On Persistent Reachability in Petri Nets *

Transcription

1 On Persistent Rehility in Petri Nets * Kmil Brylsk 1, Łuksz Mikulski 1, Edwrd Ohmnski 1,2 1 Fulty of Mthemtis nd Computer Siene, Niolus Copernius University, Torun 2 Institute of Computer Siene, Polish Ademy of Sienes, Wrszw, Polnd {khm,frodo,edoh}@mtunitorunpl Astrt The notion of persisteny, sed on the rule no tion n disle nother one is one of the lssil notions in onurreny theory In this pper, we del with ritrry ple/trnsition nets, ut onentrte on their persistent omputtions It leds to n interesting deision prolem: Is given mrking rehle with persistent run? In order to study the persistent-rehility prolem we define lss of nets, lled nonviolene nets We show tht inhiitor nets n e simulted y the nonviolene nets, nd tht rehility nd overility prolems re undeidle in the lss of the nonviolene nets Then we prove more: nonviolene nets n e simulted y the inhiitor nets, thus they re omputtionlly equivlent to Turing mhines 1 Introdution An tion of onurrent system is sid to e persistent if, whenever it eomes enled, it remins enled until exeuted This lssil notion, introdued y Krp/Miller [9], is one of the most frequently disussed issues in the Petri net theory (ppers [1,2,3,6,8,11,12] mo) A net is sid to e persistent if eh of its tions is persistent And most of the ppers out persisteny del with this sulss of ple/trnsition nets) In this pper, we del with ritrry ple/trnsition nets, ut onentrte on their persistent omputtions It leds to n interesting persistentrehility prolems: Is given mrking rehle (overle) with persistent run? It is well known tht the lssil versions of the prolems (Is given mrking rehle (overle) in given ple/trnsition net?) re deidle (overility: Krp/Miller [9], Hk [8]; rehility: Myr [13], Kosrju [10]) In order to study the persistent-rehility prolem we introdue lss of nets, lled nonviolene nets (Definition 31) They differ from ple/trnsition nets only y the exeution rule Nmely, only persistent exeutions re permitted We show tht inhiitor nets n e simulted y nonviolene nets (Proposition 44) Using this ft we prove tht the rehility nd overility prolems re undeidle in the lss of the nonviolene nets (Propositions 45 nd 47, respetively) Then we prove more: nonviolene nets n e simulted y the inhiitor nets (Proposition 48), thus the oth re omputtionlly equivlent to Turing mhines * The reserh supported y Ministry of Siene nd Higher Edution of Polnd grnt N N Reent Advnes in Petri Nets nd Conurreny, S Dontelli, J Kleijn, RJ Mhdo, JM Fernndes (eds), CEUR Workshop Proeedings, ISSN , Jn/2012, pp

2 374 Petri Nets & Conurreny Brylsk, Mikulski, nd Ohmnski Mny extensions of Petri nets re known to e Turing powerful: inhiitor nets, priority nets (Hk [7]), self-modifying nets (Vlk [17]), for instne There is lso Turing powerful model restriting the stndrd exeution rules to mximl onurrent steps (Burkhrd [4], see lso Strke [16]) But ll the models llow fight for shring resoures (tokens), wheres our model works in ompletely peeful wy In the onluding setion we notie tht the free-hoie nonviolene nets re esy trnsformle to ple/trnsition nets (not neessrily free-hoie ones) Hene, the overility nd rehility prolems re deidle in the lss of the free-hoie nonviolene nets 2 Petri Nets Bsi Definitions The set of non-negtive integers is denoted y Given set X, the rdinlity (numer of elements) of X is denoted y X, the powerset (set of ll susets) y 2 X, the rdinlity of the powerset is 2 X Multisets over X re memers of X, ie funtions from X into For onveniene, if the set X is finite, multisets of X will e represented y vetors of X 21 Petri Nets nd Their Computtions The definitions onerning Petri nets re mostly sed on Desel/Reisig [5] Net is triple N = (P,T,F), where: P nd T re finite disjoint sets, of ples nd trnsitions, respetively; F P T T P is reltion, lled the flow reltion For ll T we denote: = {p P (p,) F} the set of entries to = {p P (,p) F} the set of exits from Petri nets dmit nturl grphil representtion Nodes represent ples nd trnsitions, rs represent the flow reltion Ples re depited y irles, nd trnsitions y oxes The set of ll finite strings of trnsitions is denoted y T*, the empty string is denoted y ε, the length of w T* is denoted y w, numer of ourrenes of trnsition in string w is denoted y w Ple/trnsition net (shortly, p/t-net) is qudruple S = (P,T,F,M 0 ), where: N = (P,T,F) is net, s defined ove; M 0 P is multiset of ples, nmed the initil mrking; it is mrked y tokens inside the irles, pity of ples is unlimited Multisets of ples re nmed mrkings In the ontext of ple/trnsition nets, they re mostly represented y nonnegtive integer vetors of dimension P, ssuming tht P is stritly ordered The nturl generliztions, for vetors, of rithmeti opertions + nd, s well s the prtil order, ll defined omponentwise, re well known nd their forml definitions re omitted

