ll -lbelled trnsitions long with plces r 1 nd s 1 form cluster of the net shown in Figure 4. Definition 8 (Free choice nets [3]). A cluster C is clled free choice (FC) if ll trnsitions in C hve the sme pre-set. A net is clled free choice if ll its clusters re free choice. In lbelled net N, for free choice cluster C = (S C, T C ) define the - lbelled trnsitions C = {t T C λ(t) = }. If the net hs n S-decomposition then we ssocite post-product π(t) = Π i loc() (t S i ) with every such trnsition t. This is well defined since by the S-net condition every trnsition will hve t most one post-plce in S i. Let post(c ) = π(t). We lso define t C the post-projection of the cluster C [i] = C S i nd the post-decomposition postdecomp(c ) = Π i loc() C [i]. Clerly post(c ) postdecomp(c ). The following definition from [16, 15] provides the wy to direct product representbility. Definition 9 (distributed choice property). An S-decomposble free choice net N = (S, T, F, λ) is sid to hve distributed choice property (DCP) if, for ll in Σ nd for ll free choice clusters C of N, postdecomp(c ) post(c ). r 1 s 1 r 1 s 1 r 2 r 3 s 2 s 3 b c d e Fig. 3. Lbelled Free Choice Net system with distributed choice 2 2 r s r 3 s 3 b c d e Fig. 4. S-cover of the net in Fig. 3 Exmple 1 (Free choice net system without distributed choice nd hving productcceptnce). Consider distributed lphbet Σ = (Σ 1 = {, b, c}, Σ 2 = {, d, e}) nd the lbelled net system N shown in Figure 1 nd lbelled over Σ. Its (only
82 PNSE 18 Petri Nets nd Softwre Engineering possible) S-cover hving two S-components with sets of plces S 1 = {r 1, r 2, r 3 } nd S 2 = {s 1, s 2, s 3 } respectively, is given in Figure 2. For the cluster C of r 1, we hve the set of -lbelled trnsitions C = {t 1, t 2 } with its post-projections C [1] = {r 2, r 3 } nd C [2] = {s 2, s 3 }. So we get postdecomp(c ) = {(r 2, s 2 ), (r 2, s 3 ), (r 3, s 2 ), (r 3, s 3 )}. The post-products re π(t 1 ) = {(r 2, s 2 )} nd π(t 2 ) = {(r 3, s 3 )} so post(c ) = {(r 2, s 2 ), (r 3, s 3 )}. Since postdecomp(c ) post(c ), this cluster does not hve distributed choice, so the net system does not hve it. With the set of finl mrkings {(r 1, s 1 ), (r 1, s 2 ), (r 2, s 1 )(r 2, s 2 )} stisfying product condition, the lnguge L p ccepted by this net system is r [ε + + b + d] where r = ((bd + db) + (ce + ec)). Exmple 2 (Free choice net system without distributed choice nd not stisfying product condition of the set of finl mrkings). Consider the net system of Exmple 1 whose underlying net is shown in Figure 1. With set of finl mrkings {(r 1, s 1 ), (r 2, s 2 )}, which do not stisfy product condition, the lnguge L s ccepted by this net system is r [ε + ] where r = ((bd + db) + (ce + ec)). Exmple 3 (A net with distributed choice property nd product cceptnce condition). Consider the lbelled net system (N, (r 1, s 1 ), G)) of Figure 3, defined over distributed lphbet Σ = (Σ 1 = {, b, c}, Σ 2 = {, d, e}), nd where G = {(r 1, s 1 ), (r 1, s 2 ), (r 2, s 1 ), (r 2, s 2 )} is the set of finl mrkings stisfying product condition. Its two S-components with sets of plces S 1 = {r 1, r 2, r 3 } nd S 2 = {s 1, s 2, s 3 }, re shown in Figure 4. For cluster C of r 1, we hve C = {t 1, t 2, t 3, t 4 }, C [1] = {r 2, r 3 } nd C [2] = {s 2, s 3 }, hence postdecomp(c ) = {(r 2, s 2 )(r 2, s 3 )(r 3, s 2 )(r 3, s 3 )}. The post-products for trnsitions re π(t 1 ) = {(r 1, s 2 )}, π(t 2 ) = {(r 2, s 3 )}, π(t 3 ) = {(r 3, s 2 )}, π(t 4 ) = {(r 3, s 3 )}, giving us post(c ) = {(r 2, s 2 )(r 2, s 3 )(r 3, s 2 )(r 3, s 3 )}. Therefore we hve postdecomp(c ) = post(c ). For ll other clusters this holds trivilly, becuse ech of them hve only one trnsition nd only one post-plce, hence the net hs distributed choice. Lnguge L 3 ccepted by the net system is r [ε + + (b + c) + (d + e)] where r = ((bd + db) + (be + eb) + (cd + dc) + (ce + ec)). Exmple 4 (A net with distributed choice nd subset-cceptnce). Consider the net system of Exmple 3, with the underlying net shown in Figure 3 with set of finl mrkings {(r 1, s 1 ), (r 2, s 2 )}. The lnguge L 4 ccepted by this net system is r [ε + ] where r = ((bd + db) + (be + eb) + (cd + dc) + (ce + ec)). 3 Product systems We define product systems over fixed distribution (Σ 1, Σ 2,..., Σ k ) of Σ. First we define sequentil systems. Definition 10. A sequentil system over set of ctions Σ i is finite stte utomton A i = P i, i, G i, p 0 i where P i re clled sttes, G i P i re finl sttes, p 0 i P i is the initil stte, nd i P i Σ i P i is set of locl moves.
