Petri automata for Kleene allegories

Transcription

1 Petri utomt for Kleene llegories Pul Brunet, mien Pous To ite this version: Pul Brunet, mien Pous. Petri utomt for Kleene llegories. Logi in omputer Siene, Jul 205, Kyoto, Jpn. I, pp.68-79, 205, < <0.09/LIS.205.7>. <hl v3> HL I: hl Sumitte on 6 Mr 205 HL is multi-isiplinry open ess rhive for the eposit n issemintion of sientifi reserh ouments, whether they re pulishe or not. The ouments my ome from tehing n reserh institutions in Frne or ro, or from puli or privte reserh enters. L rhive ouverte pluriisiplinire HL, est estinée u épôt et à l iffusion e ouments sientifiques e niveu reherhe, puliés ou non, émnnt es étlissements enseignement et e reherhe frnçis ou étrngers, es lortoires pulis ou privés.

2 Petri utomt for Kleene llegories Pul Brunet n mien Pous Plume tem LIP, NRS, NS e Lyon, Inri, UBL, Université e Lyon, UMR 5668 {pul.runet,mien.pous}@ens-lyon.fr strt Kleene lger xioms re omplete with respet to oth lnguge moels n inry reltion moels. In prtiulr, two regulr expressions reognise the sme lnguge if n only if they re universlly equivlent in the moel of inry reltions. We onsier Kleene llegories, i.e., Kleene lgers with two itionl opertions whih re nturl in inry reltion moels: intersetion n onverse. While regulr lnguges re lose uner those opertions, the ove hrteristion reks. Inste, we give hrteristion in terms of lnguges of irete n lelle grphs. We then esign finite utomt moel llowing to reognise suh grphs, y tking inspirtion from Petri nets. This moel llows us to otin eiility of ientity-free reltionl Kleene ltties, i.e., the equtionl theory generte y inry reltions on the signture of regulr expressions with intersetion, ut where one foris unit. This restrition is use to ensure tht the orresponing grphs re yli. The eiility of grph-lnguge equivlene in the full moel remins open. I. INTROUTION We onsier inry reltions n the opertions of union ( ), intersetion ( ), omposition ( ), onverse ( ), trnsitive losure ( + ), reflexive-trnsitive losure ( ), n the onstnts ientity () n empty reltion (0). This moel gives rise to n (in)equtionl theory: pir of terms e, f me from those opertions n some vriles,,... is vli eqution, enote Rel e = f, if the orresponing equlity hols universlly. Similrly, n ineqution Rel e f is vli when the orresponing ontinment hols universlly. Here re vli equtions n inequtions: they hol whtever the reltions we ssign to vriles,, n. Rel ( ) ( ) = ( ) + () Rel + (2) Rel ( ) (3) Rel + ( ) + (4) Vrious frgments of this theory hve een stuie in the literture: Kleene lger [7], where one removes intersetion n onverse, so tht terms re plin regulr expressions. The theory is eile y Kleene s work [], n tully PSP-omplete [4], [5]. qution () lies in this frgment, n one n notie tht the two expressions reognise the sme lnguge. Kleene lger with onverse, where one only removes intersetion, is lso eile frgment [3]. It remins PSP [5]. Ineqution (2) elongs to this frgment. (representle, istriutive) llegories [8], sometimes lle positive reltion lgers, where trnsitive n reflexive-trnsitive losures re not llowe. They re eile [8, pge 208]; Ineqution (3) is known s the moulrity lw in this setting. To the est of our knowlege, the eiility of the whole theory, Kleene llegories, is open. Here we otin severl importnt steps towrs the resolution of this prolem: ) we give hrteristion of the full (in)equtionl theory in terms of grph lnguges; 2) we esign n utomt moel inspire y Petri nets, tht mkes it possile to reognise suh grphs; 3) we show how to ssoite suh grph utomton to ny term of Kleene llegories; 4) using these grph utomt, we give eision proeure for the frgment where onverse n ientity re forien. The ltter frgment ws stuie reently y nrék et l. []; its eiility ws open s fr s we know. The restrition to this frgment llows us to exploit simplifying ssumptions out the proue utomt, n to otin oinutive lgorithm for lnguge inlusion (Setion V). We tully show tht lnguge inlusion for these utomt is XPSPomplete (Setion VI). The next prolem, whih remins open, onsists in otining the eiility of lnguge inlusion in the full utomt moel: together with the presente results, this woul entil eiility of Kleene llegories. We outline some of the iffiulties rising with onverse or unit in presene of intersetion in Setion V-. We ontinue this introutory setion y n informl esription of the grph lnguge hrteristion n our utomt moel.. Lnguges In the simple se of Kleene lger, i.e., without onverse n intersetion, the (in)equtionl theory generte y reltions n e hrterise y using regulr lnguges. Write e for the lnguge enote y regulr expression e; for ny two regulr expressions e, f, we hve Rel e f if n only if e f. (5) (This result is esy n folklore; proving tht this is lso equivlent to provility using Kleene lger xioms [4], [2], [3] is muh hrer.) While regulr lnguges re lose uner intersetion n onverse, the ove hrteristion oes not exten to those opertions. For intersetion, onsier two istint vriles

3 (): (( ( )) ): (( ) ( )): (( ) ): ( ): Figure : rphs ssoite to some groun terms. n. The extene regulr expressions n 0 oth reognise the empty lnguge, while Rel / = 0: one n interpret n with interseting reltions. For onverse, the extene regulr expressions n oth reognise the singleton lnguge onsisting of the single-letter wor. Yet Rel / =, s there re non-symmetri reltions. B. rphs Frey n Serov eision proeure for representle llegories [8, pge 208] relies on notion of irete, lelle, 2-pointe grph. The sme notion ws propose inepenently y nrék n Breikhin [2], in more omprehensive wy. ll groun terms the terms in the syntx of llegories (omposition, intersetion, onverse, n unit). groun term u n e represente s lelle irete grph (u) with two istinguishe verties lle the input n the output. We give some exmples in Figure, see efinition for preise efinition. These grphs n e enowe with preorer reltion: we write F when there exists grph homomorphism from F to preserving lels, inputs, n outputs. For instne the grph orresponing to ( ( )) is smller thn the grph of ( ) ( ), thnks to the homomorphism epite in Figure 2 using otte rrows. Notie tht the homomorphism nees not e injetive or surjetive, so tht this preorer hs nothing to o with the respetive sizes of the grphs: grph my very well e smller thn nother in the sense of, while hving more verties or eges (n vie vers). The key result from Frey n Serov [8, pge 208], or nrék n Breikhin [2, Theorem ], is tht for ny two groun terms u, v, we hve Rel u v if n only if (u) (v). (6) The grphs re finite so tht one n serh exhustively for homomorphism, whene the eiility result.. rph lnguges To exten the ove grph-theoretil hrteristion to Kleene llegories, we nee to hnle union n (reflexive-) trnsitive losures. It suffies for tht to onsier sets of grphs: to eh expression e, we ssoite set of grphs (e). This set is most often infinite when the expression e ontins (reflexive-)trnsitive losures. Writing X for the ownwr losure of set of grphs X y the preorer on grphs, we otin the following generlistion of oth (5) n (6): for ny two expressions e n f, Rel e f if n only if (e) (f). (7) This is Theorem 6 in the sequel, n this result is lmost there in the work y nrék et l. [], [2]. To the est of our knowlege this expliit formultion is new, s well s its use towrs eiility results. When e n f re groun terms, we reover the hrteristion (6) for representle llegories: (e) n (f) re singleton sets in this se. For plin regulr expressions, the grphs re just wors n the preorer reues to isomorphism. We thus reover the hrteristion (5) for Kleene lger. This result lso generlises the hrteristion provie y Ésik et l. [3] for Kleene lger with onverse: grphs of expressions without intersetion re just wors over uplite lphet, n the orresponing restrition of the preorer preisely orrespons to the wor rewriting system they use (see Remrk 7).. Petri utomt In orer to exploit the ove hrteristion n otin eiility results, one hs first to represent grph lnguges in finitry wy. We propose for tht new finite utomt moel, lrgely se on Petri nets [6] [8]. We esrie this moel elow, ignoring onverse n unit for the ske of lrity. Rell tht Petri net onsists of finite set of ples, enote with irles; set of trnsitions, enote with retngles; for eh trnsition, set of input ples n set of output ples, enote with rrows; n initil ple, enote y n entrnt rrow; set of finl mrkings, enote y otte oxes ( mrking, or onfigurtion, eing set of ples). The exeution moel is the following: strt y putting token on the initil ple; hoose trnsition whose input ples ll ontin token, remove those tokens n put new tokens in the output ples of the trnsition; repet this proess until finl mrking is rehe. The otine sequene of trnsitions is lle n epting run. (We tully restrit ourselves to sfe Petri nets, to ensure tht there is lwys t most one token in given ple when plying this gme.) Petri utomton is just sfe Petri net with vriles lelling the outputs of eh trnsition. The utomton epite For every rehle mrking in sfe net, there is t most one token in ny given ple. 2

