COMPUTING SCIENCE. University of Newcastle upon Tyne. A Note on the Well-Foundedness of Adequate Orders Used for Truncating Unfoldings

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1 UNIVERSITY OF NEWCASTLE University of Newcstle upon Tyne COMPUTING SCIENCE A Note on the Well-Foundedness of Adequte Orders Used for Truncting Unfoldings T Chtin, V Khomenko TECHNICAL REPORT SERIES No CS-TR-998 Jnury, 2007

2 TECHNICAL REPORT SERIES No CS-TR-998 Jnury, 2007 A Note on the Well-Foundedness of Adequte Orders Used for Truncting Unfoldings Thoms Chtin nd Victor Khomenko Astrct Petri net unfolding prefixes re n importnt technique for forml verifiction nd synthesis In this pper we show tht the requirement tht the dequte order used for truncting Petri net unfolding must e well-founded is superfluous in mny importnt cses, ie it logiclly follows from other requirements We give complete nlysis when this is the cse These results concern the very `core' of the unfolding theory 2007 University of Newcstle upon Tyne Printed nd pulished y the University of Newcstle upon Tyne, Computing Science, Clremont Tower, Clremont Rod, Newcstle upon Tyne, NE1 7RU, Englnd

3 Biliogrphicl detils CHATAIN, T, KHOMENKO, V A Note on the Well-Foundedness of Adequte Orders Used for Truncting Unfoldings [By] T Chtin, V Khomenko Newcstle upon Tyne: University of Newcstle upon Tyne: Computing Science, 2007 (University of Newcstle upon Tyne, Computing Science, Technicl Report Series, No CS-TR-998) Added entries UNIVERSITY OF NEWCASTLE UPON TYNE Computing Science Technicl Report Series CS-TR-998 Astrct Petri net unfolding prefixes re n importnt technique for forml verifiction nd synthesis In this pper we show tht the requirement tht the dequte order used for truncting Petri net unfolding must e well-founded is superfluous in mny importnt cses, ie it logiclly follows from other requirements We give complete nlysis when this is the cse These results concern the very `core' of the unfolding theory Aout the uthor Thoms Chtin received his PhD in computer science from University of Rennes 1 in 2006 He is currently doing post-doc in the Deprtment of Computer Science of the University of Alorg, Denmrk He is interested in the use of forml models for the supervision, verifiction nd control of distriuted systems In prticulr he studies true concurrency models (including timed models), prtil order semntics, unfoldings nd timed gmes Victor Khomenko Otined MSc with distinction in Computer Science, Applied Mthemtics nd Teching of Mthemtics nd Computer Science in 1998 from Kiev Trs Shevchenko University, nd PhD in Computing Science in 2003 from University of Newcstle upon Tyne From Septemer 2005 Victor is Royl Acdemy of Engineering/EPSRC Post-Doctorl Reserch Fellow, working on the DAVAC project His interests include model checking of Petri nets, Petri net unfolding techniques, self-timed (synchronous) circuits Suggested keywords ADEQUATE ORDER, WELL-FOUNDEDNESS, UNFOLDING PREFIX, PETRI NET