3 Persistent rehility Petri Nets & Conurreny 375 A trnsition T is enled in mrking M whenever M (ll its entries re mrked) If is enled in M, then it n e exeuted, ut the exeution is not fored The exeution of trnsition hnges the urrent mrking M to the new mrking M'=(M )+ (tokens re removed from entries, then put to exits) We shll denote: M for is enled in M nd MM' for is enled in M nd M' is the resulting mrking Then we sy tht MM' is step This denottion we extend to strings of trnsitions: the empty string ε is enled in ny mrking (lwys MεM), string w=u ( T, u T*) is enled in mrking M whenever MM' nd u is enled in M' Predites Mw nd MwM' re defined like those for single trnsitions If MwM' then we sy tht MwM' is omputtion from M to M' Note tht ny omputtion MwM' unmiguously defines ll intermedite mrkings etween M nd M' If MwM', for some w T*, then M' is sid to e rehle from M The set of ll mrkings rehle from M is denoted y [M Given ple/trnsition net S=(P,T,F,M 0 ), the set [M 0 of ll mrkings rehle from the initil mrking M 0 is lled the rehility set of S, nd mrkings in [M 0 re sid to e rehle in S We ssume tht the notions of rehility nd overility grphs re known to the reder Their definitions n e found in ny monogrph or survey out Petri nets (see [5,16] or ritrry else) Let us rell only tht rehility grphs represent ompletely ehviours of nets, ut re mostly infinite, while overility grphs represent ehviours only prtilly, ut re lwys finite In Exmples 22 nd 23 we lso use notion of persisteny grph the rehility grph restrited to persistent steps 22 Persistent Computtions of Ple/Trnsition Nets The notion of persisteny, proposed y Krp/Miller [9], elongs to the most importnt notions in onurreny theory It is sed on the ehviourlly oriented rule no tion n disle nother one, nd generlizes the struturlly defined notion of onflitfreeness Let S=(P,T,F,M 0 ) e ple/trnsition net, nd let M e mrking The step MM' is persistent iff ( ) if M then M' The empty omputtion MεM is persistent; the omputtion MM'uM" is persistent iff the step MM' is persistent nd the omputtion M'uM" is persistent [In words: A omputtion is sid to e persistent if ny trnsition one enled during this omputtion remins enled until exeuted] A p/t net is sid to e persistent if it dmits only persistent omputtions

4 376 Petri Nets & Conurreny Brylsk, Mikulski, nd Ohmnski Exmple 21 Non-persistent nd persistent nets Fig 1 A non-persistent (left) nd persistent (right) ple/trnsition nets 23 Persistent Rehility Prolem The prolem of persisteny ( Is ple/trnsition net persistent? ), rised y Lndweer nd Roertson in [11], hs een proved to e deidle y Growski [6] nd Myr [12] Most of p/t-nets, however, re not persistent, ut some of their omputtions re persistent In this pper, we re interested in mrkings tht re rehle with persistent omputtions Let S=(P,T,F,M 0 ) e ple/trnsition net, nd let M P e mrking Rehility Prolem: Is there omputtion M 0 wm? In other words: Is the mrking M rehle in the net S? The Rehility Prolem hs een proved to e deidle y Myr [13] nd Kosrju [10], fter yers of mny uthor s efforts A rod disussion, with detiled proof, n e found in the ook [15] of Reutenuer Let S=(P,T,F,M 0 ) e ple/trnsition net, nd let M P e mrking Persistent-Rehility Prolem: Is there persistent omputtion M 0 wm? In other words: Is the mrking M rehle in the net S with persistent run? Oviously, if p/t-net is persistent, then the persistent-rehility prolem is equivlent to the lssil one, thus deidle We shll study the prolem in generl, for ritrry p/t-nets The following exmples show differene etween omplete ehviours nd persistent ehviours