Phwde et.l.: Kleene Theorems for Lbelled Free Choice Nets 83 For locl move t = p,, p of i stte p is clled pre-stte sometimes denoted by pre(t) nd p is clled post-stte of t, sometimes denoted by post(t). This nottion is extended to set of locl moves s well. Let i denote the set of ll -lbelled moves in the sequentil system A i. The lnguge of sequentil system is defined s usul. Definition 11. Let A i = P i, i, G i, p 0 i be sequentil system over lphbet Σ i for 1 i k. A product system A over the distribution Σ = (Σ 1,..., Σ k ) is tuple A 1,..., A k. Let Π i Loc P i be the set of product sttes of A. We use R[i] for the projection of product stte R in A i, nd R I for the projection to I Loc. The initil product stte of A is R 0 = (p 0 1,..., p 0 k ). The finl sttes of A re denoted by G. 3.1 Direct products Let = Π i loc() i for the product system A. The set of ll globl moves of A is = Σ. This move is globl within the set of component sequentil mchines where lbel occurs. Then for globl move g in, we define its set of pre-sttes pre(g) s the set of pre-sttes of ll its component -moves. The set of post-sttes post(g) s the set of post-sttes of ll its component -moves; nd, let g[i] denote its i-th component locl -move belonging to A i, for ll i in loc(). When finl sttes of A re G = Π i Loc G i then we sy A is product system with product-cceptnce. These systems re clled direct products in [11]. And, if G Π i Loc G i then A is product system with subset-cceptnce, these systems re clled synchronous products in [11]. The runs of product system A over some word w re described by ssociting product sttes with prefixes of w: the empty word is ssigned initil product stte R 0, nd for every prefix v of w, if R is the product stte reched fter v nd Q is reched fter v where, for ll j loc(), R[j],, Q[j] j, nd for ll j / loc(), R[j] = Q[j]. A run of product system over word w is sid to be ccepting if the product stte reched fter w is in G. We define the lnguge Lng(A) of product system A, s the set of words on which the product system hs n ccepting run. The set of lnguges ccepted by direct (resp. synchronous) products is clled direct (resp. synchronous) product lnguges. We use chrcteriztion from [11] of lnguges ccepted by direct products. Proposition 1. L is direct product lnguge defined over distributed lphbet Σ iff L = {w Σ i {1,..., k}, u i L such tht w Σi = u i Σi }. If L = Lng(A) for direct product A = A 1,..., A k defined over distributed lphbet Σ then L = Lng(A 1 )... Lng(A k ). We lso use chrcteriztion of synchronous product lnguges [11]. Proposition 2. A lnguge over distributed lphbet Σ is ccepted by product system with subset-cceptnce if nd only if it cn be expressed s finite union of direct product lnguges.
84 PNSE 18 Petri Nets nd Softwre Engineering Definition 12 (PS-mtchings [15]). For globl Σ, mtching() is subset of tuples Π i loc() P i such tht for ll i in loc(), projection of these tuples is the set of ll pre-sttes of -moves in i, nd if stte p P i ppers in one tuple, it does not pper in nother tuple. We sy product stte R is in mtching() if its projection R loc() is in the mtching. A product system is sid to hve mtching of lbels if for ll globl Σ, there is suitble mtching(). Such system is denoted by PS-mtchings. We hve PS-mtchings with product-cceptnce, if the set of finl product sttes of it is product of finl sttes of component mchines, or PS-mtchings with subset-cceptnce, if the set of finl product sttes is subset of product of finl sttes of individul components. A run of PS-mtchings A is sid to be consistent with mtching of lbels [15] if for ll globl ctions nd every prefix of the run R 0 v R Q, the pre-sttes R loc() re in the mtching. Consistency of mtchings is behviourl property nd to check if PSmtchings A hs it nd cn be done in PSPACE [15, 16]. Following property from [15] is used to cpture free choice property. Definition 13 (conflict-equivlent mtchings for PS-mtchings). In product system, we sy the locl move p,, q 1 i is conflict-equivlent to the locl move p,, q 1 j, if for every other locl move p, b, q 2 i, there is locl move p, b, q 2 j nd, conversely, for moves from p there re corresponding outgoing moves from p. For globl ction, its mtching() is clled conflict-equivlent mtching, if whenever p, p re relted by the mtching(), their outgoing locl -moves re conflict-equivlent. strt b r 1 strt c d r 2 r 3 s 2 s 3 A 1 A 2 s 1 e Figure 5, shows product system defined over distributed lphbet Σ = (Σ 1 = {, b, c}, Σ 2 = {, d, e}). It hs two components A 1,nd A 2 with finl sttes G 1 = {r 1, r 2} nd, G 2 = {s 1, s 2}, respectively. Fig. 5. Product system (A 1, A 2) Exmple 5 (Product system with mtchings). Consider product system of Figure 5 nd reltion mtching() = {(r 1, s 1 )} reltion. This mtching is conflictequivlent nd the system is consistent with this mtching reltion. We hve PS-mtchings A = (A 1, A 2 ) with product cceptnce condition, if its set of finl sttes is G 1 G 2. With the set of finl sttes s {(r 1, s 1 ), (r 2, s 2 )} G 1 G 2, we hve PS-mtchings B = (A 1, A 2 ) hving subset-cceptnce.
