A Polynomial-Time Algorithm for Checking Consistency of Free-Choice Signal Transition Graphs

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1 Fundment Informtice XX (2004) IOS Press A Polynomil-Time Algorithm for Checking Consistency of Free-Choice Signl Trnsition Grphs Jvier Esprz Institute for Forml Methods in Computer Science University of Stuttgrt esprz@informtik.uni-stuttgrt.de Abstrct. Signl Trnsition Grphs (STGs) re one of the most populr models for the speci ction of synchronous circuits. A STG cn be implemented if it dmits so-clled consistent nd complete binry encoding. Deciding this is EXPSPACE-hrd for rbitrry STGs, nd so lot of ttention hs been devoted to the subclss of free-choice STGs, which offers good compromise between expressive power nd nlizbility. In the lst yers, polynomil time synthesis techniques hve been developed for free-choice STGs, but they ssume tht the STG hs consistent binry encoding. This pper presents the rst polynomil lgorithm for checking consistency. 1. Introduction Asynchronous circuit design is ttrcting incresing interest due to some importnt potentil dvntges, like bsence of clock skew problems nd low power consumption [2, 7]. Signl Trnsition Grphs (STGs) [5, 4, 20] re one of the most populr speci ction models for synchronous circuits. They re Petri nets in which the ring of trnsition is interpreted s rising or flling of signl in the circuit. Trnsitions corresponding to rising (flling) of signl re lbelled by + ( ). Signls re prtitioned into input nd output signls, which re supposed to be controlled by the environment nd the circuit, respectively. Given STG, the synthesis problem consists in producing n synchronous circuit exhibiting the sme behvior. Such circuit is known to exist if the stte grph of the STG dmits consistent nd complete binry encoding. A binry encoding is function tht ssigns to ech rechble stte vector of boolens, one for ech signl, indicting if the signl is up (1) or down (0). Loosely speking, n encoding is consistent if it respects the semntics of trnsition lbels: e.g., if the ring of trnsition lbelled by + leds from stte s to stte s, then signl must hve vlue 0 in s nd vlue 1 in s. In This work ws mostly done while the uthor ws t the University of Edinburgh.

2 2 J. Esprz / Checking Consistency of Free-Choice Signl Trnsition Grphs prticulr, occurrences of + nd must lternte long every ring sequence. Completeness requires the binry code of stte to contin enough informtion to determine the behvior of the circuit; if two sttes hve the sme code, then they hve to enble exctly the sme output signls. Erly techniques for the synthesis of synchronous circuits from STG speci ctions proceeded by rst constructing the stte grph of the STG, then computing consistent nd complete binry encoding (if it exists), nd then synthesizing the circuit from the encoding. This pproch is firly well understood (see e.g. [16, 22, 12]), nd there exists good tool support, e.g. like tht provided by the Petrify tool [6]). If these techniques re used, checking consistency nd completeness re minor problems, since the check cn be performed in liner time in the size of the rechbility grph. However, these pproches suffer strongly from the stte explosion problem: the number of sttes of the stte grph cn grow super-exponentilly in the size of the STG, or even be in nite. As synthesis tools for synchronous systems strt to mture, the size of STGs increses nd techniques bsed on the stte grph become obsolete. Therefore, much effort is being devoted to synthesis techniques tht void the construction of the stte grph. In this new setting consistency nd completeness become mjor problem. Checking them is EXPSPACE-hrd in the size of the STG, even if its stte grph is known to be nite, result tht follows esily from the EXPSPACE-hrdness of the rechbility problem for Petri nets [17, 11]. For 1-bounded STGs, in which plce cn contin t most 1 token (the cse most common in prctice), checking consistency nd completeness is still PSPACE-complete. PSPACEhrdness follows esily from the PSPACE-hrdness of the rechbility problem for 1-bounded Petri nets [11], while membership in PSPACE is trivil. In order to cope with this problem, syntcticlly de ned subclsses of STGs hve been studied. The clss with the best compromise between expressive power nd nlizbility re free-choice STGs [5], clss tht llows to model both nondeterminism (necessry for modelling the environment) nd concurrency (essentil for synchronous modelling), but restricts their interply. As mtter of fct, STGs were originlly de ned in [5] s free-choice nets, nd mny ppers still identify STGs with free-choice STGs. Loosely speking, STG is free-choice if for every plce p, whenever some output trnsition of p is enbled, ll output trnsitions of p re enbled, nd so it is lwys possible to freely choose which of them res. This is n dequte model of the behvior of the environment, which should be ble to freely produce ny input signl to the circuit. Mny synchronous circuits cn be nturlly speci ed using free-choice nets, lthough they re not powerful enough to model rbiters. Free-choice STGs nd subclsses thereof hve been studied in numerous ppers (see e.g. [5, 3, 18, 21]). A number of techniques exist for the utomtic implementtion of STGs with consistent nd complete encodings. In [3] it is lso shown how to trnsform free-choice STG hving consistent encoding into nother one which dmits consistent nd complete encoding. The problem of checking consistency hs been studied in [18], where two polynomilly checkble conditions for consistency, one suf cient nd the other necessry, re presented. However, the exct computtionl complexity of checking consistency nd completeness is still unknown. In this pper we ttck the consistency problem with results of the theory of free-choice Petri nets obtined in the erly 90s [9]. We show tht consistency cn be checked in polynomil time for free-choice STGs known to be bounded, dedlock-free, nd cyclic (mening tht the initil mrking cn be reched from ny other rechble mrking). These re stndrd conditions used by ll ppers on the subject since Chu s work [5], nd in prticulr by [3, 18]. These conditions hold 1 for mny rel speci ctions, they 1 Or cn be rti cilly mde to hold, for instnce by dding trnsitions tht restore the initil mrking.

