Petri Nets with Time and Cost (Tutorial)

Transcription

1 Petri Nets with Time an Cost (Tutorial) Parosh Aziz Abulla Department of Information Technology Uppsala University Sween Richar Mayr School of Informatics, LFCS University of Einburgh Unite Kingom homepages.inf.e.ac.uk/rmayr/ Introuction Petri nets [, ] are a wiely use moel for the stuy an analysis of concurrent systems. Many ifferent formalisms have been propose which exten Petri nets with clocks an real-time constraints, leaing to various efinitions of Time Petri nets (TPNs) (see [, ] for surveys). In parallel, there have been several works on extening the moel of time automata [] with prices (weights) (see e.g., [,, 8]). Weighte time automata are suitable moels for embee systems, where we have to take into consieration the fact that the behavior of the system may be constraine by the consumption of ifferent types of resources. Concretely, weighte time automata exten classical time automata with a cost function Cost that maps every location an every transition to a nonnegative integer (or rational) number. For a transition, Cost gives the cost of performing the transition. For a location, Cost gives the cost per time unit for staying in the location. In this manner, we can efine, for each computation of the system, the accumulate cost of staying in locations an performing transitions along the computation. In this tutorial, we recall, through a sequence of examples, a very expressive moel, introuce in [], that subsumes the above moels. Price Time Petri Nets (PTPN) are a generalization of classic Petri nets [] with real-value (i.e., continuous-time) clocks, real-time constraints, an prices for computations. In a PTPN, each token is equippe with a real-value clock, representing the age of the token. The firing conitions of a transition inclue the usual ones for Petri nets. Aitionally, each arc between a place an a transition is labele with a time-interval whose bouns are natural numbers (or possibly as upper boun). These intervals can be open, close or half open. Like in time automata, this is use to encoe strict or non-strict inequalities that escribe constraints on the real-value clocks. When firing a transition, tokens which are remove from or ae to places must have ages lying in the intervals of the corresponing transition arcs. We assign a cost to computations via a cost function Cost that maps transitions an places of the Petri net to natural numbers. For a transition t, Cost(t) gives the cost of performing the transition, while for a place p, Cost(p) gives the cost per time unit per token in the place. The total cost of a computation is given by the sum of all costs of fire transitions plus the storage costs for storing certain numbers of tokens in certain places for certain times uring the computation. Like in price time automata, having integers as costs an time bouns is not a restriction, because the case of rational numbers can be reuce to the integer case. This work is supporte by UPMARC, The Uppsala Programming for Multicore Architectures Research Center. M.F. Atig, A. Rezine (Es.): Infinity EPTCS 7,, pp. 9, oi:./eptcs.7. c P. A. Abulla, R. Mayr This work is license uner the Creative Commons Attribution License.