Phwde et.l.: Kleene Theorems for Lbelled Free Choice Nets 85 Lemm 1 presents lnguge not ccepted by ny direct product. Lemm 1. The lnguge L 4 = {(bd + db + be + eb + ce + ec + cd + dc) (ε + )} from Exmple 4 is not ccepted by ny direct product. We know tht the clss of synchronous product lnguges is strictly lrger thn the clss of direct product lnguges [11]. With the mtching reltions this reltionship is preserved. The PS-mtchings B of Exmple 5 ccepts lnguge L 4 which by Lemm 1, is not ccepted by ny direct product. Hence, the clss of lnguges ccepted by PS-mtchings with subset cceptnce condition, is strictly lrger, thn the clss of lnguges ccepted by PS-mtchings with productcceptnce. However, using Proposition 2 we hve the following chrcteriztion of PSmtchings with subset-cceptnce. Corollry 1. A lnguge L is ccepted by product system with subset-cceptnce nd, hving conflict-equivlent nd consistent mtchings if nd only if L cn be expressed s finite union of lnguges ccepted by product system with productcceptnce nd, hving conflict-equivlent nd consistent mtchings. Lemm 2 presents lnguge not ccepted by ny synchronous product. Lemm 2. The lnguge L s = {bd, db, ce, ec} (ε + ) of Exmple 2 is not synchronous product lnguge. So we hve lnguge L s which is not ccepted by ny PS-mtchings with subsetcceptnce. This motivtes the bigger clss of utomt over distributed lphbets, which we discuss next. 3.2 Product systems with globls Let A = A 1,..., A k, be product system over the distribution Σ = (Σ 1,..., Σ k ) nd, let globls() be subset of its globl moves, nd -globl denote n element of globls(). Definition 14. A product system with globls (PS-globls) is product system with reltions globls(), for ech globl ction in Σ. With subset-cceptnce condition these systems re clled Asynchronous (or Zielonk) utomton [17, 11]. Runs of product system with globls, re defined in the sme wy s for the direct products, with n dditionl requirement of Π j loc() ( R[j],, Q[j] globls(), to be stisfied when R Q is to be tken. With buse of nottion sometimes we use pre() to denote the set {R Q, R Q}. Following property [14] is used to cpture free choiceness, in product systems with globls. Definition 15 (sme source property). A product system with globls hve sme source property if, ny two globl moves shre pre-stte then their sets of pre-sttes re sme.
86 PNSE 18 Petri Nets nd Softwre Engineering Exmple 6 (Product system with globls). Consider the product system of Figure 5. Let globls() = { ((r 1 r2 ), (s 1 s2 )), ((r 1 r3 ), (s 1 s3 ))}. This system hs sme source property. With the given globls() we hve PS-globls C = (A 1, A 2 ) with product cceptnce condition, if its set of finl sttes is G 1 G 2. And, for the set of finl sttes {(r 1, s 1 ), (r 2, s 2 )} G 1 G 2, we hve PS-globls D = (A 1, A 2 ) with subset-cceptnce. The lnguge L s = {bd, db, ce, ec} (ε + )} of Lemm 2, is ccepted by product system with globls D of Exmple 6 with sme source property. Product systems with globls nd product-cceptnce re not considered in [11]. We give in Lemm 3, chrcteriztion of clss of lnguges ccepted by product systems with globls nd hving subset-cceptnce, in terms of PSglobls nd product-cceptnce. Lemm 3. A lnguge is ccepted by PS-globls with subset-cceptnce if nd only if it cn be expressed s finite union of lnguges ccepted by PS-globls with product-cceptnce. In the construction, trnsition structure of locl components is preserved, nd so the globl moves, so we hve Corollry 2, which is used to get syntx for product systems with subset cceptnce condition. Corollry 2. A lnguge L is ccepted by PS-globls with subset-cceptnce nd hving sme source property if nd only if L cn be expressed s finite union of lnguges ccepted by PS-globls with product-cceptnce nd sme source property. In product system with globls nd hving sme source property, globl moves for n ction cn be prtitioned into different comprtments : two globl -moves belong to sme comprtment if they hve the sme set of pre-sttes. For ny -globl g of sme source comprtment SS, we ssocite trget- ) = {π(g) g SS }. [i] = post( SS ) P i nd, [i]. configurtion π(g) = Π i loc() post(g) P i. Let post( SS We define post-projection of comprtment s SS post-decomposition s postdecomp( SS ) = Π i loc() SS Definition 16. A product system with globls nd hving sme source property, is sid to hve product moves property, if for ll in Σ, nd for ll sme source comprtments SS of -globls, postdecomp( SS ) post( SS ). Product systems C nd D of Exmple 6 do not hve product moves property. Product system of Exmple 7 hs product moves property. Exmple 7 (Product system with globls nd product moves property). Consider product system A of Exmple 5 where the set of finl sttes is G 1 G 2. With globls() = { ((r 1 r2 ), (s 1 s2 )), ((r 1 r2 ), (s 1 s3 )), ((r 1 r3 ), (s 1 s 2 )) ((r 1 r3 ), (s 1 s3 )) } reltion, we hve PS-globls A which hs product moves property. This lso hs sme source property.
Phwde et.l.: Kleene Theorems for Lbelled Free Choice Nets 87 Now consider the product system B of Exmple 5, with {(r 1, s 1 ), (r 2, s 2 )} G 1 G 2 s its finl sttes, nd hving the globls() reltion s bove, we get PS-globls B with subset cceptnce condition, hving product moves nd sme source property. A product system with globls is sid to be live, if for ny globl move g nd ny rechble product stte R, there exists product stte Q such tht g is enbled t Q. 3.3 Relting product systems with mtchings nd globls First we show in Theorem 1 how to construct product system with consistent nd conflict-equivlent mtchings from PS-globls with sme source property. Theorem 1. Let Σ be distributed lphbet nd A be product system with globls defined over it.then we cn construct product system B with mtchings, liner in the size of product system A with globls such tht, 1. if A hs sme source property then B hs conflict-equivlent mtchings, 2. in ddition, if A is live then () B is consistent with mtchings, nd (b) Lng(A) = Lng(B). Now, from PS-mtchings with consistent nd conflict-equivlent mtchings we construct product system with globls hving sme source property. Theorem 2. Let Σ be distributed lphbet nd let B be product system with conflict equivlent nd consistent mtchings. Then for the lnguge of B we cn construct product system A with globls over Σ hving sme source nd product moves property. The constructed product system A with globls is exponentil in the size of system B hving mtching of lbels. 4 Nets nd Product systems We first present generic construction of 1-bounded S-coverble lbelled net systems, from product systems with globls. Definition 17 (PS-globls to nets). Given PS-globls A = A 1,..., A k over distribution Σ, net system (N = (S, T, F, λ), M 0, G) is constructed s follows: The set of plces is S = i P i, The set of trnsitions T = globls(), for ll ctions in Σ. Define T i = {λ 1 () Σ i }. The lbelling function λ lbels by ction the trnsitions in globls(). The flow reltion is F = {(p, g), (g, q) g T, g[i] = p,, q, i loc()}, define F i s its restriction to the trnsitions T i for i loc(). Let M 0 = {p 0 1,..., p 0 k }, be the initil product stte nd G = G s the set of finl globl sttes.