3 J. Esprz / Checking Consistency of Free-Choice Signl Trnsition Grphs 3 simplify the synthesis procedures, nd cn be checked in polynomil time [9]. The pper is orgnized s follows. Section 2 contins bsic de nitions. Section 3 describes the checking procedure nd proves its correctness. The resulting lgorithm is presented in Section 4, nd its complexity is nlyzed. Finlly, Section 5 contins some brief conclusions. 2. Bsic de nitions A net is triple (P, T, F), where P nd T re disjoint sets of plces nd trnsitions, respectively, nd F is function (P T) (T P) {0, 1}. Plces nd trnsitions re genericlly clled nodes. Plces re grphiclly represented s circles, nd trnsitions s boxes. If F(x, y) = 1 then we sy tht there is n rc from x to y. The preset of node x, denoted by x, is the set of its input nodes, i.e., the set {y P T F(y, x) = 1}. The postset of x, denoted by x, contins its output nodes, i.e., the set {y P T F(x, y) = 1}. A mrking M of net (P, T, F) is mpping P IN (where IN denotes the nturl numbers including 0). Grphiclly, mrking is represented by drwing M(p) tokens on the circle representing the plce p. A mrking M enbles trnsition t if it puts t lest one token on ech plce p t, i.e. if M(p) > 0 for ech p t. If t is enbled t M, then it cn re or occur, nd its occurrence leds to new mrking L, obtined by removing token from ech plce in the preset of t, nd dding token to ech plce in its postset; formlly, L(p) = M(p) + F(t, p) F(p, t) for every plce p. M L t denotes tht t is enbled t M nd tht its occurrence leds to L. A Petri net is pir (N, M 0 ) where N is net nd M 0 is mrking of N, clled the initil mrking. A sequence of trnsitions σ = t 1 t 2...t n, n 0, is n occurrence sequence from M to M if there exist mrkings M 1, M 2,..., M n 1 such tht M t 1 t M 2 t n 1...Mn 1 M We usully omit the intermedite mrkings nd write M M σ for n occurrence sequence; M σ denotes tht σ cn be red from M. The Prikh mpping of σ T, denoted by σ, is the mpping T IN tht ssigns to ech trnsition the number of times it occurs in σ. A mrking M is rechble if there exists n occurrence sequence from M 0 to M. The rechbility grph of Petri net is lbelled grph hving the set of rechble mrkings s nodes, nd the restriction of the t reltions to the set of rechble mrkings s edges. Signl trnsition grphs. We x nite lphbet A = { 1,..., n } of signls prtitioned into input nd output signls. (All our results cn be immeditely extended to STGs with dummy trnsitions, we omit them for clrity.) Rising nd flling of signl is denoted by + nd, respectively. We cll n element of L = A {+, } lbel. Loosely speking, signl trnsition grph is Petri net whose trnsitions re lbelled with elements of L. Formlly, signl trnsition grph (STG) is triple S = (N, M 0, l), where (N, M 0 ) is Petri net nd l is surjective lbelling function tht ssigns to ech trnsition of N lbel in L. The stte grph of STG S = (N, M 0, l) hs the rechble mrkings of (N, M 0 ) s nodes. If we hve M L t in the rechbility grph of (N, M 0 ), then the stte grph contins n edge M l(t) L.

4 4 J. Esprz / Checking Consistency of Free-Choice Signl Trnsition Grphs + c+ b+ b- + c- - + c+ b+ b- + c- - b- b+ c- + Figure 1. A free-choice STG nd its stte grph Figure 1 shows STG nd its stte grph. The node t the top of the stte grph corresponds to the initil mrking of the Petri net. A STG is speci ction of the behvior of the circuit under some ssumptions on the environment. A STG S = (N, M 0, l) with k plces is implementble if there exists stte coding mpping λ tht ssocites to ech rechble mrking M of (N, M 0 ) vector of signl vlues λ(m) {0, 1} k stisfying the following two properties: Consistency. If M L t nd t is lbelled by + i, then the i-th components of λ(m) nd λ(l) re 0 nd 1, respectively, nd ll other components hve the sme vlue in λ(m) nd λ(l). If t is lbelled by i, then the i-th components of λ(m) nd λ(l) re 1 nd 0, respectively, nd ll other components hve the sme vlue in λ(m) nd λ(l). Completeness: if two different rechble mrkings M, L stisfy λ(m) = λ(l), then they enble exctly the sme output lbels. Consistency is obviously necessry for implementbility. Completeness is necessry becuse the stte of n implementtion is completely determined by the signl vlues of ll signls. Therefore, if some output signl is enbled t M but not t L, M nd L must correspond to different sttes of the implementtion, nd so they must differ in the vlue of t lest one signl.

5 J. Esprz / Checking Consistency of Free-Choice Signl Trnsition Grphs 5 We sy tht STG is consistent or complete if it hs consistent or complete binry encoding, respectively. The STG of Figure 1 is not consistent. To see why, let M be the mrking reched by ring +. Since M enbles both b + nd b, consistent encoding λ must stisfy both λ(m)(b) = 0 nd λ(m)(b) = 1, nd so it cnnot exist. If the trnsition t the top lbelled by + nd the trnsition lbelled by b swp their lbels, then the STG becomes consistent. A mrking M of STG is n-bounded if M(p) n for every plce p. A STG is n-bounded if ll its rechble mrkings re n-bounded. Since the circuit implementtion of STG cn be seen s nite object with t most 2 n sttes, where n is the number of signls, STGs used in prctice re bounded (even though in principle unbounded STGs could mke sense), nd in fct most of them re even 1- bounded. A STG is dedlock-free if every rechble mrking enbles some trnsition. It is cyclic if for every rechble mrking there is sequence σ such tht M M σ 0, where M 0 is the initil mrking. STGs specifying circuits which continuously interct with their environment re dedlock-free. Mny re lso cyclic, since fter ech round of interction the circuit usully returns, or t lest my return, to home stte. A STG is well formed if it is dedlock-free, bounded, nd cyclic. The STG of Figure 1 is well formed. If we chnge the initil mrking to the one tht puts one token on the input plces of the trnsitions lbelled by b + nd c, nd no tokens elsewhere, then the STG is still live nd bounded, but no longer cyclic. Checking well formedness of STGs or consistency of well formed STGs is EXPSPACE-hrd. Therefore we introduce syntcticlly de ned subclss of STGs. A STG is free-choice if its underlying net (P, T, F) stis es the following property: for ech plce p nd every trnsition t, if F(p, t) = 1 then F(p, t ) = 1 for every p t, t p. In free-choice STG, if some output trnsition of plce is enbled t mrking, then ll its output trnsitions re enbled, nd it is possible to freely choose mong them. An importnt property of well formed free-choice STGs is tht they re live (see for instnce Theorem 4.31 of [9]). A STG is live if for every rechble mrking M nd ech trnsition t M σt holds for some sequence σ of trnsitions. Moreover, well formedness of free-choice STGs cn be checked in polynomil time (see e.g Corollry 6.18 nd Theorem 8.12 of [9]). We sy tht STG is lternting if the signs of ll signls lternte in ll pths of its stte grph. The STG of Figure 1 is not lternting: its stte grph contins pth in which b occurs twice without ny occurrence of b + in-between. We conclude this section with simple result showing tht for live nd cyclic systems consistency is equivlent to lterntion. Lemm 2.1. (1) Consistent STGs re lternting. (2) Live, cyclic, nd lternting STGs re consistent. Proof: (1) follows directly from the de nition of consistency. For (2), let S be live, cyclic, nd lternting STG, nd ssign to ech rechble mrking M binry coding λ(m) s follows: λ(m)() = 0 (= 1) if the stte grph contins pth strting t M in which + occurs before ( before + ). We prove tht λ is well de ned nd consistent. Consistency follows immeditely from the de nition. To show tht λ is well de ned, let M be n rbitrry rechble mrking. Since S is live nd its lbelling function is surjective, its stte grph contins pth M σt where t is lbelled by + or. Assume now tht