2 Time Price Petri Nets It shoul be note that PTPN are infinite-state in several ifferent ways. First, the Petri net itself is unboune. So the number of tokens (an thus the number of clocks) can grow beyon any boun, i.e., the PTPN can create an estroy arbitrarily many clocks (unlike time automata). Seconly, every single clock value is a real number of which there are uncountably many. In [] we stuy the cost to reach a given control-state in a PTPN. In Petri net terminology, this is calle a control-state reachability problem or a coverability problem. The relate reachability problem (i.e., reaching a particular configuration) is uneciable for both continuous-time an iscrete-time TPN [], even without taking costs into account. Our goal is to compute the optimal cost for moving to a control state (equivalently for covering a set of markings). In general, a cost-optimal computation may not exist (e.g., even in price time automata it can happen that there is no computation of cost, but there exist computations of cost ε for every ε > ). We show that the infimum of the costs to reach a given control-state is computable, provie that all transition an place costs are non-negative. Outline. In the next section we introuce PTPNs. In Section we escribe a special type of computations that are sufficient to solve the cost-optimality problem. We introuce a symbolic encoing of infinite sets of markings in Section, an escribe a symbolic algorithm for solving the cost-optimality problem in Section. Finally, in Section, we give conclusions an irections for future work. Time Petri Nets In this section, we introuce Price Time Petri Nets, the set of markings, the transition relation it inuces, an the coverability problem. We use N an R to enote the sets of natural numbers (incluing ) an nonnegative reals respectively. We use a set Intrv of intervals. An open interval is written as(w : z) where w N an z N { }. Intervals can also be close in one or both irections, e.g. [w : z] is close in both irections an[w : z) is close to the left an open to the right. Moel. A Price Time Petri Net (PTPN) is a tuple N =(P,T,Cost) where P is a finite set of places. T is a finite set of transitions, where each transition t T is of the form t =(In,Out). We have that In an Out are finite multisets over P Intrv which efine the input-arcs an output-arcs of t, respectively. Cost : P T N is the cost function assigning firing costs to transitions an storage costs to places. Figure shows an example of a PTPN with five places:,,,,, an five transitions: t,t,t,t,t. The transition t has an input arc from labele with the interval [..], an two output arcs to an, labele with the intervals (..) an [..] respectively. The price (cost) associate with, is, while the price associate with t is. We let cmax enote the maximum integer appearing on the arcs of a given PTPN. In Figure, we have cmax =. Markings. A marking is a multiset over P R. The marking M efines the numbers an ages of tokens in each place in the net. In Figure, we show an example of a marking M. The marking assigns two tokens in, with ages 7.9 an.8, respectively. We will represent markings by lists of colore balls with real numbers insie. Each ball represents one token in the marking. The color escribes the place in which the token resies, while the number represents the age of the token (see Figure ). Computations. We efine two transition relations on the set of configurations: time transition an iscrete transition. A time transition increases the age of each token by the same real number. A

3 P. A. Abulla, R. Mayr (..) [..) t [..) [..) t (..) (..) (.. ) t t [..] (..) (..) (..] t (..) Figure : A Price Time Petri Net (..) t [..). [..) [..) t (..) (.. ) t t [..] (..) (..) (..). 8. (..] t (..). Figure : A marking M an its representation.

4 Time Price Petri Nets t t t..9 t t t t.. Figure : A computation π. Above each in the computation we show the transition that has fire, an below each step we show the cost of the step. iscrete transition represents the effect of firing a transition t in the PTPN. More precisely, for each input arc to the transition, we remove a token from the corresponing input place, whose age lies in the relevant interval. Also, for each input arc to the transition, we a a new token to the corresponing place. The age of the newly generate token is chosen non-eterministically from the relevant interval. Performing a iscrete transition implies paying a cost which is equal to the cost of the transition. When performing a time transition, we pay a cost per each token an time unit that is equal to the cost of the place in which the token resies. A computation is a sequence of iscrete an time transitions. The cost of a computation is the accumulate cost of all the transitions in the computation. Figure shows an example of a computation π. It starts from an initial marking where we have a single token in with age. In the seventh step of π, transition t fires removing one token from with age. The age belongs to the interval [..) (which is the interval on the arc from to t ). At the same time, it as two new tokens with ages.8 an. to the places resp.. The cost of this step is equal to. The eighth step is a time transition of length., where the ages of all tokens are increase by.. The cost of the step is etermine by the number of tokens in each place an the cost of the place, i.e.,. ( + ) =. (the cost of an are resp. ). The total cost of π is given by Cost(π)= = 8.9. For a place p, we efine M p to be the set of markings which put at least one token in the place p (regarless of the ages of the tokens). For instance, if p= then M p is the set of markings that have at least one token in. The Price Coverability Problem. We will consier two variants of the cost problem, the Cost- Threshol problem an the Cost-Optimality problem. They are both characterize by an (i) initial marking M init that places a single token (with age ) in a given initial place p init, an (ii) a set of final markings M pfin efine by a final place p fin. In other wors, we start from a marking where there is only one token with age in p init an where all the other places are empty, an then consier the cost of computations that takes us to M pfin. In the Cost-Threshol problem we ask the question whether there is a computation starting from M init an reaching a marking in M pfin with a cost that is at most v for a given threshol v N. In the Cost-Optimality problem, we want to compute the optimal (smallest) cost of reaching M pfin staring from M init. For given M init an M pfin, the optimal cost of reaching M pfin from M init may not exist. However, in