5 Persistent rehility Petri Nets & Conurreny 377 Exmple 22 Comprison of the omplete nd persistent ehviours d d d d 0-2 Fig 2 A ple/trnsition net nd its rehility nd persisteny grphs The ove net is ounded (ie its rehility set is finite) nd hs infinite set of persistent omputtions The exmple elow shows n unounded net (ie with infinite rehility set) with finite set ( singleton) of persistent omputtions Exmple 23 Unounded p/t-net with finite persisteny grph ω 1-0-ω ω Fig 3 A ple/trnsition net nd its rehility, overility nd persisteny grphs 3 Nonviolene Petri Nets In this setion, we introdue the notion of nonviolene Petri nets They differ from ple/trnsition nets only y the exeution rule Nmely, n enled trnsition n e exeuted only if it is exeutle persistently (ie if its exeution does not disle ny other enled trnsition) Therefore, we hve to distinguish the notion enled nd exeutle, tht re synonymi in ple/trnsition nets, ut not in nonviolene nets

6 378 Petri Nets & Conurreny Brylsk, Mikulski, nd Ohmnski Definition 31 Nonviolene Petri Nets Nonviolene net is qudruple S = (P,T,F,M 0 ), extly the sme s in definition of ple/trnsition nets It differs from p/t-net y exeution rules: A trnsition T is enled in mrking M whenever M (ll its entries re mrked) A trnsition T is exeutle in M if it is enled in M, nd moreover the step MM' is persistent The exeution of leds to the resulting mrking M'=(M )+ (extly sme s in p/tnets) We shll denote: M for is exeutle in M nd MM' for is exeutle in M nd M' is the resulting mrking Then we sy tht MM' is nonviolent step This denottion is nturlly extended to strings w T* If MwM' then we sy tht MwM' is nonviolent omputtion Only nonviolent steps nd omputtions re permitted in the nonviolene nets And now we n formulte the rehility nd overility prolems for the nonviolene nets Let S = (P,T,F,M 0 ) e nonviolene net, nd let M P e mrking NV-Rehility Prolem: Is there nonviolene omputtion M 0 wm in S? NV-Coverility Prolem: Is there mrking M' M nd nonviolene omputtion M 0 wm' in S? 32 From Ple/Trnsition Nets to Nonviolene Nets We shll show tht every p/t-net n e simulted y nonviolene net It will e done y joining n externl ontrol to eh trnsition of the net Let us onsider n ritrry p/t-net S We trnsform it to the nonviolene net S' in the following wy To eh trnsition in the net S we join swithing trnsition ' nd two new ples p nd q We dd the ple q to the set of entries to nd to the set of exits from ' We lso dd the ple p to the set of exits from nd to the set of entries to ' In initil mrking we dd one token to the ple p One n tret the onstruted loop s preprtion of the trnsition to exeution Net S: Net S': p ' Fig 4 Trnsforming ple/trnsition net into nonviolene net Let us lso define, for every mrking M in the net S, the mrking 10M in the net S' s follows For eh ple p in the net S we set 10M(p)=M(p), for eh new ple p we q