6 6 J. Esprz / Checking Consistency of Free-Choice Signl Trnsition Grphs there re two pths π + nd π such tht + occurs before in π +, nd before + in π. By cyclicity, π + cn be extended to cycle from M to M (which my visit mrkings more thn once). Since signls lternte long ll pths, occurs lst in this cycle. Consider the pth tht strts with this lst occurrence of, reches M, nd then continues with π. In this pth, the lbel occurs twice without ny intermedite occurrence of +, contrdicting the ssumption. 3. Checking consistency The next three sections present three-step procedure for checking consistency of well formed freechoice STGs. We will mke implicit use of Lemm 2.1 long the wy. The rst two steps check non-utoconcurrency nd blncedness, two necessry conditions for consistency. The third step is divide-nd-conquer procedure which checks consistency of STGs tht pss the rst two tests Checking non-utoconcurrency A mrking M concurrently enbles two trnsitions t nd u of net if it puts t lest one token in ll input plces of t nd u, nd t lest two tokens on every input plce of both t nd u. It follows esily from the ring rule tht if M concurrently enbles t nd u then there is mrking L such tht both M L tu nd M ut L re occurrence sequences. A STG is non-utoconcurrent if no rechble mrking concurrently enbles two trnsitions lbelled by the sme signl (even if the lbels crry opposite signs). We hve the following result: Lemm 3.1. Consistent STGs re non-utoconcurrent. Proof: Assume tht two trnsitions t nd u re concurrently enbled t rechble mrking M. Then we hve M tu nd M. ut Assume tht both t nd u re lbelled by signl. If they re both lbelled by + or, then lterntion is violted. If, sy, t is lbelled by + nd u by, then no consistent coding cn exist: signl cnnot be ssigned code in M, becuse it cn both rise nd fll from M. By Lemm 3.1, if well formed free-choice STG does not pss the non-utoconcurrency test, we cn lredy dismiss it s non-consistent. The STG of Figure 1 does not pss the test, becuse b + nd b cn be concurrently enbled. Checking non-utoconcurrency is EXPSPACE-complete for rbitrry STGs (this cn be esily proved using the techniques of [11]). Fortuntely, for well formed free-choice STGs we cn do much better: It is proved in [14] tht non-utoconcurrency cn be checked in polynomil time. For the ske of completeness, we include here summry of this result. The concurrency reltion is usully de ned s the set of pirs of trnsitions tht cn be concurrently enbled. We generlize the de nition bit. Given node x of STG S, we de ne the mrking M x s follows: if x is plce, then M x is the mrking tht puts one token on x, nd no tokens elsewhere; if x is trnsition, then M x is the mrking tht puts one token on every input plce of x, nd no tokens elsewhere.

7 J. Esprz / Checking Consistency of Free-Choice Signl Trnsition Grphs p 1 p 2 p 3 p 0 Figure 2. A well formed STG for which s The concurrency reltion contins the pirs of nodes (x 1, x 2 ) such tht M M x1 + M x2 for some rechble mrking M. In prticulr, pir (t 1, t 2 ) of trnsitions belongs to the concurrency reltion if t 1, t 2 cn occur concurrently from some rechble mrking. The structurl concurrency reltion, rst presented in [13], is the smllest symmetric reltion s on the nodes of S such tht: (i) for ll plces p, p, if M 0 M p + M p, where M 0 is the initil mrking of S, then (p, p ) s ; (ii) for ll trnsitions t, if ( t t) \ id P s, then (t t ) \ id P s, where id P denotes the identity reltion on the plces of S; (iii) for ll nodes x nd for ll trnsitions t, if {x} t s, then (x, t) s nd {x} t s. Loosely speking, condition (i) sttes tht ny two plces mrked t the initil mrking re structurlly concurrent (ctully, this is the cse for pir (p, p) only if M 0 puts t lest two tokens on p). Condition (ii) sttes tht if ll the input plces of trnsition re structurlly concurrent, then so re its output plces. Condition (iii) sttes tht if node is structurlly concurrent with ll the input plces of trnsition, then it is lso structurlly concurrent with ll its output plces. It is esy to see tht s. Figure 2 (tken from [13]) shows STG for which s. The reder cn check tht (p 1, p 0 ) /. However, since (p 1, p 2 ) s nd (p 1, p 3 ) s, we hve (p 1, p 0 ) s. So, in generl, the structurl concurrency reltion provides only suf cient, but not necessry condition for non-utoconcurrency. Fortuntely, in the free-choice cse (the STG of Figure 2 is non-freechoice), they coincide: Theorem 3.1. ([14]) For live nd bounded free-choice STGs, = s. The structurl concurrency reltion cn be esily computed s lest xed-point. In [14], two lgorithms re presented which compute s in O( P 2 T ( P + T )) time for rbitrry Petri nets nd in O( P ( P + T ) 2 ) time for free-choice Petri nets. Cortés [8] hs observed tht these lgorithms re incorrect becuse they forget to mintin the s reltion symmetric. Corrected versions kindly provided by him re shown in n Appendix.