4 A Note on the Well-Foundedness of Adequte Orders Used for Truncting Unfoldings Thoms Chtin 1 nd Victor Khomenko 2 1 Deprtment of Computer Science, Alorg University, Alorg, Denmrk E-mil: chtin@csudk 2 School of Computing Science, Newcstle University, Newcstle upon Tyne, United Kingdom E-mil: VictorKhomenko@nclcuk Astrct Petri net unfolding prefixes re n importnt technique for forml verifiction nd synthesis In this pper we show tht the requirement tht the dequte order used for truncting Petri net unfolding must e well-founded is superfluous in mny importnt cses, ie, it logiclly follows from other requirements We give complete nlysis when this is the cse These results concern the very core of the unfolding theory 1 Introduction nd sic notions McMilln s finite nd complete prefixes of Petri net unfoldings [3, 8] re prominent technique for nlysing the ehviour of rective systems modelled y Petri nets It llevites the stte spce explosion prolem, ie, the prolem tht even reltively smll system specifiction cn (nd often does) hve so mny rechle sttes tht the strightforwrd enumertion of them is infesile This technique relies on the prtil order view of concurrent computtion A finite nd complete unfolding prefix of Petri net Ω is finite cyclic net which implicitly represents ll the rechle sttes of Ω together with trnsitions enled t those sttes Intuitively, it cn e otined through unfolding Ω, y successive firing of trnsitions, under the following ssumptions: (i) for ech new firing fresh trnsition (clled n event) is generted; (ii) for ech newly produced token fresh plce (clled condition) is generted Due to its structurl properties (such s cyclicity), the rechle sttes of Ω cn e represented using configurtions of its unfolding A configurtion C is finite downwrdclosed set of events (eing downwrd-closed mens tht if e C nd f is cusl predecessor of e, denoted f e, then f C) without choices (ie, for ll distinct events e,f C, there is no condition c in the unfolding such tht the rcs (c,e) nd (c,f) re in the unfolding) Intuitively, configurtion is prtilly ordered execution, ie, n execution where the order of firing of some of its events (viz concurrent ones) is not importnt We will denote y [e] the locl configurtion of n event e, ie, the smllest (wrt ) configurtion contining e (it is comprised of e nd its cusl predecessors) A finite set of events E is suffix of configurtion C if C E = nd C E is configurtion; in such cse the nottion C E will e used to denote the ltter configurtion, clled n extension of C The unfolding is infinite whenever the originl Petri net hs n infinite run; however, if the Petri net hs finitely mny rechle sttes then the unfolding eventully strts to repet itself nd cn e truncted (y identifying set of cut-off events) without loss of informtion, yielding finite nd complete prefix Intuitively, n event e cn e declred cut-off if the lredy uild prt of the prefix contins configurtion C e (clled the corresponding configurtion of e) such tht Mrk(C e ) = Mrk([e]) (where Mrk(C) denotes the finl mrking of configurtion C) nd C e is smller thn [e] wrt some well-founded prtil order on the configurtions of the unfolding, clled n dequte order [3, 6] The importnce of the ltter condition is illustrted y the exmple in Figure 1, which is tken from [3] The mrking {2 } is rechle

5 2 T Chtin, V Khomenko t 1 t 2 e 1 t 1 e 2 t 2 p 2 p 3 p 4 p 5 p 2 p 3 p 4 p 5 t 3 t 4 t 5 t 6 e 3 t 3 e 4 t 5 e 5 t 4 e 6 t 6 p 6 p 7 p 8 p 9 p 6 p 7 p 8 p 9 p 6 p 7 p 8 p 9 t 7 t 8 e 7 t 7 e 10 t 8 e 8 t 7 e 9 t () t 9 () 2 Fig 1 A sfe Petri net () nd prefix of its unfolding () in the Petri net in Figure 1() However, one cn generte the prefix shown in Figure 1(), in which this mrking is not represented (The numers of the events indicte the order in which they were dded to the prefix) The events e 8 nd e 10 re mrked s cut-off, ecuse the finl mrkings of the corresponding locl configurtions re {p 7,p 9,0 } nd {p 6,p 8,1 }, which re lso the finl mrkings of [e 7 ] nd [e 9 ], respectively Although no events cn now e dded, the prefix is not complete, ecuse {2 } is not represented in it Efficient lgorithms exist for uilding such prefixes [3, 6], which ensure tht the numer of non-cut-off events in complete prefix cn never exceed the numer of rechle sttes of the Petri net However, complete prefixes re often exponentilly smller thn the corresponding stte grphs, especilly for highly concurrent Petri nets, ecuse they represent concurrency directly rther thn y multidimensionl dimonds s it is done in stte grphs For exmple, if the originl Petri net consists of 100 trnsitions which cn fire once in prllel, the stte grph will e 100-dimensionl hypercue with vertices, wheres the complete prefix will coincide with the net itself In mny pplictions, eg, in synchronous circuit design, the Petri net models usully exhiit lot of concurrency, ut hve rther few choice points, nd so their unfolding prefixes re often exponentilly smller thn the corresponding stte grphs; in fct, in mny of the experiments conducted in [6] they re just slightly igger then the originl Petri nets themselves Therefore, unfolding prefixes re well-suited for lleviting the stte spce explosion prolem Well-foundedness of the dequte order used to truncte the unfolding is n importnt prt of the completeness proof of [3, 7] In this pper, we show tht the requirement of wellfoundedness is superfluous in mny importnt cses More precisely, we show tht in mny cses the well-foundedness of the dequte order is implied y other requirements the dequte order must stisfy First, we introduce severl importnt definitions relted to dequte orders For convenience, their form hs een slightly chnged compred with [3, 6], ut they re esily seen to e equivlent