4 : 4 F : 0 Figure 2: grph homomorphism. elow is the utomton we will onstrut for the groun term. ny run must strt y firing the left-most trnsition, rehing the mrking {B, }; then we hve the hoie of firing the upper trnsition first, rehing the mrking {, }, or the lower one, rehing the mrking {B, }. In oth ses we reh the finl mrking {, } y firing the remining trnsition. To re grph in suh n utomton, we try to fin n epting run tht mthes the grph up to homomorphism (efinitions 0 n ). We o tht y using n evolving funtion from the tokens to the verties of the grph. We strt with the funtion mpping the unique token, in the initil ple, to the input vertex of the grph. To fire trnsition, we must hek tht ll its input tokens re mppe to the sme vertex in the grph, n tht this vertex hs severl outgoing eges, lelle oring to the outputs of the trnsition. If this is the se, we upte the funtion y removing the mppings orresponing to the elete tokens, n y ing new mppings for eh of the rete tokens (using the trget verties of the forementione outgoing eges, oring to the lels). The grph is epte y the Petri utomton if we n reh finl mrking of the Petri utomton, with ll tokens mppe to the output vertex of the grph. For instne, the previous utomton epts the grph of (F in Figure 2). We strt with the funtion { 0}. We n fire the first trnsition, upting the funtion into {B, 2} (We oul lso hoose to upte the funtion into {B 2, }, or {B, }, or {B, 2} ut this woul le to e-en). Then we n fire the upper trnsition, evolving the funtion into { 3, 2}, n we finish y firing the remining trnsition, otining the funtion {, 3}. We ll lnguge of the set of grphs L ( ) epte y Petri utomton. This lnguge is ownwr-lose: L ( ) = L ( ). For instne, the previous utomton lso epts the grph from Figure 2, whih is smller thn F. Inee, when we fire the first trnsition, we n ssoite the two newly rete tokens (in ples B n ) to the B sme vertex (5). This tully orrespons to omposing the funtions use to ept F with the homomorphism epite with otte rrows. This utomt moel is expressive enough for Kleene llegories: for ny expression e, we n onstrut Petri utomton (e) suh tht L ( (e)) = (e) (Setion IV). We give three other exmples of Petri utomt to give more intuition on their ehviour. The first trnsition in the previous Petri utomton splits the initil token into two tokens, whih re move onurrently in the reminer of the run. This orrespons to n intersetion in the onsiere expression. This is to e ontrste with the ehviour of the following utomton, whih we woul onstrut for the non-groun expression. This utomton hs two epting runs: {}, {B}, {} n {}, {}, {}, whih n e use to ept the (grphs of the) groun terms n. In sense, two trnsitions ompeting for the sme tokens represent non-eterministi hoie, i.e., union in the strting expression. Still in the first exmple, the two tokens rete y the first trnsition re lter ollete in the finl mrking. Tokens my lso e ollete n merge y trnsition. onsier for instne the following utomton for ( ). It hs only one epting run, {}, {B, }, {B, }, {}, n this run n e use to re the fourth grph from Figure. s lst exmple, onsier the following utomton for the expression +. The upper trnsition introues loop, so tht there re infinitely mny epting runs. For ny n > 0, the grph of the groun term n is epte y this utomton. oring to hrteristion (7), the next step is to eie whether the ontinment L ( ) L (B) hols, for two given Petri utomt n B. Severl iffiulties rise, tht o not pper with lssil wor utomt. Our solution nevertheless uses stnr oinutive pproh, where we efine n pproprite notion of simultion (Setion V-). B B B 3

5 () = () = (w ) = (w) (u v) = (u) (v) (u v) = (u) (v) Figure 3: Inutive onstrution of the grph of groun term. Stnr nottions. For ny sets, B, we write P (S) = {P P } for the set of susets of, B for the set of funtions from to B, n B for the set of prtil mps from to B. The omin of prtil mp f is enote y om (f). II. RPH-THORTIL HRTRISTION We onsier the signture,,, +,, 0, of Kleene llegories, where e is n revition for e +. We fix set X of vriles, n we enote y Reg X the set of expressions e, f... uilt from vriles in X with these onnetives. roun terms re the expressions u, v, w... uilt from X only with the su-signture,,,. If σ X P (S S) is n interprettion of the lphet X into some spe of reltions, we write σ for the unique homomorphism extening σ into funtion from Reg X to P (S S). n ineqution etween two expressions e n f is vli, written Rel e f, if for ny reltionl interprettion σ we hve σ(e) σ(f). We let rnge over 2-pointe lelle irete grphs, whih we simply ll grphs in the sequel. Those re tuples V,, ι, o with V finite set of verties, V X V set of eges lelle with X, n ι, o V the two istinguishe verties, respetively lle input n output. efinition (rph of groun term: (w)) To eh groun term w, we ssoite grph (w), y inution on w. The grph of X hs one ege lelle y linking its input to its output. The grph for hs only one vertex, oth input n output. The omposition of two grphs with isjoint sets of verties n e performe y ientifying the output of the first grph n the input of the seon one. The intersetion on grphs onsists in merging their inputs n merging their outputs. The onverse onsists simply in exhnging the input n the output of grph. See Figure 3 for grphil esription of this onstrution. Those grphs were introue inepenently y Frey n Serov [8, pge 208], n nrék n Breikhin [2]. efinition 2 (rph homomorphism, preorers n ) grph homomorphism from V,, ι, o to V 2, 2, ι 2, o 2 is mp ϕ V V 2 suh tht ϕ(ι ) = ι 2, ϕ(o ) = o 2, n (p, x, q) entils (ϕ(p), x, ϕ(q)) 2. We enote y the reltion on grphs efine y if there exists grph homomorphism from to. This reltion gives rise to preorer on groun terms, written n efine y u v if (u) (v). iven set S of grphs, we write S for its ownwr losure: S = {, S }. Similrly, we write S for the ownwr losure of set of groun terms w.r.t.. s expline in the introution, the ove preorer on groun terms preisely hrterises inlusion uner ritrry reltionl interprettions: Theorem 3 ([2, Theorem ], or [8, pge 208]). For ll groun terms u, v, we hve Rel u v u v. To exten this result to Kleene llegories, we introue the following generlistion of the lnguge of regulr expression. Sets of wors eome sets of groun terms. efinition 4 (Terms n grphs of n expression) The set of terms of n expression e Reg X, written e, is the set of groun terms efine inutively s follows: = {} 0 = x = {x} e f = {w w w e n w f} e f = {w w w e n w f} e f = e f e = n N {w w n i, w i e} e = {w w e}. The set of grphs proue y n expression e, enote y (e) is the set of grphs ssoite to the groun terms in e: (e) = {(w) w e}. To otin the hrteristion nnoune in the introution, we nee slight refinement of lemm estlishe y nrék, Mikulás, n Németi []: Lemm 5. For ll expression e Reg X X, n ll reltionl interprettions σ X P (S S), we hve σ(e) = σ(u) = u e w e σ(w). Proof. The first equlity is extly [, Lemm 2.]; for the seon one, we use the ft tht σ(w) σ(u) whenever w u, thnks to Theorem 3 (i.e., [2, Theorem ]). Theorem 6. The following properties re equivlent, for ll expressions e, f Reg X : (i) Rel e f, (ii) e f, (iii) (e) (f). Proof. We give etile proof in ppenix. The implition (ii) (i) follows esily from Lemm 5, n (iii) (ii) is mtter of unfoling efinitions. For (i) (iii), we minly use [2, Lemm 3]. (The ext hrteristion nnoune in the introution (7) follows: for ny sets X, Y, we hve X Y iff X Y.) 4