8 8 J. Esprz / Checking Consistency of Free-Choice Signl Trnsition Grphs 3.2. Checking blncedness A STG is blnced if for every signl, every cycle of the stte grph contins the sme number of occurrences of + nd. Clerly, blncedness is necessry condition for lterntion nd hence for consistency. We show how to check blncedness. Let S = (N, M 0, l) be STG, where N = (P, T, F). A mpping J : T Q, where Q denotes the rtionl numbers, is T-invrint of N if t p J(t) = t p J(t) for every plce p. Notice tht T-invrints form vector spce. A T-invrint J is positive (semi-positive) if J(t) > 0 (J(t) 0) for ll t T. It is esy to show tht if M M σ for some rechble mrking M, then σ is semi-positive T- invrint. However, given semi-positive (integer) T-invrint J, it my not be possible to ctivte it, i.e., there my be no cycle M M σ in the rechbility grph of (N, M 0 ) such tht J = σ. Fortuntely, for well formed free-choice STGs limited version of the converse holds. In order to stte this result we need some de nitions. Let the support of semi-positive T-invrint be the set of trnsitions t such tht J(t) > 0. A semi-positive T-invrint is miniml if there is no other semi-positive T-invrint with strictly smller support. Theorem 3.2. ([9], Theorems 5.17 nd 5.20) Let S be well formed free-choice STG. All miniml T-invrints of N cn be ctivted. We use this result to check blncedness. Given mpping J : T Q nd signl we write #(J, + ) (#(J, )) to denote the sum of J(t) over ll trnsitions t lbelled by + ( ), nd de ne bl(j): A IN by bl(j)() = #(J, + ) #(J, ) for every signl. Theorem 3.3. Let S be well formed free-choice STG. S is blnced iff bl(j) = 0 for every T- invrint J of S. Proof: ( ): We rst prove bl(j) = 0 for semi-positive T-invrints. Let J be semi-positive T-invrint. It follows esily from the de nition of miniml T-invrint tht there re miniml T-invrints J 1,...,J k such tht J = J J k. By Theorem 3.2, ll these T-invrints cn be ctivted nd so, since S is blnced, we hve bl(j 1 ) =... = bl(j k ) = 0. Since bl(j) = bl(j 1 ) bl(j k ), we get bl(j) = 0. Let J now be n rbitrry T -invrint. By Theorem 2.38 of [9], S hs positive T-invrint J. Let J = J + kj, where k is lrge enough to mke J semi-positive. Since bl(j) = bl(j ) k bl(j ) nd bl(j ) = bl(j ) = 0, we hve bl(j) = 0. ( ): Let c be n rbitrry cycle of the stte grph of S. This cycle corresponds to cycle M σ M of the rechbility grph. Then σ is T-invrint of S, nd therefore bl( σ) = 0, which mens tht c contins the sme number of occurrences of + nd for every signl. The STG of Figure 3 (tken from [19] with some modi ctions) shows tht Theorem 3.3 does not hold for rbitrry well formed STGs. The STG is blnced, but the mpping J tht ssigns 0 to the two trnsitions in the middle of the picture nd 1 to the others is T-invrint for which bl(j) 0. Notice tht this T-invrint cnnot be ctivted. T-invrints cn be computed using liner lgebr. The incidence mtrix of net N, denoted by N, is P T mtrix given by N(p, t) = F(t, p) F(p, t). It is esy to see tht M L σ implies

9 J. Esprz / Checking Consistency of Free-Choice Signl Trnsition Grphs Figure 3. A well formed nd blnced STG L = M +N σ. (For σ = t 1...t n, this is nothing but the equlities L(p) = M(p)+F(t 1, p) F(p, t 1 )+...+F(t n, p) F(p, t n ) written in mtrix form.) A little clcultion shows tht J is T-invrint if nd only if N J = 0. So we hve the following corollry of Theorem 3.3: Corollry 3.1. Let S be well formed free-choice STG. S is blnced iff every rtionl solution of the liner system of equtions N X = 0 stis es bl(x) = 0. This corollry shows tht blncedness cn be decided by computing bsis of the solutions of N X = 0 nd checking tht ech element J of the bsis stis es bl(j) = A divide-nd-conquer procedure In the rest of the section we work with well formed free-choice STGs which re both blnced nd non-utoconcurrent. We cll these STGs very well formed. The third step of our procedure to check lterntion is divide-nd-conquer lgorithm for checking lterntion of very well formed free-choice STGs (i.e., of those well formed free-choice STGs which hve pssed both the non-utoconcurrency nd the blncedness tests): we decompose the STG into two prts nd show tht checking lterntion of the whole reduces to checking lterntion of ech prt. Recll tht, by Lemm 2.1, very well formed STGs re consistent if nd only if they re lternting. The divide-nd-conquer procedure is best explined by considering the specil cse of STGs in which every trnsition hs exctly one input nd one output plce. In wht follows, these re clled stte mchine STGs 2. Figure 4 shows very well formed stte mchine STG. This gure nd the next ones use two drwing conventions. First, only the lbel of trnsition is drwn (its surrounding box is omitted). Second, plces hving only one input nd one output trnsition re lso omitted. Therefore, n edge of the form, sy, + b, indictes tht the STG contins n rc from trnsition lbelled by + to plce, nother rc from this plce to trnsition lbelled by b, nd tht the plce hs no other input or output trnsitions. Tokens on the intermedite plce re just drwn on the rc. 2 Checking consistency of stte mchine STGs is trivil problem. We use them in order to present some of the ides of the free-choice cse in simple setting.

10 10 J. Esprz / Checking Consistency of Free-Choice Signl Trnsition Grphs p 1 b b + b + + p 2 + Figure 4. A very well formed stte mchine STG Let S be very well formed stte mchine STG. Let hndle be simple pth of (the underlying net of) S stisfying the following conditions: it strts nd ends t plce, ll its intermedite nodes hve exctly one input nd one output node, nd fter removing these intermedite nodes together with their djcent rcs, the remining STG is still strongly connected. In Figure 4, the pth on the right tht visits p 1,, b +, +, p 2 is hndle. Without loss of generlity, we ssume tht the intermedite plces of S contin no tokens t the initil mrking (if this is not the cse, we cn re trnsitions to remove those tokens; by cyclicity, lterntion holds for the new STG if nd only if it holds for the old one). Let H nd R be STGs whose underlying nets re the hndle nd the remining STG, respectively; the initil mrking of H puts one token on its rst plce nd no tokens elsewhere, while the initil mrking of R is the projection of the initil mrking of S onto R. For the exmple of Figure 4, the STGs H nd R re shown in Figure 5. p 1 p 1 b b + b + + p 2 p 2 + The STG R The STG H Figure 5. Dividing the STG of Figure 4 We would like to hve tht S is lternting if nd only if H nd R re lternting. However, Figure 4 shows tht this is not true. In this exmple, S is not lternting becuse of the two consecutive t the top of the gure, even though both H nd R re lternting. Our solution consists of modifying R to mke sure tht, if S is not lternting nd H is lternting, then the new R is not lternting. In order to see how to modify R, we look t the pth of R visiting p 1, b +, p 2. This is mirror imge of H in R, i.e., pth of R with the sme strt nd end nodes s H. Let α nd β be the sequences of lbels corresponding to H nd its mirror imge, i.e., α = b + + nd β = b +. Intuitively, the modi ction of R should gurntee tht if we cn use α to produce non-lternting sequence, then we cn lso use β,