5 P. A. Abulla, R. Mayr t (..) [..) Figure : A Simple PTPN Figure : A marking in δ-form, δ =.. [], we show that the infimum of the costs of all computations is a natural number (or if M pfin is not reachable from M init ). The situation is illustrate in Figure. The optimal cost for putting a token in can be mae arbitrarily close to (but not equal to ). In such a case, we simply efine the optimal cost to be. In fact, the non-existence of an optimal cost has alreay been observe for time automata [9]. Computations in δ-form In orer to solve the Cost-Threshol an the Cost-Optimality problems, it is sufficient to consier computations of a certain form where the ages of all the tokens that appear in the computation are arbitrarily close to (within some small real number δ from) an integer. Below, we assume a real number δ : <δ <.. δ-markings. A marking M is sai to be in δ-form (Figure ) if any fractional part of the age of a token appearing in M is either smaller than δ or larger than δ. We ecompose a δ-marking into submarkings such that in every submarking the fractional parts (but not necessarily the integer parts) of the token ages are ientical. We then arrange these submarkings in a sequence M m,...,m,m,m,...,m n such that M m,...,m contain tokens with fractional parts δ in increasing orer, M contains the tokens with fractional part zero, an M,...,M n contain tokens with fractional parts< δ in increasing orer. Figure shows that partitioning of the marking M in Figure. More precisely, We start with the token with the high fractional parts, namely.9 (one token in ), followe by.9 (one token in ), followe by.97 (one token in an one token in ). Furthermore, there are two tokens with zero fractional parts (one token in an one token in ). Finally, we consier the tokens with low fractional parts, namely. (one token in, one token in, an one token in ), followe by. (one token in ), followe by.7 (one token in ). Computations in δ-form. The occurrence of a iscrete transition t is sai to be in δ-form if the ages of the newly generate tokens are close to an integer (i.e., within istance δ). This is not a property of the transition t as such, but a property of its occurrence. Figure 7 shows the result of an occurrence of t in δ-form (with δ =.) on the marking of Figure. A computation is in δ-form if:

6 Time Price Petri Nets Figure : The partitioning the marking in Figure Figure 7: An application in δ-form (δ =.) of t on the marking of Figure. The two new tokens have fractional parts that are equal to.9 resp.... Every occurrence of a iscrete transition is in δ-form, an. For every time transition, the elay is either in the interval ( : δ) or in the interval x ( δ : ). Detaile Time Transitions. We say that a time transition (from a marking M) is etaile iff at most one fractional part of any token in M changes its status about reaching or exceeing the next integer value. Figures 8 an 9 show some steps in a etaile computation. In the first transition, time passes by a positive amount but not sufficiently long to make any tokens with positive fractional parts to increase to the next integer. More precisely, the time elay is. which means that two tokens in an that have zero fractional parts, will now have positive fractional parts (.). On the other han, the two tokens in an that have the highest fractional parts (.8) will not cross to the next integer (their ages will now be.98 an.98 respectively). In the secon step, the amount of elay is. which is exactly the amount neee to allow the tokens that currently have the highest fractional parts to become integers. These tokens are the ones with ages.98 an.98 in resp.. Their new ages are. resp... In the last step, all tokens have small fractional parts. We let time pass sufficiently much (.78 time units) so that the tokens will all have high fractional parts. Every computation of a PTPN can be transforme into an equivalent one (w.r.t. reachability an cost) where all time transitions are etaile, by replacing long time transitions with several etaile shorter ones where necessary. Thus we may assume w.l.o.g. that time transitions are etaile. Detaile Computations in δ-form. In [], we show the following result. For any computation π starting from an initial marking M init (efine by a initial place p init ), an reaching a give set M pfin of final markings (efine by a final place p fin ), an for each δ : <δ <., there is a etaile computation π in δ-form where (i) π starts from the same initial marking as π, (ii) π is in δ-from, (iii) π reaches M pfin, an (iv) if π is etaile then π is etaile. This means that, to solve the Cost-Threshol an Cost-Optimality problems, it is sufficient to consier etaile computations in δ-form. Figure shows a etaile computation in δ-form for the PTPN of Figure.