88 PNSE 18 Petri Nets nd Softwre Engineering We get one to one correspondence between rechble sttes of product system nd rechble mrkings of nets becuse the set of trnsitions of resultnt net is sme s the set of globl moves in the product system, nd construction preserves pre s well s post plces. Lemm 4. The constructed net system N from PS-globls A, s in Definition 17, is S-coverble nd Lng(N, M 0, G) = Lng(A). The size of constructed net is liner in the size of product system. Applying the generic construction bove to product systems with sme source property, we get free choice net, becuse ny two globl moves hving sme set of pre-plces re put into one cluster. Theorem 3. Let (N, M 0, G) be the net system constructed from PS-globls A s in Definition 17. If A hs sme source property then N is free choice net, In ddition if A hs product moves property, then N hs distributed choice. In the construction, if the product system hs subset-cceptnce then we get net with set of finl mrkings, which my not hve product condition. Since A hs subset-cceptnce, we generlize the results obtined in [14]. For the product system D of Exmple 6, ccepting lnguge L s we cn construct the net system of Exmple 2. In the cse tht the product system A hs mtchings, trnsitions of net constructed re rechble globl moves of system A [15, 16]. Even if net is 1-bounded nd S-coverble ech component need not hve only one token in it, but when we sy tht 1-bounded net is S-coverble we ssume tht ech component hs one token. For live nd 1-bounded free choice nets, such S-covers cn be gurnteed [3]. Now we describe liner-size construction of product system from net which is S-coverble. Definition 18 (nets to PS-globls). Given 1-bounded lbelled nd n S- coverble net system (N, M 0, G), with N = (S, T, F, λ) the underlying net nd N i = (S i, T i, F i ) the components in the S-cover, for i in {1, 2,..., k}, we define product system A = A 1,..., A k, s follows. Tke P i = S i, nd p 0 i the unique stte in M 0 P i. Define locl moves i = { p, λ(t), p t T i nd (p, t), (t, p ) F i, for p, p P i }. So we get sequentil systems A i = P i, i, p 0 i, nd the product system A = A 1, A 2,..., A k over lphbet Σ. Globl moves re globls() = {Π i loc() t[i] t T }. And, the set of finl sttes is G = G. Lemm 5. From net system N with finl set of mrkings, the construction of the PS-globls A in Definition 18 bove preserves lnguge. The product system A is liner in the size of net, nd product system hs subset-cceptnce. For ech -lbelled trnsition of the net we get one globl -move in the product system hving sme set of pre-plces nd post-plces. And, for ech globl - move in product system we hve n -lbelled trnsition in the net hving sme
Phwde et.l.: Kleene Theorems for Lbelled Free Choice Nets 89 pre nd post-plces. We get one to one correspondence between rechble sttes of product system nd rechble mrkings of the net we strted with. Therefore, if we begin with free choice net, we get sme source property in the product system obtined. And, for ech trnsition in the net we hve globl trsition hence, A hs product moves property if the net hs distributed choice. Theorem 4. Let (N, M 0, G) be 1-bounded, nd n S-coverble lbelled net with set of finl mrkings G. Then if N hs free choice property, then constructed product system A with globls, hs sme source property, in ddition, if the net hs distributed choice, then A hs product moves. In construction of Definition 18, we strt with net hving distributed choice nd finl set of mrkings then we get product system with mtching with subset cceptnce condition. Note tht in this cse we do not hve to construct globls [15, 16]. For the net system of Exmple 2 nd ccepting lnguge L s we cn construct the product system D of Exmple 6. Therefore, we generlize the results from [15, 16]. Theorem 5. For 1-bounded, S-coverble lbelled net hving distributed choice nd given with set of finl mrkings. Then one cn construct product system with conflict-equivlent nd consistent mtchings nd hving subset-cceptnce. Given below is the reverse direction. Theorem 6. For product system with conflict-equivlent, consistent mtchings, nd subset-cceptnce, we get lnguge equivlent free choice net with distributed choice nd hving set of finl mrkings. 5 Expressions First we define regulr expressions nd its derivtives. 5.1 Regulr expressions nd their properties A regulr expression over lphbet Σ i such tht constnts 0 nd 1 re not in Σ i is given by: s ::= 0 1 Σ i s 1 s 2 s 1 + s 2 s 1 The lnguge of constnt 0 is nd tht of 1 is {ε}. For symbol Σ i, its lnguge is Lng() = {}. For regulr expressions s 1 + s 2, s 1 s 2 nd s 1, its lnguges re defined inductively s union, conctention nd Kleene str of the component lnguges respectively. As mesure of the size of n expression we will use wd(s) for its lphbetic width the totl number of occurrences of letters of Σ in s.