11 J. Esprz / Checking Consistency of Free-Choice Signl Trnsition Grphs 11 i.e., tht α nd β re equivlent with respect to their potentil for producing non-lternting sequences. But how could forml de nition of equivlent look like? It is not dif cult to see tht we could choose this one: for ech signl, if the projection of α on is nonempty, then the projections of α nd β on begin with the sme lbel nd end with the sme lbel. However, we hve not been ble to trnsform this de nition into polynomil divide-nd-conquer procedure, nd so we choose nother one which lso works: (1) α nd β hve the sme blnce, (2) for ech signl, if occurs in α then it lso occurs in β, nd (3) if occurs in α nd β then the rst occurrence of hs the sme sign in both sequences. In Figure 4, α nd β re not equivlent becuse condition (2) is violted. p 1 b b + b + p 2 Figure 6. A stte mchine STG So our modi ction of R should gurntee tht α nd β stisfy conditions (1), (2) nd (3). Fortuntely, it is esy to see tht if (1) is violted then the STG is not very well formed. Figure 6 shows n exmple. We hve α = b nd β = b +, nd so condition (1) is violted; the STG contins circuit with two occurrences of b nd none of b +, nd so it is unblnced. Since we ssume tht our initil STG is very well formed, we do not hve to worry bout ensuring condition (1): it holds utomticlly. In order to gurntee (2) nd (3), we insert witness in R, s shown in Figure 7 for the STG R of Figure 5. The witness records tht occurs in α, nd tht the rst occurrence of hs negtive sign. Notice tht, fter introducing the witness, R is no longer lternting. The two consecutive tht were possible in S re now possible in the modi ed R s well. p 1 b + b + + p 2 Figure 7. Inserting witness in R In the rest of the section we show how to extend these ides to free-choice STGs. In prticulr, hndles cn no longer be just pths, due to the presence of concurrency. Section introduces CP-

12 12 J. Esprz / Checking Consistency of Free-Choice Signl Trnsition Grphs subnets 3, which re generliztion of hndles. Section formlizes the insertion of witnesses. Section shows the correctness of the divide-nd-conquer steps: lterntion holds for the originl STG if nd only if it holds for the prts. Finlly, Section shows how to directly check consistency of STGs tht do not contin ny CP-subnets Hndles generlize to CP-subnets A trnsition-generted subnet of net N = (P, T, F) is net N = (P, T, F ) such tht T T, P = T T (i.e., P contins ll input nd output plces of T in the net N), nd F = F ((P T ) (T P )). Trnsition-generted subnets re just clled subnets in wht follows. A subnet N = (S, T, F ) is wekly connected if for every prtitioning of P T into two disjoint, nonempty sets X 1, X 2, there is x 1 X 1 nd x 2 X 2 such tht (x 1, x 2 ) F or (x 2, x 1 ) F. It is strongly connected if (x, y) (F ) for every x, y P T, where (F ) denotes the re exive nd trnsitive closure of F. Given subnet N, we de ne N \ N s the subnet hving T \ T s set of trnsitions. A plce p of N is clled entry (exit) plce if some trnsition of p (p ) belongs to N \ N. All other plces of N re clled internl. The output trnsitions of the entry plces re clled entry trnsitions. All other trnsitions of N re clled internl. N is CP-subnet of N if (i) it is nonempty nd wekly connected, (ii) every internl plce hs exctly one input nd one output trnsition, nd (iii) the net N \ N is strongly connected. The hndle of Figure 5 is n exmple of CP-subnet. A more interesting exmple is the subnet of Figure 1 generted by the three trnsitions of the left hlf of the net lbelled by +, b +, nd b. Finlly, Figure 8 shows free-choice STG, which in Figure 9 is divided into CP-subnet, on the right, nd the rest, on the left. (The fct tht the initil mrkings of Figure 8 nd Figure 9 re different will be explined lter). In wht follows, N denotes CP-subnet of net N. Given STG S = (N, M 0, l), we denote by S \ N the STG whose underlying net is N \ N, nd whose initil mrking nd lbelling function re the projections of M 0 nd l onto N \ N. We will use the following result of [9]: Lemm 3.2. ([9]) CP-subnets of well formed free-choice STGs hve unique entry trnsition. A ushing sequence of N in S is n occurrence sequence M σ L of S stisfying the following two properties: M nd L re rechble mrkings of S tht enble no internl trnsitions of N, nd σ = t e τ, where t e is the unique entry trnsition of N, nd τ contins no occurrences of t e. Intuitively, in ushing sequence the entry trnsition of N res once, nd then the internl trnsitions of N re for s long s possible. In the cse of hndles, ushing sequence lets token run long the hndle, from its rst to its lst plce, nd so hndles hve one single ushing sequence. This is no longer 3 This strnge nme is due to historicl resons.

13 J. Esprz / Checking Consistency of Free-Choice Signl Trnsition Grphs 13 b + + c + b + c + b Figure 8. A free-choice STG the cse for CP-subnets, becuse concurrency mong the trnsitions of N is possible, nd so we cn get new ushing sequence by chnging the order in which concurrently enbled trnsitions re. For instnce, in the CP-subnet of Figure 9, there re three possible ushing sequences, corresponding to the sequences of lbels b + + c + c, b + c + + c, nd b + c + c +. However, for very well formed STGs we cn prove tht CP-subnets enjoy mny of the properties of hndles: Proposition 3.1. Let S be very well formed free-choice STG, let N be CP-subnet of S, nd let M M σ nd L τ L be rbitrry ushing sequences of N in S. Then (1) M, M, L, L coincide on ll internl plces of N, (2) the projections of l(σ) nd l(τ) on ech signl coincide, nd (3) every trnsition of N occurs exctly once in σ nd τ. Proof: (1) This is shown in Lemm 9.2 of [9]. (2) We prove stronger result, nmely tht the projections of σ nd τ on the trnsitions lbelled by + or coincide for every signl. We strt with some preliminries. Since both M nd L enble the entry trnsition of N nd S is non-utoconcurrent, they put one token in ll the entry plces of N. So, by (1), M nd L coincide on ll input plces of ll trnsitions of N. Let K be the common projection of M nd L on these plces. Then K σ nd K τ re occurrence sequences of N. We cn now mke use of well-know property of persistent nets, clss tht contins ll nets in which plces hve t most one output trnsition [15]: if K υ 1 nd K υ 2, then K υ for some sequence υ such tht υ = mx{ υ 1, υ 2 }, where mx is de ned componentwise. After these preliminries, ssume tht for some signl the projections of l(σ) nd l(τ) on the trnsitions lbelled by + or do not coincide. Let σ nd τ be the longest pre xes of σ nd τ (possibly the empty