6 A Note on the Well-Foundedness of Adequte Orders Used for Truncting Unfoldings 3 Definition 11 (Structurl isomorphism) Two finite sets of events of the unfolding of Petri net Ω, E nd E, re structurlly isomorphic, 3 denoted E s E, if the lelled digrphs induced y these two sets of events nd their djcent conditions re isomorphic Definition 12 (Preservtion y finite extensions) A strict prtil order on the finite configurtions of the unfolding of Petri net is strongly (resp wekly) preserved y finite extensions if for every pir of configurtions C, C such tht Mrk(C ) = Mrk(C ) nd C C, nd for every finite suffix E of C nd every (resp there exists ) finite suffix E of C such tht E s E, it holds tht C E C E Definition 13 ((Pre-)dequte orders) A strict prtil order on the finite configurtions of the unfolding of Petri net Ω is clled pre-dequte if: it refines, ie, C C implies C C ; it is wekly preserved y finite extensions A pre-dequte order is clled dequte if it is well-founded We now proceed y showing tht in mny cses the requirement of well-foundedness of the dequte order is superfluous, ie, tht pre-dequte orders re utomticlly dequte We consider, in turn, severl clsses of Petri nets 2 The cse of sfe Petri nets The proposition elow sttes tht the well-foundedness requirement is superfluous for sfe Petri nets Proposition 21 (The requirement of well-foundedness is superfluous for unfoldings of sfe Petri nets) A pre-dequte order on the finite configurtions of the unfolding of sfe Petri net is dequte Proof Since for sfe Petri nets, wek preservtion y finite extensions implies strong preservtion y finite extensions, this is specil cse of Proposition 33 elow 3 The cse of ounded Petri nets The cse of ounded Petri nets differs from the previous cse since the wek nd the strong preservtions y finite extensions no longer coincide, s illustrted y the following counterexmple Counterexmple 31 (The requirement of well-foundedness is not superfluous for unfoldings of ounded Petri nets in the cse of wek preservtion y finite extensions) The pre-dequte order shown in Figure 2(c,d) is not well-founded order on the configurtions of the unfolding shown in Figure 2() Indeed, ny finite execution strts y series of firings of, nd then, optionlly, fires When fires, p 2 contins two tokens, nd cn consume either of them; in the unfolding, the corresponding conditions nd the instnces of cn e esily distinguished We denote for ll n 0 the finite configurtions s n, n nd n Note tht only the configurtions of the form n cn e extended, either to n+k+1 or n+k or n+k, k 0 Suppose m n (ie, m < n) If n is extended to n+k+1 then we cn extend (in structurlly isomorphic wy) m to m+k+1 If n is extended to n+k, k 0, then we cn extend (in structurlly isomorphic wy) m to m+k, nd, y the definition of, m+k n+k If n is extended to n+k, k 0, then we cn extend (in structurlly isomorphic wy) m to m+k, nd, y the definition of, m+k n+k Hence, is wekly preserved y finite extensions However, is not well-founded due to [3] used such n isomorphism without formlly defining it It turns out tht there re severl lterntive nturl isomorphisms which cn e used; we discuss some of them in Section 5