90 PNSE 18 Petri Nets nd Softwre Engineering For ech regulr expression s over Σ i, let Lng(s) be its lnguge nd its initil ctions form the set Init(s) = { v Σi nd v Lng(s) } which cn be defined syntcticlly. We cn syntcticlly check whether the empty word ε Lng(s). We use derivtives of regulr expressions which re known since the time of Brzozowski [2], Mirkin [10] nd Antimirov [1]. Definition 19 (Antimirov derivtives [1]). Given regulr expression s nd symbol, the set of prtil derivtives of s with respect to, written Der (s) re defined s follows. Der (0) = Der (1) = Der (b) = {ε} if b =, otherwise Der (s 1 + s 2 ) = Der (s 1 ) Der (s 2 ) Der (s 1) = { Der (s 1 ) s 1 Der (s Der (s 1 s 2 ) = 1 ) s 2 Der (s 2 ), if ε Lng(s 1 ) Der (s 1 ) s 2 otherwise Inductively Der w (s) = Der w (Der (s)). The set of ll prtil derivtives Der(s) = Der w (s), where Der ε (s) = {s}. w Σ i We hve derivtives Der (b + c) = {b, c} nd Der ((b + c)) = {b + c}. A derivtive d of s with ction Init(d) is clled n -site of s. An expression is sid to hve equl choice if for ll, its -sites hve the sme set of initil ctions. For set D of derivtives, we collect ll initil ctions to form Init(D). Two sets of derivtives hve equl choice if their Init sets re sme. As in [15] we put together derivtives which my correspond to the sme stte in finite utomton. Definition 20 ([15]). Let s be regulr expression nd L = Lng(s). For set D of -sites of regulr expression s nd n ction, we define the reltivized lnguge L D = {xy xy L, d Der x (s) D, d Der y (d) with ε Lng(d )}, nd the prefixes Pref D (L) = {x xy L D }, nd the suffixes Suf D (L) = {y xy L D }. We sy tht the derivtives in set D -bifurcte L if L D = Pref D (L) Suf D (L). We use prtitions [15] of the -sites of s into blocks such tht ech block (tht is, element of the prtition) -bifurctes L. For n ction, let Prt (s) denote such prtition. In ddition to thinking of blocks of the prtition s plces of n utomton, we cn think of blocks nd their effects s locl moves. Definition 21 ([14]). Given n ction, set of -sites B of regulr expression s nd specified set of -effects E Der (B), we define the reltivized lnguges L (B,E) = {xy L d Der x (s) B, d Der (d) E, nd d Der y (d ) with ε Lng(d )}.
Phwde et.l.: Kleene Theorems for Lbelled Free Choice Nets 91 We define the prefixes P ref (B,E) Suf (B,E) L (B,E) (L) = {y xy L (B,E) (L) = {x xy L (B,E) } nd the suffixes }. We sy tht tuple (B, E) -funnels L if = P ref B (L) Suf (B,E) (L). In such pir (B, E), if B is block in the Prt (s) nd E is nonempty subset of -effects of B, then it is clled s n -duct. For n -duct (B, E), we define its set of initil ctions Init(B, E) s Init(B, E) = Init(B), cll B s its pre-block nd cll E s its post-effect. For ll i in loc() let -ducts(s i ) denote the set of ll -ducts of regulr expression s i. For ny two - ducts (B, E) nd (B, E ) in -ducts(s i ), define (B, E) = (B, E ) if B = B nd E = E. Given n -duct d = (B, E) its post-effect E is sometimes denoted by d nd its pre-block B cn be denoted s d. For collection of ducts z, the set of ll their post-effects (resp. pre-blocks) is denoted s z (resp. z). In similr wy, we define the set of post-effects of n -cble D, s D = {D[i] i loc()} nd its set of pre-blocks s D = { D[i] i loc()}. 5.2 Connected expressions over distributed lphbets The syntx of connected expressions defined over distribution (Σ 1, Σ 2,..., Σ k ) of lphbet Σ is given below. e ::= 0 fsync(s 1, s 2,..., s k ), where s i is regulr expression over Σ i When e = fsync(s 1, s 2,..., s k ) nd I Loc, let the projection e I = Π i I s i. A connected expression e = fsync(s 1, s 2,..., s k ) over Σ, is sid to hve equl choice if, for ll globl ctions in Σ nd for ny i, j in loc(), ny -site of s i hve sme Init set s of ny -site of s j. For connected expression defined over distributed lphbet its derivtives nd semntics were given in [15], nd re given s follows. For the connected expression 0, we hve Lng(0) =. For the connected expression e = fsync(s 1,..., s k ), its lnguge is Lng(e) = Lng(s 1 ) Lng(s 2 )... Lng(s k ). The definitions of derivtives extended to connected expressions [15] is s follows. The expression 0 hs no derivtives on ny ction. Given n expression e = fsync(s 1, s 2,..., s k ), its derivtives re defined by induction using the derivtives of the s i on ction : Der (e) = {fsync(r 1,..., r k ) i loc(), r i Der (s i ); otherwise r j = s j }. 5.3 Connected expressions with pirings We recll some properties of connected expressions over distribution. which were [15], useful in construction of free choice nets. Definition 22 ([15]). Let e = fsync(s 1, s 2,..., s k ) be connected expression over Σ. For globl ction, piring() is subset of tuples Π i loc() Prt (s i ) such tht the projection of these tuples includes ll the blocks of Prt (s i ), nd