6 B H I F t t 0 B t 2 t 4 t 3 t 5 Figure 5: n epting run in the utomton from Figure 4. F Figure 4: Petri utomton. The initil ple is, n the finl onfigurtions re {I} n {F, }. lso notie tht while (f) only ontins grphs emnting from groun terms, this is not the se for its losure (f). For instne, (( ) ( )) ontins the following grph, whih is not the grph of ny groun term. e Remrk 7. The grphs ssoite to groun terms without intersetion re isomorphi to wors over uplite lphet. grph homomorphism etween two suh grphs is preisely wht Ésik et l. ll n missile mp [3]. Theorem 6 n thus e seen s generlistion of [3, Theorem 5.3]. III. PTRI UTOMT We exten the set X of vriles into set X of lels: X = X {x x X } {}. Petri utomton is Petri net whose trnsition s outputs re lelle y X. efinition 8 (Petri utomton) Petri utomton over the lphet X is tuple P, T, ι, F where: P is finite set of ples, T P (P ) P ( X P ) is set of trnsitions, ι P is the initil ple of the utomton, F P (P ) is set of finl onfigurtions, onfigurtion eing set of ples. For eh trnsition t = (t, t) T, t n t re ssume to e non-empty; t P is the input of t; n t X P is the output of t. We use the grphil nottion from the introution to represent Petri utomt; the Petri utomton from Figure 4 will e use s running exmple. From onfigurtion ξ P, trnsition t = (t, t) T is enle if t ξ. If so, one my fire t, whih proues new onfigurtion ξ = ξ t {p P x X (x, p) t}. We write ξ t ξ in this se. set of trnsitions T T is sttilly omptile (or just omptile) if their inputs re pirwise isjoint. If furthermore ll trnsitions in T re enle in onfigurtion ξ, one n oserve tht the onfigurtion ξ rehe fter firing them suessively oes not epen on the orer in whih they re fire. In tht se we write ξ T ξ. In the sequel, we ssume ll onsiere Petri utomt to e sfe. (I.e., in Petri nets terminology, suh tht ny rehle mrking hs t most one token in eh ple [6]). Formlly, with our efinitions: Petri utomton P, T, ι, F is sfe if for ll onfigurtion ξ P rehle from {ι} y firing ny numer of trnsitions, if (t, t) T is enle from ξ, p ξ, n (x, p) t, then p t. Now we explin how to use Petri utomt to efine lnguges of grphs. We first efine the runs of n utomton. efinition 9 (Run, epting run, prllel run) run is sequene ξ = (ξ k ) 0 k n, (t k ) 0 k<n of onfigurtions n trnsitions, suh tht ξ k P, t k T n t k k < n, ξ k ξk+. When ξ 0 = {ι} n ξ n F, we ll ξ n epting run. prllel run is efine similrly, s sequene Ξ = (Ξ k ) 0 k n, (T k ) 0 k<n, where the T k T re omptile T k sets of trnsitions suh tht Ξ k Ξk+. (Note tht run ξ is uniquely etermine y ξ 0 n the sequene (t k ): ll susequent onfigurtions n e ompute eterministilly.) onsier the following sequene of trnsitions from the utomton in Figure 4: with ξ = (ξ 0, ξ, ξ 2, ξ 3, ξ 4, ξ 5, ξ 6 ), (t 0, t, t 2, t 3, t 4, t 5 ), ξ 0 ={}, ξ ={B, }, ξ 2 = ξ 4 ={,, }, ξ 3 = ξ 5 ={,, }, ξ 6 ={F, }. We hve {} t0 {B, } t {,, } t 0 =({}, {(, B), (, )}), t =({B}, {(, ), (, )}), t 2 = t 4 =({}, {(, )}), t 3 =({, }, {(, ), (, )}), t 5 =({, }, {(, F )}). t 2,t 4 t 3 t 5 {,, } {F, }, n sine {} is the initil onfigurtion n {F, } F, this sequene is n epting run. It n e represente grphilly s in Figure 5. s for stnr finite stte utomt, we now nee to speify how to re grph in n utomton. s expline in the introution, this is one y linking the intermeite onfigurtions of run to verties in the grph, n y imposing onitions to mth trnsitions with lelle eges of the grph. efinition 0 (Reing, prllel reing, lnguge of run) reing of = V,, ι, o long run ξ = 5