7 P. A. Abulla, R. Mayr Figure 8: Detaile time transitions for δ =..

8 Time Price Petri Nets Figure 9: Detaile time transitions (cont.)..... t t t.. t t... t t.. Figure : A etaile computation in δ-form.

9 P. A. Abulla, R. Mayr 7 increasing fractional parts increasing fractional parts high fractional parts zero fractional parts low fractional parts Figure : A region r. Regions In this section, we introuce a symbolic encoing for infinite sets of markings. The encoing is a variant of the classical notion of regions []. The main ifference is that we here nee to eal with an unboune number of clocks. It is an aaptation of the encoing introuce in []. More precisely, we change the encoing of [] so that we can now eal with markings in δ-form. First, we give the efinition of regions, an then we show how to simulate time an iscrete transitions on regions. For each type of transition, we efine the cost of firing the transition from the region. Regions. A region characterizes a set of marking in δ-form for some δ : <δ <.. An example of (our notion of) a region r is shown in Figure. The region consists of three parts, referre to as H (for high), Z (for zero), an L (for low). The part H is a wor of multisets. Each element in a multiset is a colore ball with a natural number, representing one token. The color efines the place in which the token resies, while the number efines the integer part of the age of the token. Furthermore, tokens whose ages are larger than cmax + are all represente by one element (ages > cmax cannot be istinguishe by the transitions of the PTPN). The orering of the multisets reflects the orering of the factional parts of the corresponing tokens: elements belonging to the same multiset represent tokens with ientical fractional parts, an elements in successive multisets represent tokens with increasing fractional parts. The part Z consists of one multiset, an represents the tokens with zero fractional parts. Finally, the part L consists of a wor of multisets. It has a similar interpretation to H, except that it represents tokens with low fractional parts. Figure shows a marking M (of the Petri net of Figure ) satisfying the region r of Figure as follows: The left-most multiset in H contains a re ball with value an a green ball with value. They represent the token with age.9 in the place, an the token with age.9 in. The fractional parts of the two tokens are equal (.9) an high. The next multiset contains a blue ball with value. It represents the token with age.9 in. The fractional part of the token (.9) is high an is larger than the fractional parts of the tokens in the previous multiset. The right-most multiset in H contains a white ball with value an an orange ball with value. They represent the token with ages.97 in the place, an the token with age.97 in. The