92 PNSE 18 Petri Nets nd Softwre Engineering if block of Prt (s j ), j loc() ppers in one tuple of the piring, it does not pper in nother tuple. (For convenience we lso write piring() s subset of Π i loc() Der(s i ) which respects the prtition.) We cll piring() equl choice if for every tuple in the piring, the blocks of derivtives in the tuple hve equl choice. Derivtives for connected expressions with piring re defined s follows. A derivtive fsync(r 1,..., r k ) is in piring() if there is tuple D piring() such tht r i D[i] for ll i loc(). For convenience we my write derivtive s n element of piring(). Expression e is sid to hve (equl choice) piring of ctions if for ll globl ctions, there exists n (equl choice) piring(). Expression e is sid to be consistent with piring of ctions if every rechble -site d Der(e) is in piring(). Expression e is sid to hve equl choice property if it hs equl choice piring of ctions for ll globl ctions in Σ. Given connected expression e with pirings, checking if it is consistent with piring of ctions cn be done in PSPACE [15, 16]. 5.4 Connected expression with cbles (CE-cbles) We give some properties of connected expressions over distribution, which extend the notion of piring, nd hve been relted to product systems with globls [14]. Definition 23 ([14]). Let e = fsync(s 1, s 2,..., s k ) be connected expression over Σ. For ech ction in Σ, we define the set -cbles(e) = Π i loc() - ducts(s i ). For n ction, n -cble is n element of the set -cbles(e). We sy tht block B of Prt (s i ) ppers in n -cble D if there exists j in loc() nd there exists Y Der (B) such tht D[j] = (B, Y ), i.e. if B is pre-block of component -duct of D. For ny -cble D, its set of pre-blocks D = i loc() {B i B i ppers in D}, i.e. the set of pre-blocks of ll the of its component -ducts. For expression e, let cbles() -cbles(e), such tht for ll i in loc() 1. Ech block B in Prt (s i ), ppers in t lest one -cble of it. 2. for ll (B, E) nd (B, E ) in -ducts(s i ) with (B, E) (B, E ), if B = B = E E =, i.e. if ny two distinct -ducts of s i ppering in it hve sme pre-block then, they must hve disjoint post-effects. A connected expression with cbles (CE-cbles) is connected expression with reltions cbles() of it, for ech globl ction in Σ. Derivtives of connected expression with cbles re [14] defined s follows. The CE-cbles 0 hs no derivtives on ny ction. For expression e = fsync(s 1, s 2,..., s k ), we define its derivtives on ction, by induction, using -ducts nd the derivtives of s j s: Der (e) = {fsync(r 1, r 2,..., r k ) r j Der (s j ) if there exists n -cble D in cbles() such tht, for ll j in loc(), s j is in pre-block B j nd r j is in X j of -duct D[j] = (B j, X j ) of s j, otherwise r j = s j }.
Phwde et.l.: Kleene Theorems for Lbelled Free Choice Nets 93 We use the word derivtive for expressions such s d = fsync(r 1,..., r k ) given bove. The rechble derivtives re Der(e) = {d d Der x (e), x Σ }. A CEcbles is sid to hve equl source property if for ny pir of two cbles shring common pre-block hve sme set of pre-blocks. Lnguge of e is the set of words over Σ defined using derivtives s below. Lng(e) = {w Σ e Der w (e) such tht ε Lng(r i ), where e [i] = r i }. So we cn hve next derivtive on ction, if it is llowed by the cbles() reltion. The number of derivtives my be exponentil in k. Exmple 8 (CE-pirings nd CE-cbles). Let Σ = (Σ 1 = {, b, c}, Σ 2 = {, d, e}) be distributed lphbet. Let e = fsync((b + c), (d + e) ) be connected expression defined over Σ. Here, r 1 = (b + c) nd s 1 = (d + e). The set of derivtives of r 1 is Der(r 1 ) = {r 1, r 2 = br 1, r 3 = cr 1 } nd for s 1 it is Der(s 1 ) = {s 1, s 2 = ds 1, s 3 = es 1 }. We hve -sites(r 1 ) = r 1 nd Prt (r 1 ) = {D = r 1 }. Similrly, -sites(s 1 ) = r 1 nd Prt (s 1 ) = {D = s 1 }. The only possible piring reltion is piring() = {(D, D )}. We hve Der (e) = {fsync(r i, s j ) i, j {2, 3}}. Expression e stisfies equl choice property. Now we ssocite cbling reltion with e. The set of -effects of D is Der (D) = {r 2, r 3 }. The set of -ducts of r 1 is {(D, r 2 ), (D, r 3 ), (D, {r 2, r 3 })}. The set of -effects of D is Der (D ) = {s 2, s 3 } nd the set of -ducts of s 1 is {(D, s 2 ), (D, s 3 ), (D, {s 2, s 3 })}. A possible cbles() reltions for expression e is {((D, r 2 ), (D, s 2 )), ((D, r 3 ), (D, s 3 ))}. See tht ech block in the Prt (r 1 ) nd Prt (s 1 ) ppers t lest once in the cbles() reltion. And two -ducts of r 1 ppering in this reltion, hve sme pre-block D, we hve tht their set of post-effects r 2 nd r 3 re disjoint. This condition lso holds for -ducts of s 1. For both -cbles set of pre-blocks is identicl, therefore cbles() stisfies equl source property. We hve Der (e) = {fsync(r 2, s 2 ), fsync(r 3, s 3 )}, but expression fsync(r 2, s 3 ) is not in Der (e), becuse only post-effect of D contining r 2 is the set {r 2 } nd similrly, only post-effect of D contining s 3 is the set {s 3 } nd there does not exist n -cble with (D, {r 2 }) nd (D, {s 3 }) s its components. We hve Der(e) = {e, (r 2, s 2 ), (r 3, s 3 ), (r 1, s 2 ), (r 1, s 3 ), (r 2, s 1 ), (r 3, s 1 )}. Another such exmple of connected expression with pirings (resp. cbles) is given below. Exmple 9 (CE-pirings). Let e = fsync((b+c), (d+e) ) be connected expression defined over Σ. Let p 1 = (b + c) with lnguge L 1 nd q 1 = (d + e) with lnguge L 2. The set of derivtives re Der(p 1 ) = {p 1, p 2 = bp 1, p 3 = cp 1, p 4 = ε} nd Der(q 1 ) = {q 1, q 2 = dq 1, q 3 = eq 1, q 4 = ε}. The prtitions of -sites is Prt (p 1 ) = {B = {p 1 }} nd Prt (q 1 ) = {B = {q 1 }}. A piring reltion is piring() = {(B, B )} nd wrt to tht Der (e) = {fsync(p i, q j ) i, j {2, 3, 4}}. Expression e hs equl choice property. Now we ssocite cbling reltion with e. The set of -effects of B is Der (B) = {p 2, p 3, p 4 } nd Der (B ) = {q 2, q 3, q 4 }. The set of -ducts for p 1 is {(B, {p 2 }), (B, {p 3 }), (B, {p 2, p 4 }), (B, {p 3, p 4 })(B, {p 2, p 4, p 3 })} nd for q 1