7 0 2 3 Figure 6: rph proue y the run epite in Figure 5. (ξ k ) 0 k n, (t k, t k ) 0 k<n is sequene (ρ k ) 0 k n suh tht for ll k, ρ k is mp from ξ k to V, ρ 0 (ξ 0 ) = {ι}, ρ n (ξ n ) = {o}, n k < n, the following hols: ll tokens in the input of the trnsition re mppe to the sme vertex in the grph: p, q t k, ρ k (p) = ρ k (q); the imges of tokens in ξ k tht re not in the input of the trnsition re unhnge: p ξ k t k, ρ k (p) = ρ k+ (p); eh pir in the output of the trnsition n e vlite y the grph: p t k, (x, q) t k, 4 x X (ρ k (p), x, ρ k+ (q)), x = y n y X (ρ k+ (q), y, ρ k (p)), x = ρ k (p) = ρ k+ (q). Similrly, we efine prllel reing ρ long some prllel run Ξ = (Ξ k ) 0 k n, (T k ) 0 k<n y requiring tht: ρ 0 (Ξ 0 ) = {ι}, ρ n (Ξ n ) = {o}, n k < n the following hols: p Ξ k (t,t) Tk t, ρ k+ (p) = ρ k (p); (t, t) T k, p, q t, ρ k (p) = ρ k (q); (t, t) T k, p t, (x, q) t, x X (ρ k (p), x, ρ k+ (q)), x = y n y X (ρ k+ (q), y, ρ k (p)), x = ρ k (p) = ρ k+ (q). The lnguge of run ξ, enote y L (ξ) is the set of grphs tht n e re long ξ. The lnguge of Petri utomton is finlly otine y onsiering ll epting runs. efinition (Lnguge reognise y Petri utomton) The lnguge reognise y, written L ( ), is the following set of grphs: L ( ) = ξ epting in 5 L (ξ). The lnguge of run ξ n e hrterise y using single grph whih we ll the grph proue y ξ: grphs re epte y ξ extly when they re smller thn the grph proue y ξ oring to (Lemm 4 elow). For instne, the run presente in Figure 5 proues the grph epite in Figure 6. This grph is otine in two steps, y first onsiering notion of tre, whih is grph lelle with X rther thn X, n whih tully orrespons to the notion of pomsettre from stnr Petri nets (see Setion VII for more etils on this orresponene). The tre is onstrute y reting vertex k for eh trnsition t k = (t k, t k ) of the run, plus finl vertex n. We 6 n ege (k, x, l) whenever there is some ple q suh tht (x, q) t k, n t l is the first trnsition fter t k in the run with q mong its inputs, or l = n if there is no suh trnsition in the run. efinition 2 (Tre of run) Let ξ = (ξ k ) 0 k n, (t k, t k ) 0 k<n e run. For n inex k n n ple q, let ν(k, q) e either the smllest inex l suh tht k l n q t l, or n if there is no suh inex. The tre of ξ is the grph ξ = {0,..., n}, ξ, 0, n with ξ = {(k, x, ν(k +, q)) (x, q) t k }. To get the finl grph, whih is lelle y X, one ientifies noes linke y eges lelle y, n one reples eh ege of the form (i, x, j) y (j, x, i). Formlly: efinition 3 (rph proue y run) Let ξ = (ξ k ) 0 k n, (t k, t k ) 0 k<n e run. Let ξ e the smllest equivlene reltion on {0,..., n} ontining ll pirs (i, j) suh tht (i,, j) ξ. The grph proue y ξ, is the grph (ξ) efine y (ξ) = {[i] ξ 0 i n}, ξ, [0] ξ, [n] ξ [i] ξ = {k {0,..., n} i ξ k } ξ = ([i] ξ, x, [j] ξ ) x X n k [i] ξ, l [j] ξ (k, x, l) ξ or (l, x, k) ξ We write ( ) for the set of grphs proue y epting runs of Petri utomton. To voi onfusions with the lnguge L ( ) of, we write is proue y when ( ), reserving lnguge theoreti terminology like is epte y or reognises to ses where we men L ( ). The grph proue y the run presente in Figure 5 hppens to e equl to its tre, sine it is lelle in X only. more involve exmple is given in Figures 7 to 9. Notie tht lthough the tre of run is yli n n e enowe with prtil orer struture (simply hek tht p, ν(_, p) is inresing), it is not neessrily the se for its proue grph. Lemm 4. For ny epting run ξ, we hve L (ξ) if n only if (ξ). Proof. Suppose there exists grph homomorphism ϕ from (ξ) to. Then we n uil reing y efining ρ k (p) = ϕ([ν(k, p)] ξ ) for 0 k n n p ξ k. On the other hn, if we hve reing (ρ k ) 0 k n of, we n uil homomorphism ϕ y letting ϕ([k] ξ ) = ρ k (p) for ny p t k. s (ρ k ) k is reing, ϕ is well efine. The etils of this proof n e foun in ppenix B. s n immeite orollry, we otin the following hrteristion of the lnguge of Petri utomton. orollry 5. L ( ) = ( ). 6

8 0 0 B 2 2 Figure 7: run ξ. 3 Figure 8: The tre of ξ F Figure 9: The grph proue y ξ. The left-hn sie lnguge is efine through reings long epting runs, whih is lol n inrementl notion n whih llows us to efine simultions in Setion V-. By ontrst, the right-hn sie lnguge is efine glolly, whih eses the following onstrution of n utomton reognising the lnguge of n expression. IV. FROM XPRSSIONS TO UTOMT We now show how to ssoite to ny expression e Reg X n utomton (e) tht reognises the lnguge (e). In ft the utomton we otin hs n even stronger onnetion with e: the grphs in (e) re extly those proue y epting runs in (e). To mke the onstrution simpler, we first moify the expression so tht the opertor is only pplie to vriles, y using the following rewriting system: ( ) 0 0 ( + ) ( ) + ( ) ( ). (These rules preserve the set of grphs of the expression; lso rell tht e is shorthn for e + +, so tht we o not nee to hnle it expliitly.) The forml onstrution is inutive; it is given in efinition 6 elow. We esrie it first in informl terms. For the se of x X, we simply uil n utomton with single trnsition lelle y x, going from the initil ple to istint finl ple 2. The union onsists in putting oth utomt sie y sie, n merging their initil ples. For the omposition of n 2 on the other hn we put one utomton in front of the other: the initil ple of the resulting utomton is tht of ; the finl onfigurtions re those of 2 ; for eh finl onfigurtion f of, n for eh initil trnsition (ι 2, t) oming out of the initil ple of 2, we trnsition with input f n output t. This lst step mounts to ing epsilon trnsitions from the finl 2 Note tht this overs uniformly the se of the unit, of vriles, n of onverse vriles. onfigurtions of to the initil ple of 2, n then pply some epsilon-elimintion proeure. We lso put the two utomt sie y sie for the intersetion, ut we merge their initil ples, their initil trnsitions, n their finl onfigurtions: for ny pir of initil trnsitions of the two utomt ({ι }, t ), ({ι 2 }, t 2 ), we put in the intersetion utomton the trnsition ({ι }, t t 2 ); finl onfigurtion of this utomton is the union of finl onfigurtion from n finl onfigurtion of 2. For the trnsitive losure ( + ), we use the ies for union n omposition, ing loops from the finl onfigurtions using the initil trnsitions of the utomton. efinition 6 To eh expression e Reg X, we ssoite Petri utomton (e) efine inutively s follows: x X, (x) = {0, }, {({0}, {(x, )})}, 0, {{}} (0) = {0},, 0, (e e 2 ) = P P 2, T, ι, F F 2 with T = T T 2 {({ι }, t) ({ι 2 }, t) T 2 }. (e e 2 ) = P P 2, T, ι, F 2 with T = T T 2 {(f, t) f F n ({ι 2 }, t) T 2 }. (e + ) = P, T, ι, F with T = T {(f, t) f F n ({ι }, t) T }. (e e 2 ) = P P 2, T, ι, F with F = {f f 2 f F, f 2 F 2 } n T = {(t, t) i {, 2} (t, t) T i, ι i t} {({ι }, t t 2 ) i {, 2}, ({ι i }, t i ) T i }. (In the inutive ses, we ssume (e i ) = P i, T i, ι i, F i for i {, 2}, with P P 2 =.) We prove y inution on e tht (e) is inee sfe Petri utomton; for the sfety requirement, we to the inution hypothesis the ft tht for ny onfigurtion ξ essile in (e), if there is finl onfigurtion f F suh tht f ξ, then f = ξ. nother invrint is tht the initil ple never ppers in finl onfigurtion, nor in the output of ny trnsition. Note tht the ple ι 2 eomes unrehle y onstrution in the ses for union, omposition n intersetion, so tht it oul sfely e remove, together with the ssoite trnsitions. Theorem 7 (orretness). For ll expression e Reg X, L ( (e)) = (e). Proof. We prove stronger result: ( (e)) = (e) (up to grph isomorphisms see ppenix ). This llows us to onlue thnks to orollry 5. orollry 8. The (in)equtionl theory of Kleene llegories is o-reursively enumerle. Proof. onstrut Petri utomt for the two expressions n enumerte ll potentil ounter-exmples, i.e., grphs. grph 7