7 Persistent rehility Petri Nets & Conurreny 379 set 10M(p )=1 nd for eh new ple q we set 10M(q )=0 With suh definition, the initil mrking in the net S' is 10M 0, where M 0 is the initil mrking in S An ovious oservtion is tht if trnsition is exeutle in mrking M in the net S, then the sequene ' is exeutle in the mrking 10M in the net S' Proposition 32 A mrking M is rehle in ple/trnsition net S if nd only if the mrking 10M is rehle in the nonviolene net S' Proof ( ) Let M 0 wm e omputtion in S Then repling eh in w y ' we get omputtion 10M 0 w'10m in the nonviolene net S' ( ) Let 10M 0 w'10m in the nonviolene net S' The only differene etween ehviours of S nd S' is tht efore every trnsition trnsition ' must e exeuted Susequent exeution of two (or more) primed tions my sometimes disle the nonviolene exeution of tions tht were exeutle in S However, it would not mke ny new tion exeutle Therefore, ersing ll primed tions in w', we get omputtion w suh tht M 0 wm is omputtion in the p/t-net S 4 Comprison of Nonviolene Nets nd Inhiitor Nets In this setion, we rell the notion of inhiitor nets nd some of their properties (undeidility of the the rehility nd overility prolems) Then we show tht their omputtionl power is equl to tht of the nonviolene nets Definition 41 Inhiitor Petri Nets Inhiitor net is quintuple S = (P,T,F,I,M 0 ), where (P,T,F,M 0 ) is ple/trnsition net nd I P T is the set of inhiitor rs (depited y edges ended with smll empty irle) Sets of entries nd exits re denoted y nd, s in p/t-nets; the set of inhiitor entries to is denoted y ={p P (p,) I} A trnsition T is enled in mrking M whenever M (ll its entries re mrked) nd ( p ) M(p)= 0 ll inhiitor entries to re empty And exeutle mens enled, like in p/t-nets The exeution of leds to the resulting mrking M'= (M )+ It is known tht the inhiitor nets re omputtionlly equivlent to Turing mhines nd the rehility prolem in them is undeidle (Minsky [14], Hk [7]) Ft 42 Rehility Prolem is undeidle in the lss of inhiitor nets Also the overility prolem is known to e undeidle in the lss of inhiitor nets We rell here the proving onstrution Let S = (P,T,F,I,M 0 ) e n ritrry inhiitor net with P = {p 1,, p k }, nd let M = [i 1,, i k ] e mrking to e heked to e rehle We extend it to n inhiitor net S', s follows: We dd three new ples p 0, p k+1, p k+2 nd two new trnsitions x, y, onneted p 0 x p k+1 y p k+2 (see figure 5) Moreover, we join the

8 380 Petri Nets & Conurreny Brylsk, Mikulski, nd Ohmnski ple p 0 with every trnsition of the net S y self-loop (it is depited symolilly on Figure 5), we dd the rs from p n to x, weighted y i n (for n=1,,k), [We use here the weighted rs; see remrk elow] nd the inhiitory rs from ll originl ples of S to y And the initil mrking M' 0 in S' is the following: M' 0 (p 0 )=1, M' 0 (p n )=M 0 (p n ) for n=1,,k nd M' 0 (p k+1 )=M' 0 (p k+2 )=0 Net S: Net S': p 0 p 1 p k p 1 p k i 1 i k x y p k+1 p k+2 Fig 5 Cheking rehility with overility in inhiitor nets Remrk In this onstrution, we hve used weighted (multiple) rs They re not mentioned, for simpliity, in our definitions; we ssume tht the notion is ommonly known Moreover, (ple/trnsition or inhiitor) nets with multiple rs n e trnsformed to the equivlent nets without them (Hk [8], Strke [16]) Clerly, the mrking M = [i 1,, i k ] is rehle in S if nd only if the mrking M'(p 0 )=M'(p 1 )= =M'(p k )=M'(p k+1 )=0 nd M'(p k+2 )=1 is overle in S' Hene, euse of Ft 42, we hve got Ft 43 Coverility Prolem is undeidle in the lss of inhiitor nets 41 From Inhiitor Nets to Nonviolene Nets Let us onsider n ritrry inhiitor net S In the trnsformtion to the nonviolene net S' we will use the sme ide s in the previous trnsformtion Like in tht one, to eh trnsition, we dd trnsition ' nd ples p nd q Moreover, to eh ple p, in the net S, elonging to (ie eing n inhiitor entry to ), we dd new trnsition p Both ples, p nd q, re joined y self-loops with the new trnsition p Finlly, we remove ll inhiitor rs