7 4 T Chtin, V Khomenko p 2 p 2 () p 2 () 3 n n n m n when m < n m n when m < n m n when m < n (c) n n n n (d) Fig 2 A 2-ounded Petri net (), its unfolding () nd n order on its configurtions (c,d) Remrk 32 In severl seminl ppers on unfoldings, like [2], the initil mrking is ssumed to e sfe, ie, it should contin t most one token on ech plce The net of Figure 2() does not stisfy this requirement Nevertheless, it is esy to dpt this counterexmple s follows: t p 2 This net strts y firing t, which leds to the sme mrking s in Figure 2() Denote C 0 the initil configurtion The configurtion tht is reched fter firing t corresponds to 0 in Figure 2 We still denote it 0, nd re-use the nottions n, n nd n s efore The order is lso re-used, nd extended with C 0 0 Proposition 33 (The requirement of well-foundedness is superfluous for unfoldings of ounded Petri nets in the cse of strong preservtion y finite extensions) If pre-dequte order on the finite configurtions of the unfolding of ounded Petri net is strongly preserved y finite extensions then is dequte Proof Follows from Proposition A1 with Σ = RM T nd σ(c, e) = (Mrk(C), h(e))

8 A Note on the Well-Foundedness of Adequte Orders Used for Truncting Unfoldings 5 p 2 p 2 p 2 () () 3 n n n m when n < m n m when n > m (c) n n n (d) Fig 3 An unounded Petri net (), its unfolding () nd n order on its configurtions (c,d) 4 The cse of unounded Petri nets We complete our nlysis y considering the cse of generl (unounded) Petri nets This cse might e less interesting in prctice, since the complete prefixes of unounded nets re infinite However, this cse is interesting from the theoreticl point of view Moreover, [6] shows tht finite nd complete prefix of n unounded nets cn e otined if insted of the equivlence of finl mrkings corser equivlence is used to compre the configurtions in the cut-off criterion The definition of the preservtion y finite extensions (Definition 12) requires tht is only preserved y extensions of configurtions reching the sme mrkings The counterexmple elow shows tht in this cse the requirement of well-foundedness is not superfluous Counterexmple 41 (The requirement of well-foundedness is not superfluous for unfoldings of unounded Petri nets) Consider Figure 3 The finite configurtions of the unfolding hve the form either n or n, where n rnges over the set of integers The shown order is pre-dequte, s it refines the set inclusion nd it is trivilly preserved y finite extensions of configurtions reching the sme mrking, since no two configurtions rech the sme mrking However, is not well-founded due to Techniclly, this counterexmple settles the cse of unounded Petri nets However, one cn oserve tht this negtive result holds due to the trivil reson tht it is possile to construct n unounded Petri net such tht in its unfolding no two configurtions hve the sme finl mrking Hence, it seems resonle to strengthen the ssumptions out the pre-dequte order in the unounded cse, y requiring tht is preserved not only y configurtions tht rech the sme mrking, ut lso ech time isomorphic finite suffixes cn e dded to two comprle configurtions Definition 42 (Extendile pre-dequte order) A pre-dequte order on the finite configurtions of the unfolding of Petri net Ω is clled extendile if for ll configurtions C nd C such tht C C, nd for ll finite suffixes E nd E of C nd C, resp, such tht E s E, it holds tht C E C E