9 is ounter-exmple if it n e re in one utomton ut not in the other, whih is eile property. Remrk 9. If e is n expression without intersetion, onverse or, it n e shown tht the trnsitions in (e) re ll of the form ({p}, {(x, q)}), with only one input, one output n lel in X. s onsequene, the essile onfigurtions re singletons, n the resulting Petri utomton hs the struture of non-eterministi finite-stte utomton (NF). tully, in tht se, the onstrution we esrie ove is just vrition on Thompson s onstrution [20], with inline epsilon trnsition elimintion.. Simple utomt V. OMPRIN UTOMT The ove results hol for the whole syntx of regulr expressions with onverse n intersetion. However, in the reminer of the pper, we hve to fous on expressions without onverse or ientity. This is euse in omintion with intersetion, these two opertions introue yles in the grphs ssoite to groun terms. onsier for instne the grphs for n : ( ) = ( ) = Sine reflexive-trnsitive losure ( ) impliitly ontins n ourrene of the ientity, we lso hve to fori this opertor. Inste, we the trnsitive losure ( + ). We thus work with expressions from Reg X, efine with the following syntx: e, f Reg X = x X e f e f e f e+ 0. oringly, groun terms re restrite to the following syntx: u, v, w = x X w w w w. utomt uilt using efinition 6 from expressions without onverse or unit only hve trnsitions lelle with X. This orrespons to the notion of simple utomt. efinition 20 (Simple Petri utomton) Petri utomton = P, T, ι, F is lle simple if (t, t) T, (x, p) t, x X. For ll e Reg X, (e) is simple. Moreover for ny run ξ of simple Petri utomton, ξ = (ξ) (up to isomorphism); in prtiulr, simple utomton only proues yli grphs. B. Intuitions In this setion, we show how the notion of simultion reltion, tht llows to ompre NF, n e pte to hnle simple Petri utomt. onsier two utomt = P, T, ι, F n 2 = P 2, T 2, ι 2, F 2, we try to show tht for ny grph epte y, is reognise y 2. By Lemm 4, this mounts to proving tht for ny epting run ξ in, (ξ) is reognise y some epting run ξ in 2. Leving non-eterminism prt, the first ie tht omes to min is to fin reltion etween the onfigurtions in. n the onfigurtions in 2, tht stisfy some onitions on the initil n finl onfigurtions, n suh tht if ξ k ξ k n t ξ k ξ k+, then there is onfigurtion ξ k+ in 2 suh t 2 tht ξ k+ ξ k+, ξ k ξ k+, n these trnsition steps re omptile in some sense. However, suh efinition will not give us the result we re looking for. onsier these two runs: W B Y 2 X 2 The grphs proue y the first n the seon runs orrespon respetively to the groun terms ( ) n ( ). These two terms re inomprle, ut the reltion epite elow stisfies the previously stte onitions. {} {W } {B, } {X, Y } 2 2 B Z Y {B, } {Y, Z} The prolem here is tht in Petri utomt, runs re token firing gmes. To equtely ompre two runs, we nee to losely trk the tokens. For this reson, we will relte onfigurtion ξ k in not only to onfigurtion ξ k in 2, ut to mp η k from ξ k to ξ k. This will enle us to ssoite with eh token situte on some ple in P 2 nother token ple on. We wnt to fin reing of (ξ) in 2, i.e., run in 2 together with sequene of mps ssoiting ples in 2 to positions in (ξ). onsier the piture elow. Sine we lrey hve reing of (ξ) long ξ (y efining ρ k (p) = ν(k, p), s in the proof of Lemm 4), it suffies to fin mps from the ples in 2 to the ples in (the mps η k ): the reing of (ξ) in 2 will e otine y omposing η k with ρ k. η 0 (ξ) ξ 0 t 0 ξ t tn ξ n t n ξ n+ ξ 0 t 0 ρ 0 η ξ t ρ ρ n t n ξ n ρ n+ η n t n ξ n+ η n+ We nee to impose some onstrints on the mps (η k ) to ensure tht (ρ k η k ) 0 k n is inee orret reing in 2. First, we nee to sertin tht trnsition t k in 2 my e fire from the reing stte ρ k η k to reh the reing stte ρ k+ η k+. Furthermore, s for NF, we wnt trnsitions t k 8

10 n t k to e relte: speifilly, we require t k to e inlue (vi the homomorphisms η k n η k+ ) in the trnsition t k. This is meningful euse trnsition t k ontins lot of informtion out the vertex k of (ξ) n out ρ: the lels of the outgoing eges from k re the lels on the output of t k, n the only ples tht will ever e mppe to k in the reing ρ re extly the ples in the input of t k. This lrey shows n importnt ifferene etween the simultions for NF 0 B n Petri utomt. For NF, we relte trnsition p p to trnsi- tion q q with the sme lel. t k Y Here the trnsitions ξ k ξ k+ n X 0 ξ t k k 2 ξ k+ my hve ifferent lels. Z onsier the step represente on the right, orresponing to squre in the ove igrm. The output of 0 hs lel tht oes not pper in 0, n 0 hs two outputs lelle y. Nevertheless this stisfies the onitions informlly stte ove, inee, hols. However this efinition is not yet stisftory. onsier the two runs elow: X 0 B Y 0 Z Their proue grphs orrespon respetively to the groun terms ( ) n ( ) ( ). The prolem is tht ( ) ( ) ( ), ut with the previous efinition, we nnot relte these runs: they o not hve the sme length. The solution here onsists in grouping the trnsitions n 2 together, n onsier these two steps s single step in prllel run. This lst moifition gives us notion of simultion tht suits our nees.. Simultions efinition 2 (Simultion) reltion P (P ) P (P 2 P ) etween the onfigurtions of n the prtil mps from the ples of 2 to the ples of is lle simultion etween n 2 if: if ξ n η then the rnge of η must e inlue in ξ; {ι } {[ι 2 ι ]}; if ξ n ξ (t,t) ξ, then ξ where is the set of ll η suh tht there is some η n omptile set of trnsitions T T 2 suh tht: om (η) T 2 om (η ); (t, t ) T, η(t ) t n (x, q) t,(x, η (q)) t; p om (η), ( (t, t ) T, p t ) η(p) = η (p). T Z 2 T U α B β γ δ ɛ ζ ɛ Figure 0: meing of prllel run into the run from Figure 5. if ξ n ξ F, then there must e some η suh tht om (η) F 2. We will now prove tht the lnguge of is ontine in the lnguge of 2 if n only if there exists suh simultion. We first introue the following notion of emeing. efinition 22 (meing) Let ξ = (ξ k ) 0 k n, (t k ) 0 i<n e run in, n Ξ = (Ξ k ) 0 k n, (T i ) 0 i<n prllel run in 2. n emeing of Ξ into ξ is sequene (η i ) 0 i n of mps suh tht for ny i < n, we hve: η i is mp from Ξ i to ξ i ; the imge of T i y η i is inlue in t i, mening tht for ny (t, t) T i, for ny p t n (x, q) t, η i (p) is ontine in the input of t i n (x, η i+ (q)) is in the output of t i ; the imge of the tokens in Ξ i tht o not pper in the input of T i re preserve (η i (p) = η i+ (p)) n their imge is not in the input of t i. η i ξ i t i ξ i+ Ξ i T i Ξ i+ η i+ Figure 0 illustrtes the emeing of some prllel run, prouing ((( ) ( )) ), into the run presente in Figure 5. Notie tht is it neessry to hve prllel run inste of simple one: to fin something tht mthes the seon trnsition in the upper run, we nee to fire two trnsitions in prllel in the lower run. There is lose reltionship etween simultions n emeings: Lemm 23. Let n 2 e two Petri utomt, the following re equivlent: ) there exists simultion etween n 2 ; 2) for ny epting run ξ in, there is n epting prllel run Ξ in 2 tht n e emee into ξ. Proof. If we hve simultion, let ξ = (ξ k ) 0 k n, (t k ) 0 k<n e n epting run in. By the efinition of simultion, we n fin sequene of sets of mps ( k ) 0 k n suh tht 0 = {[ι 2 ι ]} n k, ξ k k. Furthermore, we n extrt from this η θ η ι F κ 9