9 Persistent rehility Petri Nets & Conurreny 381 Net S: Net S': p ' q p p p Fig 6 Trnsforming n inhiitor net into nonviolene net Similrly to the onstrution of figure 4, for every mrking M in the net S we define the mrking 10M in the net S', in the sme wy And the initil mrking in the net S' is 10M 0, where M 0 is the initil mrking in the net S In the sme mnner s in the previous se, if the trnsition is exeutle in the mrking M in the inhiitor net S, then the omputtion ' is exeutle in the mrking 10M in the nonviolene net S' Proposition 44 The mrking M is rehle in the inhiitor net S if nd only if the mrking 10M is rehle in the nonviolene net S' Proof The proof is similr to tht of Proposition 32 Remrk tht if n tion ' is exeuted while token resides in the ple p (so is not enled in S), then token will stuk in the ple q nd no mrking of the form 10M will e rehle Corollry 45 The NV-Rehility Prolem is undeidle Proof Diretly from Proposition 44 nd Ft 42 Proposition 46 The mrking M is overle in the inhiitor net S if nd only if the mrking 10M is overle in the nonviolene net S' Proof ( ) If M is overle in S then there is mrking M' M rehle in S Hene, y Proposition 44, the mrking 10M' is rehle in the nonviolene net S' And lerly, 10M' overs 10M ( ) Notie tht ny mrking (rehle in S') overing 10M is of the form 10M' And then M' M nd M' is rehle in S (Proposition 44) So M is overle in S Corollry 47 The NV-Coverility Prolem is undeidle Proof Diretly from Proposition 46 nd Ft 43

10 382 Petri Nets & Conurreny Brylsk, Mikulski, nd Ohmnski 42 From Nonviolene Nets to Inhiitor Nets The inverse onstrution, trnsforming nonviolene net into n inhiitor net, is more involved One more, we use the ide of prediting n exeutlement of trnsitions s Net S: Net S': ' q q x p 2 t q p p 2 t p y p Fig 7 Trnsforming nonviolene net into n inhiitor net Let S e nonviolene net; we extend it to n inhiitor net S', s follows First, we dd one glol ple s, lled the swith, whih is n exit from every trnsition of the net S Then, for every trnsition of the net S, we dd trnsition '; the swith ple s is n entry to eh of these primed trnsitions An exeution of trnsition ' mens elief in the exeutlement of trnsition in the nonviolene net S After exeuting the trnsition ', the net S' heks, if the trnsition ws relly exeutle in the nonviolene net S It mens tht trnsition is enled nd no other trnsition loks its exeution In order to hek it, we dd, for every pir (p,) suh tht nd the ple p is ommon entry to the trnsitions nd, the ples x p nd y p nd the trnsition t p, s depited on figure 7 ove The trnsition t p is le to move the token from x p to y p only if t lest two tokens reside in the ple p [We use here the weighted rs; see remrk fter figure 5] By the nonviolene rules, trnsition loks nothing if it is not enled; it mens tht one of its entries is empty In order to hek it, we dd the trnsitions t q, for every entry q to trnsition, different from p Eh of the trnsitions hs one entry x p, one exit y p nd one inhiitor r from the ple q, heking if q is empty This onstrution llows to hek, whether the trnsition loks the exeution of or not If not, then token moves from x p to y p, enling in S' if nd only if it ws exeutle in S For every mrking M in the nonviolene net S we define mrking 10M in the inhiitor net S' s follows For eh ple p inside the net S we set 10M(p)=M(p) For