14 14 J. Esprz / Checking Consistency of Free-Choice Signl Trnsition Grphs b + + c + b + c + b Figure 9. Dividing the STG of Figure 8 word) for which they coincide. Then σ = σ σ tσ nd τ = τ τ uτ, where σ nd τ do not contin - lbelled trnsitions, l(t), l(u) { +, }, nd l(t) l(u). Let X = mx{ σ, τ }, X t = mx{ σ t, τ }, nd X u = mx{ σ, τ u}. Let K, K t nd K u be the mrkings reched by the sequences hving X, X t, nd X u s Prikh vector. Then, since K t = K + N t nd K u = K + N u, the mrking K enbles both t nd u. Since plces of N hve t most one output trnsition, t nd u do not shre ny input plce. So t nd u re concurrently enbled t K, contrdicting tht S is very well formed. (3) By liveness of S, every circuit of N is mrked t M (otherwise the trnsitions of the circuit cn never occur, becuse, since N is CP-subnet, the circuit cn never get mrked). Moreover, since S is non-utoconcurrent nd free-choice, M puts exctly one token on the entry plces of N (otherwise ny output trnsition of these plces cn occur concurrently with itself). Now, let t be n internl trnsition of N. Since t is not enbled t M, it hs n input plce unmrked t M. If the input trnsition of this plce is n internl trnsition, then it is gin not enbled by M, nd so in this wy we cn construct pth whose plces re unmrked t M. Since this pth cnnot run into circuit, becuse ll circuits of N re mrked, it must end t the entry plce. Since the pth contins only one token, t cn occur t most once in the ushing sequence, becuse fter its rst occurrence the pth becomes empty. Tht t occurs t lest once follows from liveness. So, even if in very well formed free-choice STG CP-subnet my hve mny different ushing sequences, they ll strt nd end t mrkings tht coincide on ll the internl plces of N (the mrkings my differ elsewhere), ech trnsition occurs in them exctly once, nd, for every signl, their projections onto the lbels +, coincide. Proposition 3.1 justi es the following de nition. Given n rbitrry ushing sequence M L σ of N in S, we de ne the chrcteristic mrking of N, denoted by M, s the mrking tht puts one token on ech entry plce of N, coincides with M (nd L) on the internl plces, nd puts no tokens on the exit plces (unless they re lso entry plces, in which cse they get one token). The chrcteristic sequence of N with respect to signl, denoted by σ, is the projection of l(σ) onto + nd. The chrcteristic

15 J. Esprz / Checking Consistency of Free-Choice Signl Trnsition Grphs 15 mrking of the CP-subnet of Figure 9 is the one shown in the gure, nd we hve σ = +, σ b = b +, nd σ c = c + c. In the cse of stte mchine STGs we divide the STG into hndle H nd the rest R. In the freechoice cse, the rôle of H is plyed by the STG S CP hving (N, M) s underlying Petri net. By Proposition 3.1(2), checking consistency of S CP is very simple: if suf ces to construct ny ushing sequence (ll of them hve liner length), nd check if they re lternting Inserting witnesses In this section we de ne the STG plying the rôle of R. Let S be very well formed free-choice STG with CP-subnet N. Without loss of generlity, we ssume tht the initil mrking M 0 of S does not enble ny internl trnsition of N. If this were not the cse, we would let the internl trnsitions of N occur for s long s possible. For instnce, in the STG of Figure 8 we let the trnsitions lbelled by c +, c, nd + occur. Since S is cyclic, the resulting STG is lternting if nd only if S is lternting. Let A be the set of signls such tht σ is of the form + α or α + for some α L. The procedure to construct S R is best described in n informl but hopefully precise wy, since forml de nition would be dif cult to red. Strt with the STG S \ N. Let P e be the set of entry plces of N (these re the input plces of the unique entry trnsition t e ), which belong to S \ N. Let T = P e \ {t e }. Remove ll the rcs (p, t) of N \ N such tht p P e nd t T. Construct net consisting of one single pth, strting t trnsition nd ending t plce. The pth, which we cll the witness of, contins two trnsitions t, u for ech lbel A, with t preceding u. If σ is of the form + α, then t is lbelled by +, nd u is lbelled by. If it is of the form α +, then t is lbelled by nd u is lbelled by +. The order between trnsitions corresponding to different lbels cn be chosen rbitrrily. Let t f be the rst trnsition of the pth, nd let p l be its lst plce. Add rcs {(p, t f ) p P e } nd {(p l, t) t T }. (At this point, the reder my sk why do we introduce witnesses only for the ctions of the set A. While we could sfely introduce witnesses for ll signls whose chrcteristic sequence is nonempty, it turns out to be super uous.) In the cse of Figure 9 the set A contins the signls nd c. The STG S R is shown in Figure 10. Notice tht if lterntion holds for S, then it lso holds for S R. The converse, however, is not true, s shown by Figure Correctness of the divide-nd-conquer strtegy The gol of this section is to prove the two essentil correctness properties of the divide-nd-conquer procedure. First, we hve to show tht lterntion holds for S (nd for ll signls) if nd only if it holds for both S CP nd S R, s de ned in Sections nd Second, since we ssume tht S is very well formed free-choice STG, we hve to prove tht S R is lso very well-formed, so tht the procedure cn be iterted with S R. The reder not interested in these proofs cn go directly to Section

16 16 J. Esprz / Checking Consistency of Free-Choice Signl Trnsition Grphs + c + c + b + b Figure 10. The STG S R We strt by proving the second point: Proposition 3.2. S R is very well formed free-choice STG. Proof: Notice tht S R is live, bounded, cyclic, nd blnced if nd only if S \ N is. Liveness nd boundedness of S \ N is proved in Proposition 7.8 of [9]. According to Theorem 8.11 of the sme reference, cyclicity requires to prove tht ll so-clled trps of S \ N re mrked. This follows esily from the fct tht ll trps of S re mrked, nd the fct tht the initil mrking of S \ N does not enble ny internl trnsition of N. Blncedness of S \ N follows from the fct tht, due to the de nition of CP-subnet, every T-invrint of S \ N is lso T-invrint of S. It remins to show non-utoconcurrency of trnsitions of S R. Notice rst tht if two trnsitions of S cnnot be concurrently enbled, then they cnnot be concurrently enbled in in S \ N either, since ll the behviors of S \ N re lso possible in S. Therefore, S \ N is non-utoconcurrent. Furthermore, it is esy to see tht no two trnsitions introduced s witnesses cn be utoconcurrent. If they were, then, since they lie in pth strting t some entry plce of N, nd in this pth ll trnsitions hve one single input plce, some rechble mrking of S would put t lest two tokens on the entry plces of N, nd so the entry trnsition of N could be concurrently red with itself, contrdicting the non-utoconcurrency of S. Finlly, we hve to show tht witness trnsition t lbelled by + or cnnot be concurrently enbled with ny -lbelled trnsition of S \ N. For this, ssume tht t cn be concurrently enbled with some trnsition u of S \ N. Then, by the wy witnesses re inserted, there is rechble mrking tht enbles u nd puts token on ech plce of P e. From this mrking the ushing sequence of N cn be executed, nd since N contins t lest one -lbelled trnsition, sy v, we hve tht t nd v cn be concurrently enbled. This contrdicts the non-utoconcurrency of S. Let us now consider the rst point, i.e., to show tht lterntion holds for S (nd for ll signls) if nd only if it holds for both S CP nd S R. When introducing the procedure for the cse of stte