11 sequene of mps (η k ) 0 k n n sequene of prllel trnsitions (T k ) 0 k<n suh tht (η k ) is n emeing of (om (η k )) 0 k n, (T k ) 0 k<n (whih is epting) into ξ. This follows iretly from the efinitions of emeing n simultion. On the other hn, if we hve property 2., then we n efine reltion y sying tht ξ if there is n epting run ξ = (ξ k ) 0 k n, (t k) 0 k<n in suh tht there is n inex k 0 : ξ = ξ k 0 ; n the following hols: η if there is n epting prllel run Ξ = (Ξ k ) 0 k n, (T k ) 0 k<n n (η k ) 0 k n n emeing of Ξ into ξ suh tht η = η k 0. It is then immeite to hek tht is inee simultion. If η is n emeing of Ξ into ξ, we n esily hek tht (ρ i η i ) 0 i n is prllel reing of (ξ) long Ξ in 2. Thus, it is ler tht one we hve suh run Ξ with the sequene of mps η, we hve tht (ξ) is inee in the lnguge of 2. The more iffiult question is the ompleteness of this pproh: if (ξ) is reognise y 2, is it lwys the se tht we n fin run Ξ tht my e emee into ξ? The nswer is ffirmtive, thnks to Lemm 24 elow. If (ρ j ) 0 j n is reing of long ξ = (ξ k ) 0 k n, (t k, t k ) 0 k<n, we write tive(j) for the only position in ρ j (t j ) 3. We ll onsistent orering on = V,, ι, o if V, is liner orer n (p, x, q) entils p q. Lemm 24. Let L ( 2 ) n e ny onsistent orering on. Then there exists run ξ n reing (ρ j ) 0 j n of long ξ suh tht k, tive(k) tive(k + ). Proof. The proof of this result is hieve y tking ny run ξ epting, n then exhnging trnsitions in ξ oring to, while preserving the existene of reing. The etils of this proof eing it tehnil, we move them to ppenix. (Notie tht if ontins yles, this lemm nnot pply euse of the lk of onsistent orering.) Lemm 24 enles us to uil n emeing from ny reing of (ξ) in 2. Lemm 25. Let ξ epting run of. Then (ξ) is in L ( 2 ) if n only if there is n epting prllel run in 2 tht n e emee into ξ. Proof. The etile proof of this result n e foun in ppenix. For the if iretion, we uil prllel reing from the emeing, s expline ove. For the other iretion, we onsier reing of (ξ) in 2 long some run ξ. Notie tht the nturl orering on N is onsistent for (ξ); we my thus hnge the orer of the trnsitions in ξ (using Lemm 24) n group them equtely to otin prllel reing Ξ tht emes in ξ. So we know tht the existene of emeings is equivlent to the inlusion of lnguges, n we previously estlishe 3 Rell tht if (ρ j ) 0 j n is reing long ξ then for ny p, q t j, we hve ρ j (p) = ρ j (q). B Figure : Petri utomton for +. tht it is lso equivlent to the existene of simultion reltion. Hene, the following hrteristion hols: Theorem 26. Let n 2 e two simple Petri utomt. L ( ) L ( 2 ) if n only if there exists simultion reltion etween n 2. Proof. By Lemms 4, 23 n 25. s Petri utomt re finite, there re finitely mny reltions in P (P (P ) P (P 2 P )). The existene of simultion thus is eile, llowing us to prove the min result: Theorem 27. iven two expressions e, f Reg X, testing whether Rel e = f is eile. Proof. By Theorems 6, 7 n 26, n resoning y oule inlusion. In prtie, we n uil the simultion on-the-fly, strting from the pir ({ι }, {[ι 2 ι ]}) n progressing from there. We hve implemente this lgorithm in OML [6]. ven though its theoretil worst se time omplexity is huge 4, we get result lmost instntneously on simple one-line exmples.. The prolems with onverse or unit The previous lgorithm is not omplete in presene of onverse or unit. More preisely, Lemm 25 oes not hol for generl utomt. Inee, it is not possile to ompre two runs just y relting the tokens t eh step, n heking eh trnsition inepenently. onsier the utomton from Figure. This utomton hs in prtiulr n epting run reognising. Let us try to test if this is smller thn the following runs from nother utomton (we represent the trnsitions simply s rrows, euse they only hve single input n single output): x 0 x x 2 x 3 x 4 x 5 x 6 y 0 y y 2 y 3 y 4 y 5 y 6 y 7 It stns to reson tht we woul reh point where for the first run: {, } {[x 3 ]} n for the seon run: {, } {[y 3 ]}. 4 quik nlysis gives O (2 n+(n+)m ) omplexity oun, where n n m re the numers of ples of the utomt. 0