9 6 T Chtin, V Khomenko wek preservtion strong preservtion sfe nets (Proposition 21) ounded nets (Counterexmple 31) (Proposition 33) unounded nets (Counterexmple 41) unounded nets (extendile order) (Counterexmple 31) (Proposition 43) Tle 1 Summry of results Note tht extendile pre-dequte orders re strongly preserved y finite extensions The proposition elow shows tht positive result cn e otined in the cse of n extendile pre-dequte order Proposition 43 (The requirement of well-foundedness is superfluous for unfoldings of unounded Petri nets in the cse of n extendile order) An extendile pre-dequte order on the finite configurtions of the unfolding of (possily unounded) Petri net is dequte Proof Follows from Proposition A1 with Σ = T nd σ(c,e) = h(e) 5 Summry nd further considertions Our results re summrised in Tle 1, where mens tht the requirement of well-foundedness is superfluous, nd mens tht it is not superfluous Moreover, we now show tht these results re roust, ie, they re not ffected if n lterntive notion of preservtion of y extensions is used, or if s is replced y different isomorphism 51 Single-Event Extensions Definition 51 is wekly (resp strongly) preserved y single-event extensions if it is wekly (resp strongly) preserved y finite extensions with singleton suffixes One cn esily show y induction on the size of the configurtion suffixes tht strong preservtion y single-event extension coincides with strong preservtion y finite extensions, nd so Propositions 33 nd 43 still hold for single-event extensions On the other hnd, wek preservtion y single event extensions is even weker thn wek preservtion y finite extensions, nd so Counterexmple 31 lso holds for wek preservtion y single-event extensions Moreover, one cn esily show tht for sfe Petri nets, wek preservtion y single-event extensions is equivlent to strong preservtion y single-event extensions (which is in turn equivlent to wek or strong preservtion y finite extensions), nd so Proposition 21 holds for single-event extensions s well To summrise, using single-event extensions insted of finite ones does not chnge our results 52 Other Isomorphisms So fr, we considered the structurl isomorphism, s, which is in sense strongest possile, s it tkes the full structure of the net into ccount Below we consider other nturl isomorphisms, which re corser then s Definition 52 (Pomset-isomorphism nd Prikh-isomorphism) Let E nd E e two finite sets of events of the unfolding of Petri net Ω

10 A Note on the Well-Foundedness of Adequte Orders Used for Truncting Unfoldings 7 E nd E re pomset-isomorphic, denoted E p E, if the lelled digrphs induced y these two sets of events in the digrph corresponding to the cuslity reltion on the events on the unfolding re isomorphic E nd E re Prikh-isomorphic, denoted E # E, if for every trnsition t of Ω, # t E = # t E, where # t E denotes the numer of instnces of t in E Note tht s refines p, which in turn refines #, ie, E s E E p E E # E Moreover, one cn oserve tht if 1 nd 2 re two isomorphisms such tht 2 refines 1 then: wek preservtion wrt 1 is even weker thn wek preservtion wrt 2 (ie, there exists n E such tht E 1 E nd C E C E, ut mye E 2 E ); strong preservtion wrt 1 is even stronger thn strong preservtion wrt 2 (for ll E such tht E 1 E, C E C E, even for those E such tht E 2 E ) Consequently, Counterexmple 31 (for ounded or unounded nets), s well s Propositions 33 nd 43, still hold for p nd # Moreover, since in the cse of sfe Petri nets it is enough to consider only single-event extensions, nd s, p nd # coincide on such extensions, Proposition 21 holds for either of these isomorphisms Finlly, one cn oserve tht Counterexmple 41 still holds for p nd # To summrise, using p or # (or ny other isomorphism refining # nd refined y s ) insted of s does not chnge our results 6 Conclusions In this pper we hve demonstrted tht the requirement tht the dequte order must e well-founded is superfluous in mny importnt cses, ie, it logiclly follows from other requirements We hve produced complete nlysis when this is the cse, y providing either proof or counterexmple in ech sitution It is noteworthy tht even though the unfolding technique hs een round for more thn decde, these results concerning the very core of the unfolding theory hve een otined only now Acknowledgements The uthors would like to thnk Jvier Esprz nd Wlter Vogler for helpful comments This reserch ws supported y the Royl Acdemy of Engineering/Epsrc post-doctorl reserch fellowship EP/C53400X/1 (Dvc) References 1 E M Clrke, O Grumerg nd D Peled: Model Checking MIT Press (1999) 2 J Engelfriet: Brnching Processes of Petri Nets Act Informtic 28 (1991) J Esprz, S Römer nd W Vogler: An Improvement of McMilln s Unfolding Algorithm Forml Methods in System Design 20(3) (2002) K Heljnko, V Khomenko nd M Koutny: Prlleliztion of the Petri Net Unfolding Algorithm Proc of TACAS 2002, Springer-Verlg, Lecture Notes in Computer Science 2280 (2002) G Higmn: Ordering y Divisiility in Astrct Algers Proc London Mth Soc 2 (1952) V Khomenko: Model Checking Bsed on Prefixes of Petri Net Unfoldings PhD Thesis, School of Computing Science, Newcstle University (2003) 7 V Khomenko, M Koutny nd V Vogler: Cnonicl Prefixes of Petri Net Unfoldings Act Informtic 40(2) (2003) K L McMilln: Using Unfoldings to Avoid Stte Explosion Prolem in the Verifiction of Asynchronous Circuits Proc of CAV 1992, Springer-Verlg, Lecture Notes in Computer Science 663 (1992)