94 PNSE 18 Petri Nets nd Softwre Engineering is {(B, q 2 ), (B, q 3 ), (B, {q 2, q 4 }), (B, {q 3, q 4 })(B, {q 2, q 4, q 3 })}. A cbles() reltion is {((B, p 2 ), (B, q 2 )), ((B, p 3 ), (B, q 3 )), ((B, p 2 ), (B, q 2 ))}. Expression e hs equl source property nd its set of derivtives with respect to letter is Der (e) = {fsync(p 2, q 2 ), fsync(p 3, q 3 ), fsync(p 4, q 4 )}. Now we give two new properties of connected expressions. In connected expression with cbles nd hving equl source property, cbles for n globl ction cn be prtitioned into different comprtments : two -cbles belong to sme comprtment if they hve equl source. For ny such -cble D belonging to n equl source comprtment ES of -cbles, we cn ssocite set of its postblocks listed in some order s π(d) = Π i loc() (D χ i ), where χ i = {Prt (s i ) Σ}. Let post(es ) = {π(d) D ES }. We define post-projection of comprtment ES s ES [i] = post(es i ) χ i, nd its post-decomposition s postdecomp(es ) = Π i loc() ES [i]. Following property of connected expressions is relted lter to product-movesproperty of direct products nd hence to distributed-choice of nets. Definition 24 (product derivtives property). A connected expression with cbles nd hving equl source property, is sid to hve product derivtives property, if for ll in Σ, nd for ll equl source comprtments ES of -cbles postdecomp(es ) post(es ). Exmple 10. The connected expression e = fsync((b + c), (d + e) ) of Exmple 8 does not hve product derivtive property with the given cbling reltion {((D, r 2 ), (D, s 2 )), ((D, r 3 ), (D, s 3 ))}. If we ssocite the cbling reltion {((D, r 2 ), (D, s 2 )), ((D, r 3 ), (D, s 3 ), ((D, r 2 ), (D, s 3 )), ((D, r 3 ), (D, s 2 ))} with e then it hs product derivtive property. Definition 25. A connected expression e is ction-live if for ll ctions in Σ, from ny rechble derivtive of e, we cn rech n -derivtive of e. 5.5 Relting connected-expressions with pirings nd with cbles First, we show how connected expressions with equl choice nd consistent pirings cn be seen s connected expression with cbles nd hving equl source nd product-derivtives property. Theorem 7. Let Σ be distributed lphbet nd e be connected expression hving equl choice nd consistent piring of ctions, defined over Σ. Then for the lnguge of e, we cn construct connected expression e with cbles hving equl source nd product derivtives property. The constructed expression e with cbles is exponentil in the size of expression e with pirings. Now we give lnguge preserving construction of connected expression with equl choice nd consistent pirings from CE-cbles hving equl source nd product-derivtives property.
Phwde et.l.: Kleene Theorems for Lbelled Free Choice Nets 95 Theorem 8. Let Σ be distributed lphbet nd e be connected expression with cbles defined over it.then we cn construct connected expression e with pirings liner in the size of e. And, if e hs equl source then e hs equl choice property. In ddition, if e is ction-live then if e hs product moves property then e hs consistency of piring. Lng(e) = Lng(e ). 5.6 Sum of Connected Expressions (SCE) We give syntx for sum of connected expressions (SCE) defined over distribution (Σ 1, Σ 2,..., Σ k ) of lphbet Σ. e ::= 0 e 1 + + e m, where e i is connected expression (CE) over Σ For n SCE e its semntics is given s follows: For the SCE 0, we hve Lng(0) =. For the SCE e = e 1 +e 2,, e m, its lnguge is given s Lng(e) = Lng(e 1 ) Lng(e 2 ) Lng(s m ). The definitions of derivtives extended to SCEs is s given below. The expression 0 hs no derivtives on ny ction. Derivtive of n SCE with respect to letter is defined s: Der (e) = Der (e 1 ) Der (e 2 ) Der (s m ). Inductively Der w (e) = Der w (Der (e)). The set of ll derivtives Der(e) is union of sets of derivtives of e with respect to ll words w in Σi for ll i in {1,..., m}. Definition 26 (SCE with pirings)). An SCE e = e 1 + + e m where ech e i is CE-pirings, is clled sum of connected expressions with piring (SCEpirings). An SCE-pirings e is sid to hve equl choice property if ech component CE-pirings e i of the sum lso hs it. Exmple 11. The expression e = e 1 + e 2 where e 1 = fsync((b + c), (d + e) ) is the CE-pirings of Exmple 8 nd e 2 = fsync((b + c), (d + e) ) is the CE-pirings of Exmple 9 is n SCE-pirings with equl choice property, s both e 1 nd e 2 hve it. Definition 27 (SCE with cbles)). An SCE e = e 1 + + e m where ech e i is CE-cbles, then e is clled sum of connected expressions with cbles (SCE-cbles). An SCE-cbles e is sid to hve equl source property if ech component CE-cbles e i of the sum lso hs it. An SCE-cbles e hs product derivtives property if ech component CE-cbles e i of the sum hs it. Exmple 12. The expression e = e 3 +e 4 where e 3 = fsync((b+c), (d+e) ) is the CE-cbles of Exmple 8 nd e 4 = fsync((b + c), (d + e) ) is the CE-cbles of Exmple 9. As n exmple of how derivtives of SCE-cbles from derivtives of SCEpirings differ even while hving identicl components, see tht fsync(r 2, s 3 ) Der (e) (of Exmple 11) but it does not belong to Der (e ).