12 So if it were possile to relte the en of the runs just with this informtion, they shoul oth e igger thn or oth smller or inomprle. But in ft the first run (reognising ) is igger thn ut the seon (reognising ) is not. This highlights the nee for hving some memory of previously fire trnsition when trying to ompre runs of generl Petri utomt, thus preventing our lol pproh to er fruits. The sme kin of exmple oul e foun with the onverse opertion inste of. ι X ι e fn Figure 2: Squring of the utomton for e. X fn VI. OMPLXITY The previous notion of simultion tully llows us to eie lnguge inlusion of simple utomt in XPSP. We tully otin tht the prolem is XPSP-omplete. Lemm 28. ompring simple Petri utomt is XPSPesy. Proof. Our mesure for the size of n utomton here is its numer of ples (the numer of trnsitions is t most exponentil in this numer). Here is non-eterministi semilgorithm tht tries to refute the existene of simultion reltion etween n 2. : strt with ξ = {ι } n = {[ι 2 ι ]}; 2: if ξ F, hek if there is some η suh tht om (η) F 2, if not return FLS; 3: hoose non-eterministilly trnsition (t, t) T suh tht t ξ; 4: fire (t, t), whih mens tht ξ = ξ t {p P x X (x, p) t} ; 5: hve progress long (t, t) s well, oring to the onitions from efinition 2. 6: go to step 2. ll these omputtions n e one in exponentil spe. In prtiulr s ξ is set of ples in P, it n e store in spe P log( P ). Similrly,, eing set of prtil funtions from P 2 to P, eh of whih of size P 2 log( P + ), n e store in spe P + P2 P 2 log( P + ). This noneterministi XPSP semi-lgorithm n then e turne into n XPSP lgorithm y Svith theorem [9]. One n hek tht the numer of ples in (e) is liner in the size of e. (The exponentil upper-oun on the numer of trnsitions is symptotilly rehe, onsier for instne the utomton for (x y ) (x 2 y 2 ) (x n y n ).) Therefore, the previous Lemm gives us XPSP lgorithm for eiing the (in)equtionl theory of ientity-free reltionl Kleene ltties. Theorem 29. ompring simple Petri utomt is XPSP-omplete. Proof. By Lemm 28, it suffies to show hrness. We perform reution from the equlity of lnguges enote y regulr expressions with squring (e 2 = e e), n XPSPomplete prolem [4]. To voi onfusion, the regulr lnguge enote y the expression e will e written e. ny wor u n e see uniquely s liner grph λ(u). By extension, the set of grphs of wors from e will e enote y λ e. First, notie tht if u n v re just wors over X, λ(u) λ(v) is equivlent to u = v. Beuse of this, it is strightforwr to hek tht for ny e, f Reg 2 X the following hols (λ e ) = (λ f ) e = f. (8) iven n expression e on this signture, we n uil in liner time Petri utomton, with liner numer of ples n trnsitions. The losure of the lnguge enote y e is e extly the lnguge reognise y. This utomton is not simple: some outgoing rs re lelle with ; therefore, this reution only ensures tht the equivlene of ritrry Petri utomt is XPSP-hr. To get XPSP-hrness for simple Petri utomt, we nee to refine the onstrution so tht is interprete s stnr letter. (More etils re given in ppenix F.) The utomt we proue here only hve one finl onfigurtion, onsisting in singleton. The onstrution is strightforwr pttion of Thompson s lgorithm for NF [20]. The only interesting se is for omputing n utomton for e = (e ) 2. We represent it grphilly in Figure 2. The trnsitions lelle y X re shorthn for set of trnsitions, ontining for eh letter x in X trnsition with one output, lelle y x. This onstrution is liner: the utomton for e is not opie. Furthermore, run in this utomton will strt y sening one token in n one in ι, the initil stte of the utomton for e. Then it will perform run in this utomton until single token rehes the finl stte for e, fn. t this point the tokens from n fn will e sent to n ι, strting new run of e. When token is finlly sent to fn, it n e onsume together with the one in, to reh the finl onfigurtion. This proof oes not llow us to eue tht lso the (in)equtionl theory of Kleene llegories is XPSP-hr: the Petri utomt we onstrut re not ssoite to some expressions of polynomil size, priori. VII. RLTIONSHIP WITH STNR PTRI NT NOTIONS Our notion of Petri utomton is relly lose to the stnr notion of lelle (sfe) Petri net, where the trnsitions themselves re lelle, rther thn their outputs. We motivte this

13 esign hoie, n we relte some of the notions we introue to the stnr ones [6]. ny Petri utomton n e trnslte into sfe Petri net whose trnsitions re lelle y X {τ}, the itionl lel τ stning for silent tions. For eh utomton trnsition ({p,..., p n }, {(x, q ),..., (x m, q m )}) with m >, we introue m fresh ples r,..., r m n m + trnsitions: silent trnsition t 0 with preset {p,..., p n } n postset {r,..., r m }; n for eh k m trnsition t k lelle y x k, with preset {r k } n postset {q k }. The inutive onstrution from Setion IV is tully simpler to write using lelle Petri nets, s one n freely use τ- lelle trnsitions to ssemle utomt into lrger ones, one oes not nee to perform the τ-elimintion steps on the fly. On the other sie, we oul not efine n pproprite notion of simultion for Petri nets: we nee to fire severl trnsitions t one in the smll net, to provie enough informtion for the lrger net to nswer; elimiting whih trnsitions to group n whih to seprte is non-trivil; similrly, efining notion of prllel step is elite in presene of τ-trnsitions. By swithing to our notion of Petri utomt, we impose strong onstrints out how those τ-trnsitions shoul e use, resulting in more fitte moel. To esrie run in Petri net N, one my use proess p K N, where K is n ourrene net ( prtilly orere Petri net) [9]. The grphil representtion (Figures 5, 7 n 0) we use to esrie runs in n utomton re in ft mere pttion of this notion to our setting (with lels on rs rther thn on trnsitions). Our notion ξ of tre of run tully orrespons to the stnr notion of pomset-tre, vi ulistion (see ppenix H). Jtegonkr n Meyer showe tht the pomsettre equivlene prolem for sfe Petri nets is XPSPomplete [0]. However this equivlene is too strong n oes not oinie with the one isusse in the present pper, even for simple Petri utomt. The grph proue y run is its tre in this se, so tht pomset-tre equivlene for Petri nets orrespons to equivlene of the sets of grphs proue y Petri utomt ( ( ) = (B), up to grph isomorphism). However, for the equtionl theory we onsier, we nee to ompre the lnguges, whih re ownwr-lose sets of grphs, ( ( ) = (B), i.e., L ( ) = L (B)) rther thn the sets of grphs themselves. lso note tht the lss of sets of grphs proue y Petri utomt ({ ( ) Petri utomton}) is not lose uner ownwr losure. Intuitively, the with of ny grph in ( ) is oune y the numer numer of ples of, ut ( ) usully ontins grphs of ritrry with. s onsequene, one nnot esily reue our prolem to pomsettre equivlene of sfe Petri nets. VIII. IRTIONS FOR FUTUR WORK The utomt moel we introue to reognise ownwrlose grph lnguges llowe us to otin the eiility of ientity-free reltionl Kleene ltties, n (in)equtionl theory stuie reently y nrék et l. []. Thnks to this moel, we lso otine tht the (in)equtionl theory of Kleene llegories is o-reursively enumerle. We leve severl questions open. First, is the (in)equtionl theory of Kleene llegories eile? Two pprohes oul provie n ffirmtive nswer: fining n lgorithm for ompring ritrry sfe Petri utomt, or fining omplete n reursively enumerle xiomtistion. Seon, n we otin Kleene-like theorem for Petri utomt: is the lnguge of ny Petri utomton lso the lnguge of Kleene llegory term? This question n e restrite to simple Petri utomt; if one of the nswers is negtive, is there n lgeri wy of representing these utomt? Whih re the missing opertors? RFRNS [] H. nrék, S. Mikulás, n I. Németi. The equtionl theory of Kleene ltties. Theoretil omputer Siene, 42(52): , 20. [2] H. nrék n. Breikhin. The equtionl theory of union-free lgers of reltions. lger Universlis, 33(4):56 532, 995. [3] S. L. Bloom, Z. Ésik, n. Stefnesu. Notes on equtionl theories of reltions. lger Universlis, 33():98 26, 995. [4] M. Boff. Une onition impliqunt toutes les ientités rtionnelles. Informtique Théorique et pplitions, 29(6):55 58, 995. [5] P. Brunet n. Pous. Kleene lger with onverse. In Pro. RMiS, volume 8428 of Leture Notes in omputer Siene, pges 0 8. Springer Verlg, 204. [6] P. Brunet n. Pous. We ppenix to this strt, [7] J. H. onwy. Regulr lger n finite mhines. hpmn n Hll Mthemtis Series, 97. [8] P. J. Frey n. Serov. tegories, llegories. North Holln, 990. [9] U. oltz n W. Reisig. The non-sequentil ehviour of petri nets. Informtion n ontrol, 57(2):25 47, 983. [0] L. Jtegonkr n. R. Meyer. eiing true onurreny equivlenes on sfe, finite nets. Theoretil omputer Siene, 54():07 43, 996. Twentieth Interntionl olloquium on utomt, Lnguges n Progrmming. [] S.. Kleene. Representtion of vents in Nerve Nets n Finite utomt. Memornum. Rn orportion, 95. [2]. Kozen. ompleteness theorem for Kleene lgers n the lger of regulr events. In Pro. LIS, pges I omputer Soiety, 99. [3]. Kro. omplete System of B-Rtionl Ientities. In Pro. ILP, volume 443 of Leture Notes in omputer Siene, pges Springer Verlg, 990. [4]. Meyer n L. Stokmeyer. The equivlene prolem for regulr expressions with squring requires exponentil spe. In Swithing n utomt Theory, 972., I onferene Reor of 3th nnul Symposium on, pges I, 972. [5]. Meyer n L. J. Stokmeyer. Wor prolems requiring exponentil time. In Pro. M symposium on Theory of omputing, pges 9. M, 973. [6] T. Murt. Petri nets: Properties, nlysis n pplitions. Pro. of the I, 77(4):54 580, pr 989. [7].. Petri. Funmentls of theory of synhronous informtion flow. In IFIP ongress, pges , 962. [8].. Petri. Kommuniktion mit utomten. Ph thesis, rmstt Univ. of Teh., 962. [9] W. J. Svith. Reltionships etween noneterministi n eterministi tpe omplexities. Journl of omputer n system sienes, 4(2):77 92, 970. [20] K. Thompson. Regulr expression serh lgorithm.. of the M, :49 422,