11 Persistent rehility Petri Nets & Conurreny 383 dditionl swith ple s we set 10M(s)=1 nd for every ple r dded y the onstrution (ie for ll ples x p nd y p ) we set 10M(r)=0 Diretly from our onstrution, if trnsition is exeutle in mrking M in nonviolene net S then it is potentilly exeutle in the mrking 10M in inhiitor net S' (efore exeuting trnsition we exeute trnsition ' nd positively hek ll onditions to fill ll ples y p ) The initil mrking of S' is ssumed to e 10M 0 Proposition 48 The mrking M in the nonviolene net S is rehle if nd only if the mrking 10M is rehle in the inhiitor net S' Proof ( ) In the net S' we n reh mrking 10M, from the initil mrking 10M 0, y exeuting trnsition ' efore eh trnsition nd heking the onditions ( ) Exeuting ny trnsition from the originl net S is possile only y prediting this exeution y exeuting the trnsition ' If we do mistke, mking wrong predition, our net S' would reh ded mrking nd stops It mens tht if mrking 10M in the inhiitor net S' is rehle, then the only senrio of rehing tht mrking is orretly prediting nd exeuting trnsitions from the net S The orretness of our proess of prediting mens tht we ould just exeute these trnsitions in the originl, nonviolene net S, rehing mrking M Finlly, mrking M is rehle in the nonviolene net S, whih ends the proof Conlusions We hve proved (Propositions ) tht nonviolene nets re equivlent (in the mrking rehility sense) to inhiitor nets As the ltter re Turing powerful, one n sy tht the former llow to do everything wht possile without ny fight It is quite surprising, euse persistent exeutions re only prt of ritrry exeutions But the prie for the pee is undeidility We hve shown (Corollry 47) tht even overility, deidle in mny extensions of ple/trnsition nets, is undeidle in the lss of the nonviolene nets Notie tht free-hoie (if then = ) nonviolene nets n e simulted y ple/trnsition nets (Figure 8), thus the lssil deision prolems (rehility, overility) re deidle in the lss of free-hoie nonviolene nets Net S: Net S': 2 Fig 8 Trnsforming free-hoie nonviolene net into ple/trnsition net Let S e free-hoie nonviolene net We reple every r from ple, eing ommon entry of two (or more) trnsitions nd is not prt of self-loop, y two rs: n r from the ple to the trnsition, weighted with 2, nd n r from the

12 384 Petri Nets & Conurreny Brylsk, Mikulski, nd Ohmnski trnsition to the ple, weighted with 1 And self-loops remin not hnged And the initil mrking remins the sme Clerly, the ple/trnsition net S' uilt this wy works extly s the free-hoie nonviolene net S A se of the free-hoie nonviolene net is shown y Exmple 22 The ove onstrution does not work for non-free-hoie nonviolene nets, see Exmple 23, for instne It would e interesting to study some other sulsses of the lss of nonviolene nets Espeilly, to find sulss of the lss of nonviolene nets, omputtionlly equivlent to the lss of ple/trnsition nets Aknowledgments The pper ws inspired y the Exmple 22 Gret thnks re due to Ull Goltz, who relled to us tht very old exmple Referenes 1 E Best: A Note on Persistent Petri Nets Conurreny, Grphs nd Models, LNCS 5065, pp Springer E Best, P Drondeu: Deomposition Theorems for Bounded Persistent Petri Nets Petri Nets 2008, LNCS 5062, pp Springer E Best, P Drondeu: Seprility in Persistent Petri Nets Petri Nets 2010, LNCS 6128, pp Springer H-D Burkhrd: Ordered Firing in Petri Nets Elektronishe Informtionsverreitung und Kyernetik 17(2/3), pp 71-86, J Desel, W Reisig: Ple/Trnsition Petri Nets Letures on Petri Nets, LNCS 1491, pp Springer J Growski: The Deidility of Persistene for Vetor Addition Systems Informtion Proessing Letters 11(1), pp 20-23, M Hk: Petri Net Lnguges Report MIT 124, M Hk: Deidility Questions for Petri Nets Ph D Thesis, MIT RM Krp, RE Miller: Prllel progrm shemt JCSS 3, pp , SR Kosrju: Deidility of Rehility in Vetor Addition Systems Pro of the 14th Annul ACM Symposium on Theory of Computing, pp , LH Lndweer, EL Roertson: Properties of onflit-free nd persistent Petri nets Journl of ACM 25, pp , E Myr: Persistene of Vetor Replement Systems is Deidle At Informti 15, pp , E Myr: An lgorithm for the generl Petri net rehility prolem Pro of the 13th Annul ACM Symposium on Theory of Computing, pp , M Minsky: Computtion: Finite nd Infinite Mhines Prentie-Hll C Reutenuer: The Mthemtis of Petri Nets Prentie-Hll P H Strke: Petri-Netze VEB Berlin R Vlk: Self-Modifying Nets, Nturl Extension of Petri Nets ICALP 1978, LNCS 62, pp Springer 1978