11 8 T Chtin, V Khomenko 9 T Murt: Petri Nets: Properties, Anlysis nd Applictions Proceedings of the IEEE 77(4) (1989) A Vlmri: The Stte Explosion Prolem In: Lectures on Petri Nets I: Bsic Models, W Reisig nd G Rozenerg (Eds) Springer-Verlg, Lecture Notes in Computer Science 1491 (1998) Appendix A The proof of the min result Proposition A1 Let e strict prtil order on configurtions of the unfolding of Petri net Ω, Σ e finite lphet nd σ(c,e) e mpping tht ssigns letter from Σ to ech pir (C,e), where C is configurtion nd e is n event tht extends C, stisfying { } C1 C C 1,e 1,C 2,e 2 : 2 C σ(c 1,e 1 ) = σ(c 2,e 2 ) 1 {e 1 } C 2 {e 2 } Then is well-founded Proof For the ske of contrdiction, ssume tht is not well-founded, ie, there is n infinite descending sequence C 1 C n We ssume tht the configurtions hve strictly incresing sizes (n infinite susequence of C 1,,C n, stisfying this property cn lwys e extrcted) The union of the configurtions C 1,,C n, induces n infinite rnching process, nd the nlog of König s lemm for rnching processes [7] sttes tht it hs n infinite cusl chin of events p Ech configurtion C i hs finite intersection with p, since configurtions re finite y definition On the other hnd, the union of ll these configurtions hs n infinite intersection with p Hence, infinitely mny configurtions hve non-empty intersection with p, nd these intersections cn e ritrrily lrge (since if some event e of p elongs to C i then ll the preceding events of p lso elong to C i ) Therefore, without loss of generlity, we cn ssume tht p C 1 < < p C n < (n infinite susequence of C 1,,C n, stisfying this property cn lwys e extrcted) Let D n C n e the configurtion defined s the cusl pst of the events of p tht re in C n, nd E n = C n \ D n, ie, C n = D n E n We ssume tht the sizes of the E n s re non-decresing (n infinite susequence of C 1,,C n, stisfying this property cn lwys e extrcted) For ech E n, let e n,1,,e n,sn e n ritrry lineristion of the events of E n consistent with the cusl order, s n = E n, E n,k = {e n,1,,e n,k } nd C n,k = D n E n,k We define the word W n = n,1 n,sn with k = σ(c n,k 1,e n,k ) Now we cn pply Higmn s lemm [5] to W 1,,W n,, which re finite words over the finite lphet Σ This returns two integers i < j such tht W i is suword of W j Let 0 = l 0 < l 1 < < l si s j such tht for ll k {1,s i }, i,k = j,lk We hve W j = j,l1 j,l2 j,lsi = i,1 i,2 i,si Strting from C i,0 = D i D j = C j,l0, we show y induction on k tht C i,k C j,lk for ll k s i, which gives C i,si C j,lsi C j,sj, ie, C i C j, which leds to contrdiction We get the inductive step s follows: if C i,k 1 C j,lk 1, then C i,k 1 C j,lk 1 ecuse C j,lk 1 C j,lk 1; moreover σ(c i,k 1,e i,k ) = i,k = j,lk = σ(c j,lk 1,e lk ), so C i,k 1 {e i,k } C j,lk 1 {e j,lk }, ie, C i,k C j,lk