14 . Omitte proof: Theorem 6 PPNIX We enote y W X the set of groun terms. We wnt to estlish tht for ny regulr expressions with intersetion n onverse e n f, the following re equivlent: (i) Rel e f, (ii) e f, (iii) (e) (f). ) (iii) (ii): Suppose (e) (f), let u e e groun term. Neessrily there is some grph (f) suh tht (u). By efinition of (f), there is some term v f suh tht = (v). This mens tht u v, thus proving tht u f. ) (ii) (i): Let e, f Reg X two expressions suh tht e f, n σ X P (S S) some reltionl interprettion. σ(e) = σ(w) (Lemm 5) w e w f σ(w) (e f) = σ(f) (Lemm 5) ) (i) (iii): In orer to prove this lst implition, we nee the following lemm, whih is ue to nrék n Breikhin. Lemm 30 ( [2, Lemm 3]). Let S e se set, i, j S, v W X, (v) = V v, v, ι v, o v n σ X P (S S). The following re equivlent: ) (i, j) σ(v); 2) ϕ V v S suh tht ϕ(ι v ) = i; ϕ(o v ) = j n (p,, q) v (ϕ(p), ϕ(q)) σ(). Let e, f Reg X two expressions suh tht Rel e f, n u e suh tht (u) = V u, u, ι u, o u ; we n uil n interprettion σ X P (V u V u ) y speifying: σ() = {(p, q) (p,, q) u }. It is quite simple to hek tht σ(u) = {(ι u, o u )}. By Lemm 5 n Rel e f, we know tht σ(u) σ(f) = σ(v). v f Thus there is some v f suh tht (ι u, o u ) σ(v). By Lemm 30 we get tht there is mp ϕ V v V u suh tht ϕ(ι v ) = ι u ; ϕ(o v ) = o u n (p,, q) v (ϕ(p), ϕ(q)) σ(). Using the efinition of σ, we rewrite this lst onition s (p,, q) v (ϕ(p),, ϕ(q)) u. Thus ϕ is grph homomorphism from (v) to (u), proving tht (u) (v), hene (u) (f). B. Omitte proof: Lemm 4 Let = V,, ι, o, ξ = {0,..., n}, ξ, 0, n, n (ξ) = {[i] ξ 0 i j }, ξ, [0] ξ, [n] ξ. Suppose there exists grph homomorphism ϕ from (ξ) to. We uil reing (ρ k ) k of long ξ y letting ρ k (p) = ϕ([ν(k, p)] ξ ) for 0 k n n p ξ k. We now hve to hek tht ρ is truly reing of in : for the initilistion n onlusion of the reing: ρ 0 (ι ) = ϕ([ν(0, ι )] ξ ) (y efinition) = ϕ([0] ξ ) = {ι} (ϕ is homomorphism) p ξ n, ρ n (p) = ϕ([ν(n, p)] ξ ) = ϕ([n] ξ ) ( k, p, k ν(k, p) n) = {o}. (ϕ is homomorphism) for ll p t k, ρ k (p) = ϕ([ν(k, p)] ξ ) = ϕ([k] ξ ) whih oes not epen on p. for ll p ξ k t k, we hve ν(k, p) = ν(k +, p) (sine p t k ). Hene ρ k (p) = ϕ([ν(k, p)] ξ ) = ϕ([ν(k +, p)] ξ ) = ρ k+ (p). for ll p t k n (x, q) t k, we know tht ρ k (p) = ϕ([k] ξ ) n tht (k, x, ν(k +, q)) ξ. If x X, we lso hve ([k] ξ, x, [ν(k +, q)] ξ ) ξ. Beuse ϕ is homomorphism we n eue tht: (ϕ([k] ξ ), x, ϕ([ν(k +, q)] ξ )), whih n e rewritten (ρ k (p), x, ρ k+ (q)). If x = y, y X, we lso hve ([ν(k +, q)] ξ, y, [k] ξ ) ξ. Beuse ϕ is homomorphism we n get like efore (ρ k+ (q), y, ρ k (p)). If finlly x =, then we know tht k ξ ν(k +, q), thus proving tht ρ k (p) = ϕ([k] ξ ) = ϕ([ν(k +, q)] ξ ) = ρ k+ (q). If on the other hn we hve reing (ρ k ) 0 k n of, we efine ϕ {0,..., n} V y ϕ([k] ξ ) = ρ k (p) for ny p t k. s (ρ k ) k is reing, ϕ is well efine 5. Let us hek tht ϕ is homomorphism from (ξ) to : ϕ([0] ξ ) = ρ 0 (ι ) = ι; ϕ([n] ξ ) = ρ n (p) with p ξ n, n sine (ρ k ) k is reing ρ n (p) = o. if ([k] ξ, x, [l] ξ ) ξ is n ege of (ξ), then it ws proue from some ege (i, y, j) ξ, with either x = y n i, j [k] ξ [l] ξ or y = x n i, j [l] ξ [k] ξ. There is some p t i n q suh tht (y, q) t j n j = ν(i +, q). By efinition of ν we know tht i + m < j, q t m. Thus, euse (ρ k ) k is reing, ρ i+ (q) = ρ j (q) n either (ρ i (p), x, ρ i+ (q)) or (ρ i+ (q), x, ρ i (p)), n thus (ϕ([k] ξ ), x, ϕ([l] ξ )). 5 It is not iffiult to hek tht k ξ l p, q t k t l, ρ k (p) = ρ l (q). 3