96 PNSE 18 Petri Nets nd Softwre Engineering 6 Connected Expressions nd Product Systems To get product system with globls hving subset-cceptnce, from sum of connected expression, we use n erlier result from [14], where construction of PS-globls with product-cceptnce ws given from CE-cbles. For ech set of derivtives (pre-blocks nd fter-effects), of component regulr expression, we constructed n unique stte of locl component, which gives us product moves property in the constructed product system if we hve product derivtives property for given CE-cbles. Lemm 6 (CE-cbles to PS-globls with product-cceptnce [14]). Let e be CE-cbles, defined over distribution Σ. Then for the lnguge of e, we cn compute PS-globls with product-cceptnce liner in the size of expression e. Further, if e hd equl-source, then system A hs sme source property; nd, if e hd product-derivtives then A hs product moves property. Using Lemm 6, we get PS-globls with subset-cceptnce, from sum of connected expressions. Theorem 9 (SCE-cbles to PS-globls with subset-cceptnce). Let e be n sum of connected expression defined over Σ. Then we cn construct PS-globls A with subset-cceptnce for the lnguge of e. If e hd equl source property, then hs sme source property. In ddition, if e hs product derivtives property then A hs product moves property. Exmple 13. For SCE-cbles e of Exmple 12, we cn construct PS-globls with sme source property nd subset-cceptnce D of Exmple 6, ccepting lnguge L s, using Theorem 9. A lnguge preserving, construction of connected expressions with equl source property, from PS-globls with sme source property nd productcceptnce, ws given in [14]. Since ech locl stte whether source stte or trget stte of locl move is mpped uniquely to set of derivtives of component regulr expression, we hve product-derivtives property for the expression, if the product system hd product-moves property. Lemm 7 (PS-globls with product-cceptnce to CE-cbles [14]). Let Σ be distributed lphbet nd, A be product system with globls nd productcceptnce, defined over Σ. For the lnguge of A, we cn construct connected expression e with cbles, exponentil in the size of the given product system. Further, if product system hs sme source property then connected expression with cbles hs equl source property, in ddition, if it hs product-moves property then connected expression with cbles hs product-derivtives property. Now using Lemm 7, we get sum of connected expressions for PS-globls with subset-cceptnce.
Phwde et.l.: Kleene Theorems for Lbelled Free Choice Nets 97 Theorem 10 (PS-globls with subset-cceptnce to SCE-cbles). Let A be product system with globls nd subset-cceptnce, defined over distribution Σ. For the lnguge of A, we cn construct sum of connected expression e with cbles. And, if product system hs sme source property then e hs equl source property. Also, in ddition, if A hs product moves property then e hs product derivtives property. Exmple 14. For PS-globls with sme source property nd subset-cceptnce D of Exmple 6, ccepting lnguge L s, we cn construct n SCE-cbles e of Exmple 12 using Theorem 10. Using equivlence of PS-mtchings with product-cceptnce nd CE-cbles, from [15, 16], nd Corollry 1 we get lnguge equivlent SCE-pirings for PSmtchings with subset-cceptnce, nd vice-vers. Theorem 11 (PS-mtchings with subset-cceptnce to SCE-pirings). Let A be product system with conflict-equivlent nd consistent mtchings, hving subset-cceptnce. For the lnguge of A, we cn construct sum of connected expression e with equl choice nd consistent pirings. We hve the reverse trnsltion given s below. Theorem 12 (SCE-pirings to PS-mtchings with subset-cceptnce). Let Σ be distributed lphbet nd sum of connected expression e defined over it, with equl choice nd consistent pirings. Then for its lnguge we cn construct product system with conflict-euivlent nd consistent mtchings, hving subset-cceptnce. 7 Conclusion In this pper, we hve given lnguge (L s of Exmple 2) which cn be ccepted by free choice Zielonk utomt. This lnguge is not ccepted by ny synchronous product or direct product. We hve lso given lnguge (L 4 of Exmple 4) which cn be ccepted by free choice synchronous product nd not by ny direct product. A lnguge which cn be ccepted by free choice direct product (L 3 of Exmple 3) ws given in [15]. With this we hve hierrchy of lbelled free choice nets similr to utomt over distributed lphbets. In ddition we hve defined Zielonk utomt with product cceptnce condition nd its free choice restriction. We hve given lnguge (L p of Exmple 1) of this clss. We used this intermedite utomt to obtin Kleene theorem for free choice Zielonk utomt. We give below the summry of correspondences estblished for the nets, utomt over distributed lphbets nd expressions. To get n expression, for the lnguge of lbelled 1-bounded nd S-coverble free choice net (with or without distributed choice) hving finite set of finl mrkings, we use Theorem 4 nd Theorem 10. In the reverse direction, we use Theorem 9 to get product sytem