10 8 Time Price Petri Nets Figure : A marking M satisfying the region of Figure. fractional parts of the two tokens are equal (.97). The fractional parts of these tokens (.97) are high an are larger than the fractional part of the token in the previous multiset. The part Z consists of a single multiset. It contains a blue ball with value an a re ball with value. They represent the token with age. in the place, an the token with age in. The fractional parts of the two tokens are zero. The left-most multiset in L contains an orange ball with value an a green ball with value. They represent the token with age. in the place, an the token with age 8. in. The fractional parts of the two tokens are equal (.) an low. The age of the token in is 8. cmax+ which means that it is represente by in r. The next multiset contains a white ball with value. It represents that token with age. in. The fractional part of the token (.) is low an is larger than the fractional parts of the tokens in the previous multiset. The next multiset contains a re ball with value. It represents that token with age. in. The fractional part of the token (.) is low an is larger than the fractional part of the token in the previous multiset. We use [[r]] to enote the set of markings satisfying r. Time Transitions. We will escribe how to encoe the effect of etaile time transitions on regions. To o that, we efine ifferent types of transitions on regions. Type I This simulates a small elay where the tokens of integer age now have a positive fractional part, but no tokens reach an integer age. An example of such a transition is shown in Figure. Here, the elay is. which is not sufficient to make the tokens with the highest fractional parts (the token with age.97 in, an the token with age.97 in ) to become integers. Notice that the tokens with zero fractional parts (the token with age. in, an the token with age. in ) will now have have low fractional parts (in fact, they will have the smallest fractional parts, namely., among all tokens in the marking). At the region level, the two elements in Z will move to L, forming the left-most multiset in L (reflecting the fact that they have the lowest fractional parts). Type II Transition. This simulates a small elay in the case where there were no tokens of integer age an the tokens with the highest fractional parts just reach the next integer age. An example of such a transition is shown in Figure. Here, the elay is., which is sufficient to make the tokens with the highest fractional parts (the token with age.98 in, an the token with age.98 in ) to become integers, i.e., an respectively. At the region level, the right-most multiset in H will move to Z, an the value of each element in the multiset is incremente by one to reflect the fact that the ages of the token moves to the next integer. Type III Transition. This simulates a elay close to (but smaller than) where the tokens with low fractional parts will now either have high fractional parts, or they have reache (an passe) the

11 P. A. Abulla, R. Mayr Figure : Type I Transition Figure : Type II Transition. next integer an thus have low fractional parts again. The tokens that alreay ha high fractional parts will all have passe the next integer an will now have high fractional parts again. No token will have an integer value after the transition (the case where some tokens have integer ages is covere in Type IV transitions, see below). Here, the elay is.9. We have three types of tokens: Tokens that have low fractional parts both before an after the transition (the token with age. in, an the token with age.7 in ). The ages of these tokens are. an. after the transition. Thus, the elay is sufficient to make their ages go beyon the next integer. After the transition, these tokens will be the only ones with low fractional parts. The relative orering of their fractional parts will not be change. The integer part of their ages will have increase by one. At the region level, these two tokens are represente by the two right-most multisets in L. After the transition, they will be the only multisets in L, an their values are incremente by each. Notice that the relative orering of these tokens insie the region will be preserve. Tokens that have low fractional parts before the transition an high fractional parts after the transition (the token with age. in, the token with age. in, the token with age. in, an the token with age 8. in ). The ages of these tokens are.98,.98,.99, resp after the transition. These tokens have the highest fractional parts among all tokens in the marking. The relative orering of the fractional parts of these tokens will not be change. Also, the elay is not sufficiently long to make their values reach (or pass) to the next integer. At the region level, the corresponing multisets move from L to H, an will now be the right-most multisets in H. The orering of these multisets is preserve. Tokens that have high fractional parts both before an after the transition (the token with

12 Time Price Petri Nets Figure : Type III Transition Figure : Type IV Transition. age.98 in, the token with age.98 in, an the token with age.99 in ). The ages of these tokens are 7.9,.9, resp..9 after the transition. The elay is sufficiently long both to make their values pass the next integer integer, an to make their fractional parts high again. However, these tokens have now the lowest fractional parts among all tokens with high fractional parts. The relative orering of the fractional parts of the tokens will not be change. At the region level, the corresponing multisets will be the left-most multisets in H. The orering of these multisets is preserve. Their values are incremente by one (to reflect that they have reache the next integer). Notice that the new value of the token in is represente by since the value is cmax +. Type IV Transition. This is similar to a Type III transition, except that some of the tokens that have low fractional parts will have integer values after the transition (see Figure ). Discrete Transitions. Figure 7 shows the firing of transition t (Figure ), an escribes how the firing of the transition may be simulate at the region level. We remove a token from whose age is in the interval [..). This is one at the region level by removing the re ball with value from Z (the ball represents a token in whose age is exactly ). We a one to token to whose age is in the interval (..), an one to token to whose age is in the interval [..). In Figure 7, this is one at the region level by aing a white ball to a multiset in H with value (the ball represents a token in whose age is in the interval (..)), an aing a blue ball to a multiset in L with value (the ball represents a token in whose age is in the interval (..)).