17 J. Esprz / Checking Consistency of Free-Choice Signl Trnsition Grphs 17 mchine STGs we spoke of the mirror sequences of the ushing sequence of hndle. Recll tht the gol of introducing witnesses ws to ensure tht the ushing sequence nd its mirror imge re equivlent, mening tht they hve the sme blnce, contin the sme signls, nd, for ech signl, their projections on strt by the sme letter. In the free-choice cse even the existence of mirror sequences is nontrivil. The next proposition shows tht they indeed exist, nd tht they stisfy the required properties.(loosely speking, proving the existence of mirror sequences mounts to showing tht if we cn rech L from M in S by executing ushing sequence of N, then we cn lso rech L from M without using trnsitions of N.) Proposition 3.3. Let S be very well formed free-choice STG, let N be CP-subnet of S, nd let M σ L be ushing sequence of N in S. (1) S R hs n occurrence sequence M τ L, where M nd L re the projections of the mrkings M nd L onto the plces of S R. (2) bl(σ) = bl(τ). (3) Let σ nd τ be the projections of σ nd τ onto +,. If σ nd τ re lternting nd σ is nonempty, then σ nd τ strt nd end with the sme lbel. Proof: (1) By Lemm 3.1, M nd L coincide on the internl plces of N, nd every trnsition of N occurs exctly once in τ. So we cn dd new trnsition t to N \ N which removes one token from the entry plce of N, nd dds one token to ech of its exit plces (the plces hving no output trnsition in N). We then hve L t K. Let C nd C t be the incidence mtrices of N \N, nd of the result of dding t to N \N, respectively. By Theorem 6.7 of [10] (see lso Theorem 7.13 of [9]), there is vector X 0 such tht C X = C t t. Since L t K, we hve L = K +C t t, nd so K = L +C X. By Theorem 9.6 nd 9.7 of [9], K is rechble from L in S \ N, nd we re done. (2) By cyclicity, there is sequence L M. υ So the rechbility grph of S contins the circuits M L σ M υ nd M L τ υ M. By blncedness, we hve bl(συ) = bl(τυ) = 0, nd so bl(σ) = bl(τ). (3) Consider two cses: bl(σ) = 0. Since σ is lternting, we cn ssume without loss of generlity tht σ = + α for some α. Since S R contins witness for signl, the projection τ lso strts with +. Now we hve: τ is lternting, τ strts with +, nd bl(τ) = bl(σ) = 0. It follows tht τ ends with. bl(σ) 0. Since σ is lternting, we cn ssume without loss of generlity tht σ = + α + for some α. Since τ is lternting nd bl(τ) = bl(σ), we hve τ = + β + for some β.

18 18 J. Esprz / Checking Consistency of Free-Choice Signl Trnsition Grphs The next lemm llows us to obtin cnonicl form for the non-lternting sequences of S (if they exist). Lemm 3.3. Let S be very well formed free-choice STG with initil mrking M 0, nd let N be CP-subnet of S. If S is non-lternting, then it hs non-lternting sequence of the form such tht: M 0 σ 0 L0 τ 0 M1...M n σ n Ln τ n Mn+1 the σ i sequences only contin trnsitions of S R, the τ i sequences only contin trnsitions of S CP, nd L 0 τ 0 M1,..., L n τ n Mn+1 re ushing sequences of N in S. Proof: σ Assume tht S is non-lternting. Then it hs non-lternting occurrence sequence M 0 M. Let t nd u be n rbitrry internl trnsition of N nd trnsition of N \ N, respectively. By the de nition of CP-subnet, t nd u hve disjoint preset. So for every occurrence sequence L L τtuτ, we hve tht L L τutτ is lso n occurrence sequence. Moreover, since S is non-utoconcurrent, L L τtuτ is lternting if nd only if L L τutτ is lternting. σ By exhustively swpping internl trnsitions of N nd trnsitions of N\N, we reorgnize M 0 M σ into n occurrence sequence M 0 τ 0 σ 0 L0 M1... M n τ n τ n Ln Mn+1, where ll L i i Mi+1 but τ possibly L n τ n Mn+1 re ushing sequences. If L n n Mn+1 is not ushing sequence, we extend it to one by letting internl trnsitions of M n occur s long s possible. We re now redy to prove the min result: Theorem 3.4. Let S be very well formed free-choice STG, nd let N be CP-subnet of S. S is lternting if nd only if S CP nd S R re lternting. Proof: ( ): Esy. σ ( ): Assume tht S is non-lternting. Let M 0 τ 0 σ 0 L0 M1... M n τ n n Ln Mn+1 be the nonlternting sequence with respect to some signl, whose existence is gurnteed by Lemm 3.3. If τ some L i i Mi+1 is non-lternting, then S CP is non-lternting, nd we re done. So ssume tht ll τ i s re lternting sequences. Let M i, L i be the projections of M i, L i onto N \ N. By repeted pplictions of Proposition 3.3, there exist sequences υ 0,...υ n 1 such tht M 0 σ 0 σ n L υ 0 0 M 1...M n L υ n n M n+1 Observe tht the projection of ll the M i, L i onto the non-entry plces of N is equl to M, since they re ll initil or nl mrkings of ushing sequence (Lemm 3.1). Therefore, M 0 σ 0 L0 υ 0 M1...M n σ n Ln υ n Mn+1