13 P. A. Abulla, R. Mayr Figure 7: Firing the transition t. Costs. At the region level, the cost of performing a type I or type II transition is, since we can assume the time elay to be arbitrarily small. The cost of performing a type III or type IV transition is equal to the cost of performing a time transition of time unit, since we can make the elay arbitrarily close to. Thus, the cost of performing the transition in Figure or Figure is. The cost of performing a iscrete transition at the region level is the same as the cost of performing the transition on concrete markings. Thus, the cost of performing the transition in Figure 7 is. Solving the Cost-Optimality Problem In this section we explain our solution for the Cost-Optimality problem. Here, we give an informal overview of the main ieas. The (quite complicate) technical etails can be foun in []. First, we show that the Cost-Optimality problem can be reuce to the Cost-Threshol problem. Then, we introuce a general framework of orere transition systems, which we then instantiate to the case of regions. Finally, we present an algorithm that allows to solve the Cost-Threshol problem. From Cost-Optimality to Cost-Threshol. Consier an instance the Cost-Optimality problem, efine by M init an M pfin (see Section ). The task is to compute the optimal cost of reaching M pfin from M init, i.e., the infimum of the costs of all computations reaching M pfin from M init. To compute this value, it suffices to solve the Cost-Threshol problem for any given threshol v N, i.e., to ecie whether there is any computation from M init to M pfin with cost v. To see this, we first ecie whether M pfin is reachable from M init in the unerlying time Petri net (without consiering costs). This can be reuce to the Cost-Threshol problem by setting all place an transition costs to zero an solving the Cost-Threshol problem for v=. If the answer is no, then we can efine the optimal cost to be (M pfin is not reachable form M init ). If yes, then we can fin the optimal cost v by solving the Cost-Threshol problem for threshol v=,,,,... until the answer is yes. We solve the Cost-Threshol problem using regions as symbolic encoings of sets of markings. Orere Transition Systems. An orere transition system is a triple T =(S, A, ) where S is a (potentially) infinite set of configurations (or states), is a transition relation on S, an is an orering on S. We say that is monotone wrt. if the following hols for all configurations c,c,c S: if c c an c c then there is a c such that c c an c c.

14 Time Price Petri Nets free Figure 8: Orering on Regions. For a set S S of configurations, we efine Pre(S) to be the set of preecessors of S wrt., i.e., the set of configurations from which we can reach a configuration in S through a single application (a single step) of. We efine Pre to be the reflexive transitive closure of Pre, i.e., Pre (S) is the set of configurations from which we can reach a configuration insthrough any number of steps of. A sets S is sai to be upwar-close if for any two configurations with c c, it is the case that c Simplies c S. The upwar closures of a setsof configurations is the set of configurations that are larger than or equal to some configuration inswrt., i.e., S :={c S c S.c c }. Below, we will consier ifferent transition systems that are inuce by ifferent sets of configurations an ifferent transition relations. Instantiation. Consier an instance of the Cost-Threshol problem, efine by M init, M pfin, an a threshol v. Define a configuration c to be a pair (r,u) where r is a region, an u v. Intuitively, u enotes the maximal allowe cost of the remainer of a computation that passes through r. Let S be the set of all configurations. Let C be the set of configurations of the form (r,u) where r contains only tokens in the costs places (places whose costs are larger than ), an where the number of tokens in r is smaller than u. Notice that C is finite. Consier regions r,r. We write r all r if we can obtain r from r by aing a number of tokens to r. We write r free r if we can obtain r from r by aing a number of tokens to the free places (places whose costs are ). Notice that free all. Figure 8 shows an example of two regions (interprete over the PTPN of Figure ) relate by free. For configurations c =(r,u ) an c =(r,u ), we use c all c resp. c free c to enote that u = u an that r all r resp. r free r. For a set S S of configurations, we use S free to be the upwar closure of S with respect to free, i.e., it contains all configurations that are larger than or equal to some configuration ins wrt. free. We efines all in a similar manner. Let i enote the time transition relation of type i {I,II,III,IV}, an let Disc be the iscrete transition relation. Define A := Disc, i.e., a transition of type A is either a time transition of type I or II, or a iscrete transition. Define B :=, i.e., a transition of type B is a time transition of type III or IV. For a setm, we efine Pre A (M) to be the set of markings from which we can reach a marking in M through a single application of a transition of type A. We efine Pre B (M) analogously. Algorithm. We give an overview of an algorithm to solve the reachability problem. We notice that M pfin ( ) + is reachable from M init with a cost v iff M init A B A Mpfin an the accumulate cost of all involve transitions is v. Furthermore, we observe that M pfin can be characterize by the upwar closure (wrt. all ) of a finite set of regions. Therefore, it is sufficient to give an algorithm that, given a region r fin an threshol v, checks whether there is a region r init where M init is inclue in the enotation ( ) of r init such that (r init,) +(rfin A B A,v ) all. To o that, we generate a sequence of sets of configurations V,U,V,U,..., as follows:

15 P. A. Abulla, R. Mayr ( ( V := min free Pre A (rfin,v) all ) (C free) ). This set is possible to compute as follows. The set ( r fin,v ) all is (obviously) upwar-close wrt. all. The relation A is monotone wrt. all. We can then use the backwar ( reachability ( algorithm (introuce in []) for well quasiorere systems to compute min all Pre A (rfin,v) all )) ( (. The result follows from the fact that both min all Pre A (rfin,v) all )) an C are finite. U := min free (Pre B (V free)). This set can be compute by a straightforwar application of B on the elements of V. Notice that U C free, an that it is a finite set. For k >, given the finite set U k, we compute V k := min free (Pre A (U k free) (C free)). Notice that we here are solving a reachability problem rather than coverability problem, since U k free is not upwar-close wrt. all. In fact, this problem has an extremely complicate solution (escribe in []). The construction to compute it uses many calls to a subroutine which relies on the eciability of the reachability problem for Petri nets with one inhibitor arc [, 7]. In a sense, this is unavoiable, since the reverse reuction also hols. The reachability problem for Petri nets with one inhibitor arc can be reuce to the zero-cost coverability problem for PTPN, i.e., Cost-Threshol with threshol. For k >, we compute U k := min free (Pre B (V k free)) in a similar manner to U. U k C free, an that it is a finite set. Notice that The sequence U free,u free,... is a monotone-increasing sequence of upwar-close (wrt. free ) subsets of C free. This sequence converges, because free { is a well-quasi-orering on C free. Therefore, we get U n = U n+ for some finite inex n an U n free= c c( B A ) r fin }, because the transition B is only enable in C free. Finally, we compute the (finite) set of configurations, min all (Pre A (U n free)), an check whether the set contains a configuration of the form (r init,u) such that M init belongs to the enotation of r init. Conclusions an Future Work We have given an informal escription of a metho for computing the infimum of the costs of placing a token in given place of a time Petri net, starting from a given initial marking. Interesting irections for future work inclue augmenting time with other infinite-state iscrete moels such as push-own systems an asynchronously communicating processes, an to a other quantitative parameters such as probabilistic behaviors. References [] P.A. Abulla & B. Jonsson (): Moel checking of systems with many ientical time processes. Theoretical Computer Science 9(), pp., oi:./s-97()-9. [] P.A. Abulla & R. Mayr (): Computing optimal coverability costs in price time Petri nets. In: Logic in Computer Science (LICS), th Annual IEEE Symposium on, IEEE, pp. 99 8, oi:.9/lics... [] Parosh Aziz Abulla, Karlis Cerans, Bengt Jonsson & Yih-Kuen Tsay (99): General Deciability Theorems for Infinite-State Systems. In: LICS, pp., oi:.9/lics [] R. Alur & D. Dill (99): A Theory of Time Automata. TCS, pp. 8, oi:./-97(9)9-8.

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