19 J. Esprz / Checking Consistency of Free-Choice Signl Trnsition Grphs 19 is n occurrence sequence of S. Consider two cses: (1) Some σ i or υ i is non-lternting. Then S \ N is non-lternting, nd so S R is non-lternting. (2) All σ i nd υ i re lternting. We show tht in this cse S R is non-lternting. Assume without loss of generlity tht (σ 0 τ 0...σ n τ n ) contins the word + +. Since ll σ i nd τ i re lternting, none of them contin + +. Therefore, the rst nd the second + must be the lst nd the rst letter, respectively, of two different elements of {σ 0, τ 0,...,σ n, τ n }. There re four possible cses, of which we consider only one, the other three being similr: σ i ends with +, τ i+k strts with +, nd τ i σ i+1...σ i+k is the empty word. By Proposition 3.3(3), υ i+k lso strts with +. By Proposition 3.3(2), the word υ i...υ i+k 1 hs blnce 0. Since the υ i s re lternting, this word is either empty, strts with +, or ends with +. In ll cses, σ i υ i...υ i+k contins the word The bse cse Theorem 3.4 nd Proposition 3.2 llow to itertively remove CP-subnets from S until either consistency is disproved, or no CP-subnets cn be found. It is esy to show tht one of the two will eventully be the cse: every time CP-subnet is removed, the sum over ll plces p of p 1 strictly decreses. If this sum becomes 0, then every plce hs exctly one output trnsition, nd so no CP-subnets cn exist. It is shown in [9] (Proposition 7.11) tht if no CP-subnets exist, then ech plce hs exctly one output nd exctly one input trnsition. Such nets re clled T-nets or mrked grphs, nd we cll the corresponding STGs mrked grph STGs. The rest of the section contins only folklore results, but we include them for the ske of completeness. In order to check consistency of very well formed mrked grph STGs, we reuse result tht we proved for CP-subnets. It cn be extended without effort to ll non-utoconcurrent mrked grph STGs. Lemm 3.4. Let S be non-utoconcurrent mrked grph STG, nd let M σ nd M τ be two occurrence sequences of S. For ech signl, l(σ) is pre x of l(τ), or vice vers. Proof: The proof of Lemm 3.1 works with minor modi ctions. We construct sequence υ such tht υ = mx{ σ, τ}, nd show tht if the desired property does not hold then S is not utoconcurrent. It is well known tht well formed mrked grphs hve the following property: there exists n occurrence sequence M 0 M 0 in which ll trnsitions occur exctly once. This sequence cn obviously be σ iterted n rbitrry number of times. We hve the following result: Theorem 3.5. Let S be very well formed mrked grph STG with initil mrking M 0. Let M 0 σ M 0 be the sequence in which ech trnsition occurs exctly once. S is consistent if nd only if σ is lternting. Proof: The only if direction is trivil. For the if direction, ssume S is not consistent. Since S is very well

20 20 J. Esprz / Checking Consistency of Free-Choice Signl Trnsition Grphs τ formed, it is live nd cyclic. By Lemm 2.1 there is non-lternting sequence M 0. Choose number n such tht σ n σ is longer thn τ. By Lemm 3.4, M n 0 is non-lternting. Since S is blnced nd σ is T-invrint, we hve bl( σ) = 0. But then σ must be non-lternting. 4. The lgorithm In this section we present the complete lgorithm for checking consistency. Input: well formed free-choice STG S. Output: consistent or not consistent. (1) Check if S is non-utoconcurrent. If it is utoconcurrent, return not consistent, otherwise go to step 2. (2) Check if S is blnced. If not, return non consistent, otherwise go to step 3. (3) If S is mrked grph STG, construct sequence M 0 σ M 0 in which every trnsition occurs exctly once; if σ is lternting for every signl then return consistent, otherwise return not consistent. If S is not mrked grph, go to step 4. (4) Select CP-subnet N of S, nd let its trnsitions occur s long s possible (ech trnsition cn occur t most once). Cll this new STG (new becuse of the different initil mrking) S. (5) Consider N with the following initil mrking: one token on ech entry plce, s mny tokens on ech internl plce s in the initil mrking of the STG S, nd no tokens on exit plces. Execute ushing sequence in which ech trnsition occurs exctly once. Check if ll signls lternte. If not, return not consistent, otherwise go to step 6. (6) Remove from S ll internl plces nd trnsitions of N. Let S := S R s de ned in Section (inserting witnesses where necessry), nd go to step 3. We show tht the lgorithm runs in O(( P + T ) 3 ) time: The lgorithm of [14] for Step (1) tkes O( P ( P + T ) 2 ) time. Step (2) requires to compute bsis of the solutions of the system N X = 0, where N hs dimension P T. This cn be done in time O(M(mx{ P, T })), where M(n) denotes the complexity of mtrix multipliction, nd so certinly in time O(( P + T ) 3 ). The loop of the divide-nd-conquer procedure cn run for t most T itertions, since ech itertion reduces the number of output trnsitions of one plce by 1. Moreover, t ech step the STG S R hs t most s mny nodes s S (the inserted pth of witnesses contins t most s mny nodes s the removed CP-subnet). Steps (3) nd (5) tke O(( P + T ) 2 ) time, even if we ssume tht it tkes liner time to nd n enbled trnsition.

21 J. Esprz / Checking Consistency of Free-Choice Signl Trnsition Grphs 21 Step (4) tkes O(( P + T ) 2 ) time: there re t most T cndidtes to be the entry trnsition of CP-subnet, nd we cn check in O( P + T ) time for ech of them if they indeed re (using Trjn s lgorithm for checking strong connectedness). Step (6) tkes O( P + T ) time. So the execution of ll the itertions of the loop s body tke together O( T ( P + T ) 2 ) time, nd the whole lgorithm runs in O(( P + T ) 3 ) time. 5. Conclusions We hve presented the rst polynomil lgorithm for checking consistency of well formed free-choice STGs. The correctness proof uses mny results from Petri net theory. Almost ll of the min theorems of the monogrph [9] re directly or indirectly used. There could be much simpler proof, lthough experience shows tht results bout free-choice nets often need long rguments. Together with the techniques of [3] nd [18], our lgorithm cn be used to produce free-choice STGs with consistent nd complete encodings without hving to construct their stte grph, which could be exponentilly lrger thn the STG itself. We suspect tht it is possible to get rid of the blncedness check t the price of more complicted divide-nd-conquer procedure. Exploring this is left for future reserch. The consistency problem is n instnce of the more generl problem of determining synchronic distnces between sets of trnsitions of Petri net (how often cn trnsitions in one set occur without the others occurring?). These problems pper lso in other pplictions of free-choice nets in the re of work ow processes [1]. We think tht our results will lso be useful there. 6. Acknowledgements Mny thnks to Jordi Cortdell nd Alex Ykovlev for mny helpful comments, pointers to the literture, nd discussions. The uthor is very indebted to the nonymous referees for their creful reding nd very helpful comments. References [1] vn der Alst, W.: The Appliction of Petri Nets to Work ow Mngement, The Journl of Circuits, Systems nd Computers, 8(1), 1998, [2] vn Berkel, C., Josephs, M., Nowick, S.: Scnning the Technology: Applictions of Asynchronous Circuits, Proceedings of the IEEE, 87(2), 1999, [3] Crmon, J., Cortdell, J., Pstor, E.: A structurl encoding technique for the synthesis of synchronous circuits, Proc. Int. Conf. on Appliction of Concurrency Theory to System Design, IEEE Computer Society, [4] Chu, T.-A.: On the models for designing VLSI synchronous digitl systems, Integrtion: the VLSI journl, 4, 1986,